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52 AG13 Abstracts IP1 How Applied Algebraic Geometry is Useful in Pure Mathematics For historical reasons, the culture of pure algebraic geom- etry has often been quite distant from applications, and ”applied” methods. While there are some good reasons for the distinction between the pure and applied side of the subject, this division is, happily, gradually eroding. I will describe some examples of where in theoretical advances have been built on insights and experiences coming from the more applied side of the subject. For this reason, it is worth our time to learn to talk to people in different parts of the subjects, even if the cultural and linguistical differences sometimes make it challenging. Ravi Vakil Department of Mathematics Stanford University [email protected] IP2 A Tale of Two Theorems I will explain and draw connections between the follow- ing two theorems: (1) Hilbert’s theorem on nonnegative polynomials and sums of squares, and (2) Classification of varieties of minimal degree by Del Pezzo and Bertini. This will result in the classification of all varieties on which non- negative polynomials are equal to sums of squares. Along the way I will provide an introduction to Convex Algebraic Geometry. The talk is based on joint work with Greg Smith and Mauricio Velasco. Greg Blekherman Georgia Institute of Technology [email protected] IP3 On k-apart Configuration Spaces k-apart - or no-k-equal configuration spaces - are formed by tuples of points in a topological space with no more than k coinciding. They appeared as a model problem in theoretical computer sciences, and are very useful in other applications, such as motion planning in robotics. I will survey some old and new results in the area. Yuliy Baryshnikov University of Illinois at UrbanaChampaign Department of Mathematics [email protected] IP4 Numerics and Algebraic Geometry Numerical methods have increasingly proven to be both helpful and necessary in Algebraic Geometry. Conversely geometrical and algebraic tools have led to new algorithms providing advances in more applied areas such as engi- neering and medical science. We will present a few ex- amples showing this fruitful interplay. Classical theory from algebraic geometry can be used in Kinematics. Es- tablished tools form topology apply to give a numerical cell-decomposition of solution sets. Numerical methods, on the other hand, are essential tools for efficient algorithms to compute invariants of algebraic varieties, like Chern classes or the Euler characteristics. The talk is based on joint work with Besana, Eklund, Hauenstein, Peterson, Sommese and Wampler. Sandra Di Rocco Department of Mathematics KTH [email protected] IP5 Algebraic Geometry in System Biology Systems biology aims to understand complex systems and the mechanisms that are responsible for specific behaviors, such as multi-stationarity or oscillation. Typical mathe- matical models of biological systems produce polynomial systems of equations. In recent years tools from algebraic geometry are increasingly being applied to understand such polynomial systems and extract information that are rele- vant for the design of experiments/systems and the analy- sis of experimental data. In this talk I will review recent results, and discuss some of the challenges we are facing. Carsten Wiuf University of Copenhagen [email protected] IP6 Cluster Algebra and Complex Volume of Knots The cluster algebra was introduced by Fomin and Zelevin- sky around 2000. The characteristic operation in the alge- bra called ‘mutation’ is related to various notions in math- ematics and mathematical physics. In this talk I review a basics of the cluster algebra, and introduce its appli- cation to study the complex volume, (hyperbolic volume) + i (Chern-Simons invariant), of knot complements in S 3 . We formulate the ideal tetrahedral decomposition of hyper- bolic 3-manifolds in terms of the cluster algebra, where a mutation produces an ideal tetrahedron. This talk is based on joint work with Kazuhiro Hikami (Kyushu University). Rei Inoue Chiba University [email protected] IP7 Speeding up Lattice Reduction with Numerical Linear Algebra Techniques A lattice is the set of all integer linear combinations of some linearly independent vectors. Visually, it is an infinite grid of regularly spaced points. Lattices have many applica- tions in computer science. For example, they frequently appear in computer algebra (e.g., to factor rational poly- nomials), and in cryptography (both to break and design cryptographic protocols). The LLL algorithm, named after its authors Arjen Lenstra, Hendriz Lenstra and Lszl Lovsz, enables the computation of a good representation, or ba- sis, of a given lattice: This representation provides decent intrinsic information on the lattice under scope. Numer- ous applications were found right after the discovery of the LLL algorithm, which motivated the search of algo- rithmic improvements. Today, the most efficient approach relies, internally, on low precision floating-point computa- tions, leading to a numeric-algebraic hybrid algorithm. In this talk, I will first give an introduction to lattices, and then describe the hybrid numeric-algebraic approach un- derlying the modern variants of the LLL algorithm. This talk relies on joint works with Xiao-Wen Chang, Phong

AG13 Abstracts · 2018. 5. 25. · AG13 Abstracts 53 Nguyen,AndrewNovocin,IvanMorelandGillesVillard DamienStehl´e EcoleNormaleSup´´ erieuredeLyon [email protected] IP8

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Page 1: AG13 Abstracts · 2018. 5. 25. · AG13 Abstracts 53 Nguyen,AndrewNovocin,IvanMorelandGillesVillard DamienStehl´e EcoleNormaleSup´´ erieuredeLyon damien.stehle@ens-lyon.fr IP8

52 AG13 Abstracts

IP1

How Applied Algebraic Geometry is Useful in PureMathematics

For historical reasons, the culture of pure algebraic geom-etry has often been quite distant from applications, and”applied” methods. While there are some good reasons forthe distinction between the pure and applied side of thesubject, this division is, happily, gradually eroding. I willdescribe some examples of where in theoretical advanceshave been built on insights and experiences coming fromthe more applied side of the subject. For this reason, itis worth our time to learn to talk to people in differentparts of the subjects, even if the cultural and linguisticaldifferences sometimes make it challenging.

Ravi VakilDepartment of MathematicsStanford [email protected]

IP2

A Tale of Two Theorems

I will explain and draw connections between the follow-ing two theorems: (1) Hilbert’s theorem on nonnegativepolynomials and sums of squares, and (2) Classification ofvarieties of minimal degree by Del Pezzo and Bertini. Thiswill result in the classification of all varieties on which non-negative polynomials are equal to sums of squares. Alongthe way I will provide an introduction to Convex AlgebraicGeometry. The talk is based on joint work with Greg Smithand Mauricio Velasco.

Greg BlekhermanGeorgia Institute of [email protected]

IP3

On k-apart Configuration Spaces

k-apart - or no-k-equal configuration spaces - are formedby tuples of points in a topological space with no morethan k coinciding. They appeared as a model problem intheoretical computer sciences, and are very useful in otherapplications, such as motion planning in robotics. I willsurvey some old and new results in the area.

Yuliy BaryshnikovUniversity of Illinois at UrbanaChampaignDepartment of [email protected]

IP4

Numerics and Algebraic Geometry

Numerical methods have increasingly proven to be bothhelpful and necessary in Algebraic Geometry. Converselygeometrical and algebraic tools have led to new algorithmsproviding advances in more applied areas such as engi-neering and medical science. We will present a few ex-amples showing this fruitful interplay. Classical theoryfrom algebraic geometry can be used in Kinematics. Es-tablished tools form topology apply to give a numericalcell-decomposition of solution sets. Numerical methods, onthe other hand, are essential tools for efficient algorithms tocompute invariants of algebraic varieties, like Chern classesor the Euler characteristics. The talk is based on joint workwith Besana, Eklund, Hauenstein, Peterson, Sommese and

Wampler.

Sandra Di RoccoDepartment of [email protected]

IP5

Algebraic Geometry in System Biology

Systems biology aims to understand complex systems andthe mechanisms that are responsible for specific behaviors,such as multi-stationarity or oscillation. Typical mathe-matical models of biological systems produce polynomialsystems of equations. In recent years tools from algebraicgeometry are increasingly being applied to understand suchpolynomial systems and extract information that are rele-vant for the design of experiments/systems and the analy-sis of experimental data. In this talk I will review recentresults, and discuss some of the challenges we are facing.

Carsten WiufUniversity of [email protected]

IP6

Cluster Algebra and Complex Volume of Knots

The cluster algebra was introduced by Fomin and Zelevin-sky around 2000. The characteristic operation in the alge-bra called ‘mutation’ is related to various notions in math-ematics and mathematical physics. In this talk I reviewa basics of the cluster algebra, and introduce its appli-cation to study the complex volume, (hyperbolic volume)+ i (Chern-Simons invariant), of knot complements in S3.We formulate the ideal tetrahedral decomposition of hyper-bolic 3-manifolds in terms of the cluster algebra, where amutation produces an ideal tetrahedron. This talk is basedon joint work with Kazuhiro Hikami (Kyushu University).

Rei InoueChiba [email protected]

IP7

Speeding up Lattice Reduction with NumericalLinear Algebra Techniques

A lattice is the set of all integer linear combinations of somelinearly independent vectors. Visually, it is an infinite gridof regularly spaced points. Lattices have many applica-tions in computer science. For example, they frequentlyappear in computer algebra (e.g., to factor rational poly-nomials), and in cryptography (both to break and designcryptographic protocols). The LLL algorithm, named afterits authors Arjen Lenstra, Hendriz Lenstra and Lszl Lovsz,enables the computation of a good representation, or ba-sis, of a given lattice: This representation provides decentintrinsic information on the lattice under scope. Numer-ous applications were found right after the discovery ofthe LLL algorithm, which motivated the search of algo-rithmic improvements. Today, the most efficient approachrelies, internally, on low precision floating-point computa-tions, leading to a numeric-algebraic hybrid algorithm. Inthis talk, I will first give an introduction to lattices, andthen describe the hybrid numeric-algebraic approach un-derlying the modern variants of the LLL algorithm. Thistalk relies on joint works with Xiao-Wen Chang, Phong

Page 2: AG13 Abstracts · 2018. 5. 25. · AG13 Abstracts 53 Nguyen,AndrewNovocin,IvanMorelandGillesVillard DamienStehl´e EcoleNormaleSup´´ erieuredeLyon damien.stehle@ens-lyon.fr IP8

AG13 Abstracts 53

Nguyen, Andrew Novocin, Ivan Morel and Gilles Villard

Damien Stehle

Ecole Normale Superieure de [email protected]

IP8

Multivariate Polynomial Interpolation providesSurprising Combinatorial Insights: Zonotopal Al-gebra and Beyond

I will survey recent developments connecting multivariatepolynomial interpolation, special polynomial ideals, geom-etry and combinatorics of hyperplane arrangements, vectorpartition functions and matroid theory. These connectionsoriginated in the theory of multivariate splines and led tothe construction of so-called zonotopal algebra(s), which inturn shed light on enumerative problems related to graphsand matroids. However, many interesting open questionsremain.

Olga HoltzUniversity of California, BerkeleyTechnische Universitaet [email protected]

CP1

Support Function Based Description of Topologyand Approximation of Real Algebraic Curves

The support function representation can considerable sim-plify each of main phases of the problem of describing thetopology of real algebraic curves (the determination of crit-ical points and the reconstruction of their connectivity).Critical points of each type can be described as the solu-tions of a system of equations and their connectivity canbe determined in a geometrical way. This representationallow us to find an approximation which also approximatethe curvature, specially preserves cusps.

Eva CernohorskaFaculty of Mathematics and PhysicsCharles University in [email protected]

Zbynek SirCharles University in PragueCzech [email protected]

CP1

A Dynamical System Which Produces MutuallyUnbiased Bases and An Application of PersistentHomology

We present a novel algorithm, thought of as a discrete-timedynamical system on a space of unitary matrices with fixeddimension, which (often) converges to a (numerically) mu-tually unbiased basis. This algorithm serves as an efficientmethod to sample elements from the space of mutuallyunbiased bases in a fixed dimension. The persistent ho-mology of these sample sets is computed to extract coursetopological features of the underlying spaces from whichthe samples are drawn.

Francis C. MottaColorado State [email protected]

CP1

Free Tilings on Genus-3 Surfaces and ResultingCrystalline Patterns

It is well established that crystalline 3-periodic nets can besystematically catalogued using Delaney-Dress tiling the-ory. One approach is to enumerate two-dimensional hyper-bolic (H2) tilings then project those patterns into three-dimensional Euclidean space (E3) via triply periodic min-imal surfaces (see epinet.anu.edu.au). We extend this toinvestigate 3d structures that emerge from the systematicenumeration of “free tilings’ of H2. These hyperbolic pat-terns consist of unbounded tiles with translational symme-try. The results in 3d are novel multicomponent interwovenperiodic nets and filaments. Some of the simpler examplesof these structures are known to chemists as highly porousmetal-organic frameworks and rod packings.

Vanessa RobinsAustralian National [email protected]

Myfanwy EvansUniversitaet [email protected]

Stuart Ramsden, Stephen HydeRSPhysSEAustralian National [email protected], [email protected]

CP1

Construction of Lorentz-Conformal CoordinateTransformations

We use solutions of the wave equation to construct Lorentz-orthogonal coordinate transformations of the plane as wellas triply-orthogonal coordinate systems in Lorentz space.This construction may be translated to an approach inwhich the initial data for constructing such a system con-sists of a spectral curve, which may be singular and re-ducible. Such systems have applications to systems of dif-ferential equations with Lorentzian symmetries.

Patrick ShipmanDepartment of MathematicsColorado State [email protected]

CP1

Geometrically Minimal Realizations for LinearControl Systems over Boolean Semiring

A problem of description of all (minimal) realizations ofgiven i-o map in a category of systems over semiring withfree state semimodule is studied. Our approach is based onthe study of morphisms in the category of realizations. Wehave classified all reductions realizable in the category ofboolean systems with free states and studied some reduc-tions in category of free systems, arising from the non-freecase. Realizations for various classes of i-o maps are stud-ied.

Oleg O. Vasil’evInstitute of Control Sciences [email protected]

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54 AG13 Abstracts

CP2

Elegant Expressions and Formulae for RiemannZeta, Dirichlet Beta, Euler Numbers and OtherMathematical Functions

New techniques are being developed to derive simple for-mulae for calculating the values of Riemann zeta, Dirichletbeta, Euler numbers and other special functions. Thesetechniques may be used to provide simple proofs of thevalues of zeta(2n) and beta(2n+ 1). A new concept for de-riving the values of Euler numbers is also discovered. Thisnew concept is being used to establish some new identities.We attempt to find a closed form expression for ζ(2n + 1)and discover a most intriguing result.

Michael A. IdowuUniversity of AbertayDundee, United [email protected]

CP2

List Decoding of Repeated Codes

We present a decoding algorithm for hard-decision decod-ing of repeated codes, based on the soft-decision algorithmof Koetter and Vardy. Lower bounds for decoding per-formance are given for best case, worst case and expectedcase. Computer simulations show that the performance issomewhat better than these bounds, particularly for higherrates. We compare performance and decoding time of arepeated Reed-Solomon code and standard Reed-Solomoncode (over a larger field). The decoding performance isrelatively close, but the repeated code requires much lesstime.

Michael E. O’SullivanSan Diego State [email protected]

Fernando HernandoUniversitat [email protected]

Diego RuanoAalborg [email protected]

CP2

Doubly Adapted Bases for the Symmetric Group

Adapted bases play an important computational role inmany applications of the representation theory of finitegroups. In this talk, I will describe an interesting “doublyadapted” (with respect to the usual left action and right ac-tion) basis for the regular representation of the symmetricgroup. I will then explain why we think such bases mightbe the key to a new approach to creating fast Fourier trans-forms for finite groups.

Michael E. OrrisonHarvey Mudd [email protected]

Michael HansenN/[email protected]

Masanori KoyamaUniversity of Wisconsin-Madison

[email protected]

MS1

Identifiability and Parameter Estimation in Mod-eling Disease Dynamics

Connecting differential equation models with data to yieldpredictive results requires a variety of parameter estima-tion, identifiability, and uncertainty quantification tech-niques. Identifiability analysis addresses the question ofwhether it is possible to uniquely recover the model pa-rameters from a given set of data. In this talk, I will dis-cuss some recent work using and developing identifiabilitymethods based on tools from computational differential al-gebra and systems theory, and present several clinical andpublic health applications to problems in human disease,including cholera and other infectious disease transmissionprocesses.

Marisa EisenbergUniversity of [email protected]

MS1

Identifiability of Mechanical Systems in Cardiovas-cular Modeling

Cardiovascular (CV) modeling of the blood pressure regu-lation system requires mathematical description of the vis-coelastic mechanical properties of the arterial wall as wellas the coupling of the nerve endings of various receptors(e.g. baroreceptors) to the wall. Those mechanical mod-els usually involve springs (elastic elements) and dash-pots(viscous elements) in various configuration, which can bedescribed using algebraic ODEs. An essential step in solv-ing the inverse problem i.e. estimating model’s parame-ters is to establish whether a model is structurally identifi-able. The importance of this property is that it guaranteesthat for the noise-free observations and perfect model itsunknown parameters can be uniquely estimated from theinput/output experiments. In this talk we consider thestructural identifiability problem related to the importantclass of mechanical models used in the CV modeling. Wealso show how algebraic methods can be used to designdynamically ”optimal” model.

Adam MahdiNorth Carolina State [email protected]

Nicolette MeshkatUniversity of California, Los [email protected]

Seth SullivantNorth Carolina State [email protected]

MS1

Identifiable Reparameterizations of Linear OdeSystems

Identifiability concerns finding which unknown parametersof a model can be quantified from given input-output data.Many linear ODE models, used primarily in Systems Biol-ogy, are unidentifiable, which means that parameters cantake on an infinite number of values and yet yield the sameinput-output data. We study a particular class of uniden-

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AG13 Abstracts 55

tifiable models and find conditions to obtain identifiablereparameterizations of these models. In particular, we usea graph-theoretic approach to analyze the models and showthat graphs with certain properties allow a monomial scal-ing reparameterization over identifiable functions of the pa-rameters.

Nicolette MeshkatUniversity of California, Los [email protected]

Seth SullivantNorth Carolina State [email protected]

MS1

Differential Algebra Techniques for Identifiabilityof Biological Systems

The aim of this talk is to stress the role of differentialalgebra techniques for checking identifiability of dynamicmodels described by nonlinear ordinary differential equa-tions, as those arising in chemical kinetics and biologicalsystems. Identifiability is a well-known prerequisite for thewell-posedness of model parameter estimation. The pro-posed methodology, which applies to systems of rationalODE’s, is based on computer algebra algorithm to computea Grobner basis of certain families of parameter-dependentpolynomials which can be extracted from the model by Rittalgorithm.

Maria Pia SaccomaniUniversity of [email protected]

MS2

Utilising New CAD Developments for Simplifica-tion in Computer Algebra

Formulae involving inverse functions (e.g. square root)may not universally hold when restricting to single-valuedfunctions (e.g. the positive root). This may lead com-puter algebra systems to be overly cautious. One solutionis to construct CADs according to the branch cuts of thefunctions, checking the formula in each cell. We report onrecent progress on the calculation of cuts and implemen-tation of this approach utilising new theory (truth tableinvariant CAD).

Matthew England, Russell Bradford, James Davenport,David J. WilsonUniversity of [email protected], [email protected],[email protected], [email protected]

MS2

Automatic Proofs of Transcendental Function In-equalities and Their Applications

We shall present an overview of some recent applicationsof real algebraic geometry to automatic proofs of transcen-dental function inequalities. In the process, we shall de-scribe how these techniques, as implemented in the theo-rem proving tools MetiTarski and Z3, are being used in theformal verification of real-world cyber-physical systems,e.g., to verify the safety of geometric maneuvers at theheart of collision avoidance algorithms in next-generation

autonomous (self-driving) cars.

Grant O. PassmoreCarnegie Mellon UniversityUniversity of [email protected]

MS2

Beyond Equational Constraints in CAD

We discuss the new concept of truth table invariant CADs(TTICADs). These have cells with invariant truth valuesfor a list of unquantified Tarski formulae, getting closer tothe minimal information required for applications. Theseare constructed by extending the theory of equational con-straints so it applies for each clause, even if there is nooverall equational constraint. We also consider how bestto formulate problems for this and other CAD algorithms.

Russell Bradford, James Davenport, Matthew England,David J. WilsonUniversity of [email protected], [email protected],[email protected], [email protected]

MS3

Random Points on Curves in Rn with Applicationto Parameterizing QSIC

A method for finding a random point on a curve in Rn

will be described. Once a point is known the irreduciblecomponent containing this point can be found. Fractionallinear transformations can then be applied which will oftengive the component in a simple form allowing parameteri-zation. As an application a completely numerical methodfor finding parameterizations of Quadric surface intersec-tion curves (QSIC) in R3 is given that does not requirelooking at many separate cases.

Barry H. DaytonNortheastern Illinois [email protected]

MS3

Computing H-Bases to Precondition PolynomialSystems for Homotopy Continuation

When solving polynomial systems using homotopy continu-ation, it is sometimes the case that we have a large numberof paths going to ∞. We want to remove the pieces of thesystem that tend to ∞ in order to get a smaller number ofpaths with better conditioning. We present an algorithmto compute an H-basis for our polynomial system in orderto remove the pieces at ∞. Examples are given to showthe removal of the infinite paths in order to simplify thehomotopy continuation run.

Steven L. IhdeColorado State [email protected]

Daniel J. BatesColorado State UniversityDepartment of [email protected]

Jonathan Hauenstein

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56 AG13 Abstracts

North Carolina State [email protected]

MS3

A Numerical Algorithm for the Topological EulerCharacteristic of Algebraic Varieties

Recent work on numerical computation of degrees of Chernand Segre classes can be used to numerically compute thetopological Euler characteristic of both non-singular andsingular algebraic varieties. I will explain the idea behindthe algorithm and its motivation from algebraic statistics

Christine JostStockholm [email protected]

MS3

Non-convex Optimization and Numerical Homo-topies

This talk will consider a couple of problems in (semi-definite) optimization and will indicate how numerical al-gebraic geometry can be used to help in their solution.

Chris PetersonColorado State [email protected]

MS4

The Number of Nonsimple Principally PolarizedAbelian Surfaces over a Finite Field

There are approximately 2q3 principally polarized abeliansurfaces over a finite field Fq. We show that the num-ber of these surfaces that are nonsimple is bounded aboveby c q2.5(log q)d for some constants c > 0 and d, and isbounded below by an expression of the same form. Thissuggests a heuristic estimate for the number of primesp < x for which a given Jacobian over Q has nonsimplereduction; we present evidence in support of this heuristic.

Everett W. HoweCenter for Communications [email protected]

Jeff AchterColorado State [email protected]

MS4

Computing Discrete Logarithms in the Jacobian ofHigh-Genus Hyperelliptic Curves and Applications

We describe novel and improved versions of index-calculusalgorithms for solving discrete logarithm problems in Jaco-bians of medium-genus to high-genus hyperelliptic curves.One important improvement is based on the effective adap-tation of sieving techniques known from the number fieldsieve, the function field sieve, and the arithmetic of numberfields to the curve setting. We will also discuss paralleliza-tion techniques and their realization in hardware, such asgraphic cards. Our new algorithms are applied to con-crete problem instances arising from the Weil descent at-tack methodology for solving the elliptic curve discrete log-arithm problem, demonstrating significant improvementsin practice. This is joint work with Mark D. Velichka and

Michael J. Jacobson.

Andreas SteinCarl von Ossietzky Universitat [email protected]

MS4

Vertical Brauer Groups and Del Pezzo Surfaces ofDegree 4

Del Pezzo surfaces X of degree 4 are smooth intersec-tions of two quadrics in four-dimensional projective space.They are some of the simplest surfaces for which there canbe cohomological obstructions to the existence of ratio-nal points, mediated by the Brauer group Br X. I willexplain how to construct, for every non-constant elementA ∈ Br X , a rational genus-one fibration X → P 1 suchthat A is “vertical’ for this map. This implies, for ex-ample, that if there is an obstruction to the existence ofa point on X arising from A, then there is a genus-onefibration X → P 1 where none of the fibers are locally sol-uble, giving a geometric way of “seeing’ a Brauer-Maninobstruction. The construction gives a fast, practical algo-rithm for computing the Brauer group of X. Conjecturally,this gives a mechanical way of testing for the existence ofrational points on X.

Tony Varilly-AlvaradoRice [email protected]

Bianca VirayBrown [email protected]

MS4

2-torsion Brauer Classes on Surfaces with Hyper-elliptic Fibrations

Using descent techniques and the purity theorem, we givea method to compute the representatives of all 2-torsionclasses in the Brauer group of a surface with a hyperellipticfibration. We will describe the method in the example of aK3 surface which is a double cover of a quadric surface inP 3.

Brendan CreutzUniversity of [email protected]

Bianca VirayBrown [email protected]

MS5

Arithmetic Codices and Applications to Cryptog-raphy

In this talk we present the notion of arithmetic codices.These are combinatorial objects which generalize some no-tions studied in the areas of information-theoretically se-cure cryptography and algebraic complexity. Known con-structions of “asymptotically good’ families of codices usetowers of function fields with many rational places. Fur-thermore, we define and study another parameter of atower, that we call torsion limit, which also plays an im-

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AG13 Abstracts 57

portant role in the asymptotics of codices.

Ignacio CascudoCentrum Wiskunde & [email protected]

Ronald CramerCWI Amsterdam and Leiden University, The [email protected]

Chaoping XingNTU [email protected]

MS5

On Retrieving a Representation of An AlgebraicGeometry Code

Code-based cryptography is an interesting alternative toclassic number-theory PKC. Some families of codes havebeen proposed for these cryptosystems such as algebraicgeometry codes. For the so called very strong algebraicgeometry codes C = CL(X ,P , E), where X is an algebraiccurve over Fq and P is an n-tuple of mutually distinct Fq-rational points of X and E is a divisor of X with disjointsupport from P we construct an equivalent representation.This equivalent representation given by C = CL(Y,Q,F)can be found as follows; The n-tuple of points are obtaineddirectly from a generator matrix of C, where the columnsare viewed as homogeneous coordinates of these points.The curve Y is given by I2(Y), the homogeneous elementsof degree 2 of the vanishing ideal I(Y).

Edgar Martınez Moro, Irene Marquez-CorbellaUniversidad de [email protected], [email protected]

Ruud PellikaanTechnical University of [email protected]

Diego RuanoAalborg [email protected]

MS5

Point Compression for the Trace Zero Variety

The hardness of the (hyper)elliptic curve discrete logarithmproblem over extension fields lies in the trace zero variety.A compact representation of the points of this abelian vari-ety, computed by a point compression algorithm, is neededin order to accurately assess the hardness of the discretelogarithm problem there. Such algorithms have been pro-posed by Lange and Silverberg. We present a new represen-tation that is optimal in size, compatible with the structureof the variety, and allows efficient point compression anddecompression.

Maike MassiererUniversity of Basel and University of [email protected]

Elisa GorlaUniversity of [email protected]

MS5

Exponentiating in Pairing Groups

Bilinear maps derived from the Weil or the Tate pairingon elliptic curves over finite fields are versatile tools incryptography. Finding the right pairing-friendly curve iscrucial for practical performance. This talk discusses theselection of curve and field parameters that enable secureand fast implementations of cryptographic pairings for themost common security levels. Based on these choices, westudy exponentiations in pairing groups and show that,although the Weierstrass model is preferable for pairingcomputation, it can be worthwhile to map to alternativecurve representations for the non-pairing group operationsin protocols.

Michael Naehrig, Craig Costello, Joppe W. BosMicrosoft [email protected], [email protected],[email protected]

MS6

Toric Embedding and Birational Geometry

Given a toric variety, a small modification on it, the so-called flip, is described by a change of the fan structure.Considering the Cox construction of a toric variety, thistype of birational modification can be computed combina-torially and have a particularly meaningful picture whenthe Picard number of the variety is low. I describe thisbriefly and show how one can get interesting results in bi-rational classification of 3-folds using a suitable toric em-bedding and performing these maps.

Hamid AhmadinezhadRadon Institute (RICAM)Austrian Academy of [email protected]

MS6

(Convex) Normal Lattice Polytopes

We show that for lattice polytopes P there is no differencebetween being 3− and k-convex normal (k ≥ 3) and im-prove the bound of how long edges of a lattice polytope Phave to be, so that P is known to be normal, by a factorof 2. The original bound was proved by Gubeladze in his2009 paper ”Convex normality of rational polytopes withlong edges”.

Jan Hofmann, Petra MeyerGoethe University Frankfurt am [email protected], [email protected]

MS6

Combinatorial Mutations and Fano Manifolds

Given a Laurent polynomial f , one can form the period πf :a function of one complex variable that plays an importantrole in Mirror Symmetry for Fano manifolds. Mutationsare a particular class of birational transformations actingon Laurent polynomials in two variables that preserve theperiod and, at least in two-dimensions, are closely con-nected with cluster algebras. I will give a combinatorialdescription of mutation acting on the Newton polytope Pof f , and and use this to establish many basic facts aboutmutations.

Alexander M. Kasprzyk

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58 AG13 Abstracts

Imperial College [email protected]

MS6

New Developments in LattE Integrale

The software package LattE has been known as the first im-plementation of Barvinok’s 1994 algorithm for computinggenerating functions of polyhedra, counting lattice points,and computing Ehrhart series. We report on the new de-velopments in the latest version, LattE integrale, whichcomplements these discrete generating function techniquesby new implementations of continuous and intermediate(“mixed integer’) generating function algorithms for com-puting integrals of polynomial functions over polytopes andcomputing the top k coefficients of generalized Ehrhartquasipolynomials.

Velleda BaldoniU. Roma Tor [email protected]

Nicole BerlineEcole [email protected]

Jesus A. De Loera, Brandon E. DutraDepartment of MathematicsUniv. of California, [email protected], [email protected]

Matthias KoeppeDept. of MathematicsUniv. of California, [email protected]

Michele VergneUniversite Paris 7 DiderotInstitut Mathematique de [email protected]

MS6

Perturbation of Transportation Polytopes

We introduce a perturbation method that can be used toreduce the problem of finding the multivariate generatingfunction (MGF) of a non-simple cone to computing theMGF of simple cones. We then give a universal perturba-tion that works for any transportation polytope. We applythis perturbation to the family of central transportationpolytopes of order kn × n, and obtain formulas for theMGFs of the feasible cones of vertices of the polytope andthe MGF of the polytope. The formulas we obtain are enu-merated by combinatorial objects. A special case of theformulas recovers the results on Birkhoff polytopes givenby the author and De Loera and Yoshida.

Fu LiuUC [email protected]

MS7

Tensor Ranks

In this talk I will present various notions of tensor ranksthat are of interest in applications. I will range from clas-sical and more modern definitions. I will also show what isknown on their mutual relations by associating them some

algebraic varieties/sets.

Alessandra BernardiUniversity of [email protected]

MS7

Projective Methods for the Identifiability of Ten-sors II

In this talk, effective results on identifiability of tensors,based on projective methods, are shown. Further develop-ments are proposed.

Cristiano BocciUniversita di [email protected]

MS7

Projective Methods for the Identifiability of Ten-sors I

I will introduce methods of Projective Algebraic Geometry,from the theory of secant varieties, that can be applied tothe study of the uniqueness of the decomposition of generaltensors. Examples of tensors for which the identifiabilityfails, and their Geometric explanation, will be given in thetalk.

Luca ChiantiniUniversita di [email protected]

MS7

Decomposition of Infinite-dimensional Tensors

We discuss a technique that allows blind recovery of signalsor blind identification of mixtures in instances where theywere thought to be impossible: (i) closely located or highlycorrelated sources in antenna array processing, (ii) highlycorrelated spreading codes in CDMA radio communication,(iii) nearly dependent spectra in fluorescent spectroscopy.To provide theoretical underpinnings for this technique,we discuss separable approximations of functions and anappropriate notion of rank and nuclear norm.

Lek-Heng LimUniversity of [email protected]

Pierre ComonGIPSA-Lab CNRSGrenoble, [email protected]

MS8

Effective Calculations of Cohomology via SpectralSequences

We will discuss approaches for computing cohomologygroups by considering examples from algebraic geometry,commutative algebra, and algebraic topology. Emphasiswill be placed on interactions between spectral sequences,arising from filtered complexes, and computer calculations.Technical details will be kept to a minimum so as to makethe talk accessible to a general audience.

Nathan Grieve

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Queen’s [email protected]

MS8

Computation in the Intersection Ring of Flag Bun-dles and Isotropic Flag Bundles

We give an explicit description of the Groebner basis ofthe ideal of relations defining the intersection ring of a flagbundle of a vector bundle on a nonsingular variety, andalso of isotropic bundles. This justifies a key algorithm inthe Schubert2 package in Macaulay2 which supports com-putation in intersection theory and enumerative geometry.

Dan [email protected]

Alexandra SeceleanuUniversity of Nebraska - [email protected]

Mike StillmanCornell [email protected]

MS8

State Polytopes of Ideals and Syzygies and Geo-metric Invariant Theory for Moduli of Curves

A major open question in algebraic geometry is to investi-gate the birational geometry of the moduli space of stablegenus g algebraic curves Mg. I will summarize how cal-culations of state polytopes of ideals and syzygy modulesusing Macaulay2, gfan, and polymake have helped us con-struct and explore new birational models of Mg .

Anand DeopurkarColumbia [email protected]

Maksym FedorchukBoston [email protected]

David SwinarskiFordham [email protected]

MS8

Fixed Point Sets in Affine Buildings

Let G be a semisimple group over a complete, discretelyvalued non-Archimedean field. The Bruhat-Tits buildingassociated to G is an infinite simplicial complex on whichG acts. For example, for SL2, the building is an infinitetree. The building is glued from apartments, each of whichis an affine space. We study fixed point sets in the buildingunder the G-action. In each apartment, a fixed point set isan alcoved polytope associated to a root system. We willdiscuss various computational problems and experimentsin Macaulay 2.

Annette WernerGoethe-Universitat [email protected]

Josephine Yu

School of MathematicsGeorgia Institute of [email protected]

MS9

On Hyperbolicity Cones and Spectrahedra

The generalized Lax conjecture asserts that each hyperbol-icity cone is a slice of the cone of positive semidefinite ma-trices. We will talk about recent combinatorial approachesto the conjecture.

Petter BrandenKTH [email protected]

MS9

Containment Problems for Polytopes and Spectra-hedra

In this talk we study the computational question ofwhether a given polytope or spectrahedron, the feasibleregion of a semidefinite program, is contained in anotherone. Extending results on the polytope/polytope-case byGritzmann and Klee, we first classify the computationalcomplexity. To overcome the situation that the generalcontainment problem for spectrahedra is co-NP-hard, re-laxation techniques are of particular interest. The aim ofthis talk is to present some semidefinite conditions to cer-tify containment.

Kai KellnerGoethe University [email protected]

Thorsten TheobaldJ.W. Goethe-UniversitatFrankfurt am Main, [email protected]

Christian TrabandtGoethe University [email protected]

MS9

Positive Polynomials on Non-Compact Sets

The difference between positive polynomials and polynomi-als representable via sums of squares is best understood forcompact sets (with many positive results) or highly non-compact sets (with corresponding negative results). Stillrelatively little is known for sets that lie in between. Inthis talk, we will give some partial results and discuss rel-evant examples.

Daniel PlaumannUniversitat [email protected]

MS9

Bounds on the Equivariant Betti Numbers of Sym-metric Semi-Algebraic Sets

Abstract not available at time of publication.

Cordian RienerAalto [email protected]

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Saugata BasuPurdue [email protected]

MS9

When is Every Nonnegative Quadric a Sum ofSquares?

We identify a numerical invariant which is an obstructionfor equality and show that it vanishes iff X is a varietyof minimal degree. These result generalize a well knownTheorem of Hilbert characterizing the pairs (d,k) for whichevery nonnegative homogeneous polynomial of degree 2d ink variables is a sum of squares.

Mauricio VelascoUniversidad de los [email protected]

Gregoriy BlekhermanGeorgia [email protected]

Gregory G. SmithQueen’s [email protected]

MS10

Invited. Participation Uncertain 1

Abstract not available at time of publication.

Graziano ChesiDepartment of Electrical and Electronic Engineering(CYC BldUniversity of Hong [email protected]

MS10

Bounded Symbolic-Numeric Cylindrical AlgebraicDecomposition for Solving Optimization Problems

With many applications in engineering and in scientificfields, quantifier elimination (QE) has been attracting moreattention these days. Cylindrical algebraic decomposition(CAD) is used as a basis for a general QE algorithm. Wepropose an effective symbolic-numeric cylindrical algebraicdecomposition algorithm for solving polynomial optimiza-tion problems. Our approach constructs CAD only in re-stricted admissible regions to remove redundant projectionfactors and avoid lifting cells where truth values are con-stant over the region.

Hidenao IwaneFUJITSU LABORATORIES LTD / Kyushu University4-1-1 Kamikodanaka, Nakahara-ku, Kawasaki 211-8588,[email protected]

Hirokazu AnaiFUJITSU LABORATORIES LTDKyushu [email protected]

MS10

Polynomial Optimization with Real Varieties

We study the optimization problem

min f(x) s.t. h(x) = 0, g(x) ≥ 0

with f a polynomial and h, g two tuples of polynomialsin x ∈ Rn. Lasserre’s hierarchy is a sequence of sum ofsquares relaxations for finding the global minimum fmin.Let K be the feasible set. We prove the following results:i) If the real variety VR(h) is finite, then Lasserre’s hierar-chy has finite convergence, no matter the complex varietyVC(h) is finite or not. This solves an open question in Lau-rent’s survey. ii) If K and VR(h) have the same vanishingideal, then the finite convergence of Lasserre’s hierarchy isindependent of the choice of defining polynomials for thereal variety VR(h). iii) When K is finite, a refined versionof Lasserre’s hierarchy (using the preordering of g) has fi-nite convergence.

Jiawang NieUniversity of California, San [email protected]

MS10

Sums of Squares of Polynomials with Rational Co-efficients

We consider multivariate polynomials f with coefficients inthe field Q of rational numbers which are sums of squaresof polynomials with real coefficients. Such f has a sum ofsquares representation with coefficients in some finite realextension K of Q, but not necessarily in Q itself. We shallattempt to provide more information on possible fields K,and to give sufficient conditions that imply the existenceof a representation over Q.

Claus ScheidererFachbereich Mathematik und StatistikUniv. Konstanz, [email protected]

MS11

Euler-Mahonian Statistics Via Polyhedral Geome-try

Permutations are some of the most fundamental objectsof mathematics, and a basic combinatorial statistics of apermutation π ∈ Sn is the number of descents des(π) :={j : π(j) > π(j + 1)}. Euler realized that

∑k≥0

(k + 1)ntk =

∑π∈Sn

tdes(π)

(1 − t)n+1

and there has been various generalization of this identity,most notably when Sn gets replaced by another Coxetergroup. We will illustrate how one can view Euler’s identity(and its generalizations) geometrically through enumerat-ing integer points in certain polyhedra. This gives rise to”short” proofs of known theorems, as well as new identities.This is joint work with Ben Braun (Kentucky).

Matthias BeckSan Francisco State [email protected]

Benjamin BraunUniversity of Kentucky

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[email protected]

MS11

Towards a Classification of Restricted LatticeWalks

Completing the Classification of Walks with Small Steps inthe Quarter Plane The enumeration of walks in the quarterplane with small steps is an elegant problem in combina-torics. Recently, much work has been done attempting toclassify the seventy nine non-equivalent walks of this typeaccording to analytic properties of their generating func-tions. We discuss this classification, its usefulness, and out-line a method of classifying the final three walks not yetproven by other means. In particular, our method givesa parametrization of the generating function which illumi-nates the nature of its singularities, in turn proving thatthey are infinite in number.

Stephen MelczerSimon Fraser [email protected]

Marni MishnaSimon Fraser [email protected]

MS11

Automated Asymptotics of Multivariate Generat-ing Functions

I begin with a description of the field of Analytic Combi-natorics in Several Variables, drawing on my recent bookwith Mark Wilson. It is well known that coefficient asymp-totics of a multivariate generating function depend mainlyon the geometry of the algebraic surface Q=0. One of thechallenges in moving from theorems that handle most casesin practice to automated asymptotics is to combinatorial-ize the geometric data. This talk concerns some of theproblems that arise when implementing automation.

Robin PemantleUniversity of [email protected]

Mark WilsonUniversity of [email protected]

MS11

On the Summability of Bivariate Rational Func-tions

We present criteria for deciding whether a bivariate ratio-nal function in two variables can be written as a sum of two(q-)differences of bivariate rational functions. Using thesecriteria, we show how certain double sums can be evalu-ated, first, in terms of single sums and, finally, in terms ofvalues of special functions.

Shaoshi Chen, Michael F. SingerDepartment of MathematicsNorth Carolina State [email protected], [email protected]

MS12

The Dynamics of Chip-firing on Abstract TropicalCurves

Chip-firing has been studied for nearly 25 years since itsindependent introductions in statistical physics and graphtheory. Recently, it has been employed as a combinatoriallanguage for describing linear equivalence of divisors ongraphs and tropical curves. In this talk we will addresssome dynamical questions which arise in this subject in thecontext of greedy algorithms and generalized chip-firing viaopen covers.

Spencer BackmanGeorgia Institute of [email protected]

MS12

Combinatorics of the Tropical Moduli Space ofCurves

A tropical curve is a metric graph marked with integervertex weights obtained as a metrized dual graph of asemistable model of a curve over a valued field. We will re-view the construction tropical moduli space M trop

g,n of genusg tropical curves with n marked points and report on somecombinatorial properties of these spaces.

Melody ChanHarvard [email protected]

MS12

Tropical Geometry and Combinatorics in Inte-grable Cellular Automata

The box-ball system (BBS) is an cellular automaton givenby a simple rule to move finite number of balls in boxes ina line. The integrability of the BBS is clarified by crystalbase theory and tropical geometry. In this talk I will reviewthe BBS mainly from the view point of tropical geometry.I also give an overview of the talks in this session.

Rei InoueChiba [email protected]

MS12

Rigged Configurations and Box-Ball Systems

I will explain algebraic methods for the studies on the box-ball systems. The topics include Kashiwara’s crystal basesfor the quantum affine algebras, the rigged configurationsand the tropical Riemann theta function. The purposeof the talk is to present main constructions for the sim-plest possible situation as we did in an introductory review:arXiv:1212.2774

Reiho SakamotoTokyo University of [email protected]

MS13

Polynomial Inequalities for Bistability in a DoublePhosphorylation Network

Double phosphorylation mechanisms are important build-ing blocks of mammalian signaling systems and bistabil-

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ity is considered an important property of ODE models ofthese systems. Bistability often arises from a saddle-nodebifurcation. Establishing such a bifurcation is difficult, asparameter values are either unknown or confined to largeintervals. Hence conditions guaranteeing a saddle-node bi-furcation are desirable. We present such a condition inform of polynomial inequalities in the Km and Kcat val-ues.

Carsten ConradiMax-Planck-Institut Dynamik komplexer [email protected]

Maya MinchevaNorthern Illinois [email protected]

MS13

Identification of Multistationary Reaction Net-works Modeled with Power-law Kinetics

Multistationarity in cellular systems provides a mechanismfor switching between different responses and can be cru-cial for cellular decision making. It is in general difficultto decide whether a particular reaction network has thecapacity to exhibit multiple steady states. In this talk Iwill present our most recent results concerning both thepreclusion and identification of multistationarity in reac-tion networks modeled with power-law kinetics (which in-clude mass-action kinetics).

Elisenda Feliu, Carsten WiufUniversity of [email protected], [email protected]

MS13

Calculating Detailed-Balanced Equilibrium byFixed-Point Iterations and Cell Exclusion

We will report on our quest for an unconditionally conver-gent algorithm to compute the (detailed-balanced) equi-librium of complete networks of reversible binding reac-tions. Several networks used in pharmacology for simula-tion and parameter estimation are in this class, and meth-ods used to compute equilibrium are often questionable oreven inapplicable. We turned the algebraic equation forequilibrium into a fixed-point problem. We will presentnetwork-structural conditions that guarantee the conver-gence of fixed-point iterations, and a cell-exclusion algo-rithm that applies more generally.

Gilles [email protected]

MS13

Characterization of Steady States of General MassAction Systems by Correspondence to Weakly Re-versible Networks

In order to analyze the dynamical behavior of chemicalreaction networks, it is important to first of all characterizethe steady states permitted by the mechanism. In this talk,I will relate a recent steady state characterization, calledtoric steady states, to a classical characterization related toweakly reversible networks. I will present a graphical trickfor reformulating arbitrary networks into weakly reversible

networks and present conditions under which the steadystates are guaranteed to be preserved under the operation.

Matthew JohnstonUniversity of Wisconsin [email protected]

MS13

Chemical Reaction Networks As CompartmentalSystems

Consider a chemical readtion network with general mono-tone rate functions. Conditions are given for the sytem tobe reducable to a compartmental system. Consequently,every solution will approach an equilibrium point. Com-parison is made with other approaches.

David SiegelUniversity of WaterlooDepartment of Applied [email protected]

MS14

Computational and Statistical Tradeoffs Via Con-vex Relaxation

Processing massive datasets is usually viewed as a substan-tial computational challenge. However, if data are a statis-ticians main resource then access to more data should beviewed as an asset rather than as a burden. We describe aframework based on convex relaxation to reduce the com-putational complexity of an inference procedure when onehas access to increasingly larger datasets, and we demon-strate the efficacy of this methodology in a class of denois-ing problems.

Venkat ChandrasekaranCalifornia Institute of [email protected]

Michael JordanUniversity of California at [email protected]

MS14

Differentiable, Continuous, and CombinatorialHodge Theories

The usual Hodge theory on Riemannian manifolds, whichmay be described as “differentiable Hodge theory,” hasbeen very useful in physical problems like fluid dynamicsand electromagnetics. In data analytic problems, we oftenhave only some notion of proximity of the data points andsome knowledge of the distribution of the data set. Wediscuss recent developments in “continuous Hodge theory”on metric spaces and “discrete Hodge theory” on simplicialcomplexes for such problems.

Lek-Heng LimUniversity of [email protected]

MS14

It is Hard to be Strongly Faithful

Many algorithms for inferring causality rely heavily on thefaithfulness assumption. The main justification for impos-ing this assumption is that the set of unfaithful distribu-

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tions has Lebesgue measure zero, since it can be seen as acollection of hypersurfaces in a hypercube. However, dueto sampling error the faithfulness condition alone is notsufficient for statistical estimation, and strong-faithfulnesshas been proposed and assumed to achieve uniform or high-dimensional consistency. In contrast to the plain faithful-ness assumption, the set of distributions that is not strong-faithful has non-zero Lebesgue measure and in fact, canbe surprisingly large as we show in this talk. Taking ageometric point of view we give bounds on the Lebesguemeasure of strong-faithful distributions for various classesof directed acyclic graphs. Our results imply fundamentallimitations for algorithms inferring causality based on par-tial correlations, with the PC-algorithm as its most promi-nent example.

Caroline UhlerInstitute of Science and Technology [email protected]

Garvesh RaskuttiDepartment of Statistics and Operations ResearchUniversity of North [email protected]

Peter BuehlmannETH [email protected]

Bin YuUniversity of California, [email protected]

MS15

On a Family of Determinantal Varieties Arising asCritical Loci in a Classical Computer Vision Prob-lem

Scene reconstruction is a classical problem in computer vi-sion. It is well known that there exist critical configurationsof camera positions and sample points that make recon-struction not possible. Such configurations live on a familyof determinantal varieties that will be introduced.

Gianmario BesanaDePaul [email protected]

MS15

Certifiable Numerical Computations in SchubertCalculus

Numerical methods for solving polynomial systems maybe advantageous in avoiding the complexity of calculatinga Grobner basis in many applications. However, numericalsolutions are approximate, and algorithms which certifythem require that the system be square. Many geomet-ric problems are presented algebraically as overdeterminedsystems. We use Schubert problems as prototypical overde-termined systems. Using local coordinates and duality, wegive a method for formulating these problems as squaresystems.

Nickolas HeinUniversity of Nebraska at [email protected]

Jonathan HauensteinNorth Carolina State University

[email protected]

Frank SottileTexas A&M [email protected]

MS15

Macaulay Dual Space and Numerical Primary De-composition

Numerical algorithms for an irreducible decomposition of apolynomial ideal fall short of a full decomposition into pri-mary components. In particular, an irreducible decomposi-tion does not detect embedded components. The Macaulaydual space provides a numerical alternative to standard ba-sis algorithms for computing local information at a point.We employ the Macaulay dual to examine points of interestand distinguish embedded components from other singularpoints on the variety.

Robert KroneSchool of MathematicsGeorgia Institute of [email protected]

Jonathan HauensteinNorth Carolina State [email protected]

Anton LeykinSchool of MathematicsGeorgia Institute of [email protected]

MS15

Determinantal Representations of HyperbolicCurves via Polynomial Homotopy Continuation

A smooth curve in the real projective plane is hyperbolicif its ovals are maximally nested. By the Helton-VinnikovTheorem, any such curve admits a definite symmetric de-terminantal representation. We compute such representa-tions numerically by tracking a specially designed homo-topy in a space of regular real polynomial systems. (Jointwork with Daniel Plaumann)

Anton LeykinSchool of MathematicsGeorgia Institute of [email protected]

MS16

Arithmetic Occult Periods

For certain types of complete intersections, the associatedcomplex moduli space is actually a quotient of a complexball. I will survey recent work, particularly involving thecase of cubic surfaces, which suggests that this unexpectedstructure is the complex realization of a morphism of inte-gral moduli spaces.

Jeff AchterColorado State [email protected]

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MS16

Crystalline Cohomology of the Igusa Tower

We relate the crystalline cohomology of the Igusa tower toclassical cusp forms of finite slope. In the ordinary setting,we obtain a canonical isomorphism between the crystallinecohomology of the Igusa tower and the Iwasawa module ofordinary Λ-adic cuspforms.

Bryden CaisUniversity of [email protected]

MS16

Sato-Tate Groups of Abelian Surfaces and Three-folds

We classify the Sato-Tate groups of abelian surfaces overnumber fields, and give partial results towards the caseof threefolds. We then compare with numerical evidencesupporting the generalized Sato-Tate conjecture.

Francesc FiteUniversity of [email protected]

Kiran S. KedlayaUniversity of California at San [email protected]

Vıctor RotgerUniversitat Politecnica de [email protected]

Andrew V. [email protected]

MS16

Colmez’s Product Formula for CM Abelian Vari-eties

We complete a proof of Colmez, showing that the standardproduct formula for algebraic numbers has an analog forperiods of CM abelian varieties with CM by an abelianextension of the rationals. The proof depends on explicitcomputations with the De Rham cohomology of Fermatcurves, which in turn depends on explicit computation oftheir stable reductions.

Andrew ObusColumbia [email protected]

MS16

Rational Points on Twists of Modular Curves

In this talk, I will present results about points on certaintwists of the classical modular curve. Some of these twistedcurves violate the Hasse principle and this violation is ex-plained by the Brauer-Manin obstruction. In some othercases, it is possible to study Hasse principle violations vialocal-global trace obstructions.

Ekin OzmanUniversity of Texas at [email protected]

MS17

Short Algebraic-Geometry Codes and TheirWeight Distribution for Diffusion in Block Ciphersand Hash Functions

We study the notion of diffusion in block ciphers, in thecase of substitution permutation networks. This is for-malized with the branch number of a linear matrix, whichshould be high for good resistance against linear and dif-ferential cryptanalyses. A maximal branch number is ob-tained with MDS codes. Non maximality introduces codesof low genus, and their weight distribution is needed tostudy the resistance of the cipher.

Daniel [email protected]

MS17

On the Design of Wiretap Codes

Wiretap codes are families of codes that promise both re-liability between the legitimate users Alice and Bob, andconfidentiality in the presence of an eavesdropper Eve. Wesurvey an error probability point of view to the design ofwiretap codes for several continuous channels, and discussthe resulting code design criteria. They give rise to a se-rie of interesting problems, such as the understanding of anew lattice invariant, or the search for number fields withparticular discriminant and regulator.

Frederique Oggier, Jerome DucoatNanyang Technological [email protected], [email protected]

MS17

Orders of Central Simple Algebras as a Tool forWireless Communications

We demonstrate how central simple algebras and their or-ders can be used to construct efficient space-time codes thatimprove the quality of wireless communications. We willprovide the essential algebraic background and familiar-ize the audience with the transmission model for wirelessfading channels. Examples of space-time codes designedespecially for digital video broadcasting, an application ofmajor current interest, will be given.

Camilla HollantiAalto [email protected]

MS17

New Matrix-Based Lattice Construction Tech-niques

Let m and n be integers greater than 1. Given point-lattices A and B of dimensions m and n, respectively, atechnique for constructing a new lattice from them of di-mension either m+n−1 or m+n−2 is proposed. The tech-nique is based on the existence of a common sub-lattice ofdimension 1 or 2. Denser sphere packings than previouslyknown ones in dimensions 52, 68, 84, 248, 520, and 4098 areobtained.

Carmelo InterlandoSan Diego State [email protected]

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MS17

Probability Bounds for Algebraic Lattice Codes

Lattices which arise from totally real algebraic numberfields have many applications to the coding theory of fadingchannels, which arise naturally in the context of wirelesscommunications. One can naturally attach to the ring ofintegers of a number field K of degree n over Q a lattice inRn, from which one can carve a finite codebook in a naturalway. We will show how familiar number theoretic invari-ants of K, such as its regulator and values of its Dedekindzeta function, control the probability that the correspond-ing codebook provides reliable transmission over a fadingchannel. We will also discuss generalizations to codes builtfrom central simple algebras.

David KarpukAalto [email protected]

MS18

Light-weight Methods for Automatic Recognitionin Mobile Applications

I will describe some automatic recognition problems whichI came across as part of my research on portable device ap-plications. These problems, I will argue, justify the needfor a mathematical theory of shape that is amenable to dis-crete, noisy data. As a tentative first step towards such atheory, I will define the Pascal Triangle of a discrete gray-scale image as a pyramidal arrangement of complex-valuedmoments and explore its geometric significance. In particu-lar, we will observe that the entries of row k of this pyramidcorrespond to the Fourier series coefficients of the order kmoment of the Radon transform of the image. Group ac-tions on the plane can be naturally prolonged onto theentries of the Pascal triangle; we will propose simple testsfor equivalence and self-equivalence under some commongroup actions.

Mireille BoutinSchool of ECEPurdue [email protected]

MS18

Estimating Radar Target Invariants

Radar imaging methods require precise knowledge of therelative locations between the measurement antenna(s) andthe target to be imaged. Over the past few decades, meth-ods for determining the relative motion, based on exploit-ing the targets underlying geometric invariance, have beendeveloped in the setting of a single antenna. Extension ofthese methods to multiple antennas is not straightforward.We discuss our attempts to address the problem by usinga specific matrix of Euclidean invariants.

Matthew Ferrara, Gregory ArnoldMatrix Research, [email protected],[email protected]

Jason T. ParkerAFRL/[email protected]

MS18

The Ideal of the Trifocal Variety

Trifocal tensors are constructed from a bilinear map builtfrom the three view camera setup. It is natural to ask whatare the minimal generators of the ideal of the algebraicvariety of trifocal tensors. We answer this question using avariety of numerical and symbolic techniques from Algebra,Geometry, and Representation Theory. This is joint workwith Chris Aholt (Washington)

Chris AholtUniversity of [email protected]

Luke OedingUniversity of California BerkeleyUniversita degli Studi di [email protected]

MS18

Invariant Histograms and Signatures for ObjectRecognition and Symmetry Detection

I will survey recent developments in the use of group-invariant histograms, using distances, areas, etc., and sig-natures, using differential invariants, joint invariants, in-variant numerical approximations, etc., for object recogni-tion and symmetry detection in images, including recentapplications to automated jigsaw puzzle assembly.

Peter OlverMathematics DepartmentUniversity of [email protected]

MS19

On Best (r1, . . . , rd) Approximation of d-Mode Ten-sors

In this talk we show that for a generic real d-mode ten-sor the best (r1, . . . , rd) approximation is unique. Further-more, for a generic symmetric d-mode tensor the best rankone approximation is unique, hence symmetric. Similarresults hold for partially symmetric tensors. The talk isbased on a recent joint paper with Giorgio Ottaviani anda recent paper of the speaker.

Shmuel FriedlandDepartment of MathematicsUniversity of Illinois, [email protected]

MS19

Computational Complexity of Tensor Problems

Frequently, a problem in science or engineering can be re-duced to solving a linear (matrix) system of equations andinequalities. Other times, solutions involve the extractionof certain quantities from matrices such as eigenvectors orsingular values. Recently, there has been a flurry of workon multilinear analogues to these basic problems of linearalgebra. These tensor methods have found applications inmany fields, including computational biology, neuroimag-ing, phylogenetics, signal processing, spectroscopy, wirelesscommunications, and other areas. Thus, tensor general-izations to the standard algorithms of linear algebra havethe potential to substantially enlarge the arsenal of core

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tools in numerical computation. We explain what is knownabout the computational complexity of tensor problems. Inparticular, we show that many naturally occurring tensorproblems are NP-hard, both to decide whether solutionsexist and to approximate them when they do.

Christopher HillarUniversity of California, [email protected]

Lek-Heng LimUniversity of [email protected]

MS19

Tensor Decomposition, Low Rank Structured Ma-trix Approximation and Applications

The tensor decomposition problem can be interpreted as arank completion problem on structured matrices. We willdetail this transformation and show how it can be exploitedin a decomposition algorithm, using a flat extension prop-erty. In the case of small rank, this yields a direct methodto compute the unique decomposition. The problem of ap-proximation by tensors of small rank will be discussed andapplications in numerical computation will be given.

Bernard MourrainGALAADINRIA Sophia [email protected]

MS19

Counting Singular Vectors of a MultidimensionalTensor

A rectangular matrix of size m×n has in general min(m,n)singular vector pairs. The singular vector pairs correspondto the critical points of the function which contracts thematrix over the product of spheres. For a tensor of size2 × 2 × n there are 6 singular vector triples for n = 2and 8 singular vector triples for n > 2. We give a for-mula counting the number of singular vector d-ples for anygeneral multidimensional tensor, considering also the (par-tially) symmetric case. The technique is to reduce to aChern class computation of a suitable vector bundle. Thisis joint work with Shmuel Friedland.

Giorgio OttavianiUniversity of [email protected]

Shmuel [email protected] of illinois at chicago

MS19

Direct Sum Decomposability of Polynomials

A form F = F (x1, . . . , xn) is decomposable as a di-rect sum if, possibly after a linear change of coordinates,F = F1(x1, . . . , xk) + F2(xk+1, . . . , xn). For example,xy = 1

4(x + y)2 − 1

4(x − y)2 and the 2 × 2 determinant

ad− bc are direct sums. General forms are indecomposableas direct sums, but this can be hard to show for partic-ular forms. We give an interesting necessary criterion fora form to be a direct sum and use it to answer a ques-tion of Shafiei. We investigate the indecomposable forms

satisfying our criterion.

Weronika Buczynska, Jaros�law [email protected], [email protected]

Zach TeitlerBoise State [email protected]

MS20

Effective Computing in Rings with Infinite Num-bers of Variables

Computing with ideals in polynomial rings having infinitenumbers of variables is usually assumed to be an impossi-ble task. For instance, algorithmic termination guaranteestypically require Noetherianity (finite generation of ideals)hypotheses, which are violated when there are more than afinite number of indeterminates. However, if the ideals inquestion have extra structure, such as being invariant un-der a group action, it is sometimes possible for symbolic al-gorithms to terminate. We discuss this phenomenon in thecontext of ideals invariant under the symmetric group anddemonstrate software that can effectively solve nontrivialproblems of symbolic algebra involving infinite numbers ofvariables.

Christopher HillarUniversity of California, [email protected]

Robert Krone, Anton LeykinSchool of MathematicsGeorgia Institute of [email protected], [email protected]

Seth SullivantNorth Carolina State [email protected]

MS20

Geometry of Wachspress Surfaces

Let Pd be a convex polygon with d vertices. The asso-ciated Wachspress surface Wd is a fundamental object ingeometric modeling, defined as the image of the rationalmap from the projective plane to projective d − 1 space,determined by the Wachspress barycentric coordinates. Weshow that the Wachpress surface is always smooth, and ford > 4 is isomorphic to the blowup of the projective planeat points corresponding to intersections of lines determin-ing the edges of the polygon, which are not vertices of thepolygon. We determine the ideal of polynomials vanishingon Wd, and other algebraic invariants (betti numbers andregularity).

Corey IrvingMathematics DepartmentSanta Clara [email protected]

Hal SchenckMathematics DepartmentUniversity of [email protected]

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MS20

Bounds on Projective Dimension

In commutative algebra, computing resolutions (e.g. byusing computer algebra systems) has become essential. Inorder to understand the complexity of these computations,it is important to be able to bound the homological invari-ants. There has been a lot of interest, motivated both froma theoretical and a computational perspective, in findingbounds on the projective dimension of homogeneous idealsof polynomial rings in terms of data readily apparent be-fore one computes a resolution. Hilbert’s syzygy theorem isa classical example. More recently, Stillman asked whetherthe projective dimension of an ideal could be bounded interms of the degrees of the generators, without involvingthe number of variables of the ambient ring. This andthe similar question for regularity are still open in com-plete generality. I will discuss some approaches and recentprogress on these questions.

Alexandra SeceleanuUniversity of [email protected]

MS20

Ghosts of the Jacobian Ideal and Graphic Arrange-ments

The Jacobian ideal J(f) = ( ∂∂xi

f)ni=1 in S = k[x1, . . . , xn]

encodes much more than just the singularities of a hyper-surface f . The ring S/J(f) is Cohen-Macaulay exactlywhen the module of logarithmic 1-forms Ω1 is a free S-module. We study families of arrangements where the codi-mension of some Exti(S/J(f), S) is greater than i. We callthe associated primes of these Ext-modules ghosts. Thegoal is to find necessary and/or sufficient combinatorialconditions for J to have a ghost.

Max WakefieldUS Naval [email protected]

MS21

Dimensional Differences Between Faces of Nonneg-ative Polynomials and Sums of Squares

We study dimensions of the faces of the cone of nonneg-ative polynomials and sums of squares. In particular, weestablish dimensional differences between these faces yield-ing nonnegative polynomials that are not sums of squares.In several important special cases we will give an explicitand constructive answer to the question of when these gapsoccur for the first time and how to use them to build non-negative polynomials that are not sums of squares. (Basedon joint work with G. Blekherman and M. Kubitzke)

Sadik IlimanGoethe University [email protected]

Grigoriy BlekhermanGeorgia [email protected]

Martina KubitzkeGoethe University [email protected]

MS21

The A-Truncated K-Moment Problem

Let A be a finite subset of Nn, and K be a compact semi-algebraic set. An A-tms is a vector y indexed by elementsin A. The A-TKMP problem studies whether a given yadmits a K-measure or not. This paper proposes a numer-ical algorithm for solving A-TKMPs. It is based on findinga flat extension of y by solving a hierarchy of semidefiniterelaxations, whose objective R is generated in a certainrandomized way. If y admits no K-measures and R[x]Ais K-full, we can get a certificate for the nonexistence ofrepresenting measures. If y admits a K-measure, then foralmost all generated R, we prove that: i) we can asymptot-ically get a flat extension of y; ii) under a general conditionthat is almost sufficient and necessary, we can get a flat ex-tension of y. The SOEP and CP decomposition problemscan be solved as an A-TKMP.

Jiawang NieUniversity of California, San [email protected]

MS21

Computing Upper Bounds for Densest PolytopePackings

I will discuss the problem of finding upper bounds for themaximum density of a packing of translative or congruentcopies of polytopes in Euclidean space. In the last years,the case of regular tetrahedra got quite some attention (butsee also Hilbert’s 18th problem) and only very weak upperbounds are known here. In the talk I will present compu-tational strategies on how to compute upper bounds. Forthis techniques from combinatorial optimization (Lovasztheta number), harmonic analysis (positive type functionsfor locally compact groups), and polynomial optimization(sums of squares invariant under Coxeter groups) will beused. (based on joint work with Cristobal Guzman andFernando Mario de Oliveira Filho)

Frank VallentinDelft University of [email protected]

MS21

A Concrete Approach to Hermitian DeterminantalRepresentations

If a Hermitian matrix of linear forms is positive definite atsome point, then its determinant is a hyperbolic hypersur-face. In 2007, Helton and Vinnikov proved a converse inthree variables, namely that every hyperbolic plane curvehas a definite Hermitian determinantal representation. Inthis talk I will discuss a concrete proof of this statementand a method for computing these determinantal represen-tations in practice. This involves relating the definitenessof a matrix to the real topology of its minors and extend-ing a classical construction of Dixon from 1902. This isjoint work with Daniel Plaumann, Rainer Sinn, and DavidSpeyer.

Cynthia VinzantUniversity of MichiganDept. of [email protected]

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MS22

Polar Varieties and Algebraic Certificates

Polar varieties are traditionally used to compute real so-lutions of polynomial systems of equations. We show howto modify their definitions so that modified polar varietiesallow to compute algebraic certificates of positivity (bymeans of sums of squares decompositions) of non-negativepolynomials over real algebraic sets.

Aurelien GreuetUniv. [email protected]

Feng [email protected]

Mohab Safey El DinUniversity Pierre and Marie [email protected]

Lihong ZhiAcademia [email protected]

MS22

Inequality Proving and Global Optimization Via aSimplified CAD Projection

Let Xn = (x1, . . . , xn) and f ∈ R[Xn,k]. The problemof finding all k such that f(Xn, k) ≥ 0 for all Xn ∈ Rn isconsidered in this paper, which obviously takes as a specialcase the problem of computing the global infimum or prov-ing the semi-definiteness of a polynomial. For solving theproblems we propose a simplified Brown’s CAD projectionoperator, called Nproj, of which the projection scale is al-ways no larger than that of Brown’s. For many problems,especially when n ≥ 3, the scale of Nproj is much smallerthan that of Brown’s. As a result, the lifting phase is alsosimplified. Some new algorithms based on Nproj for solv-ing those problems are designed and proved to be correct.Comparison to some existing tools on some examples is re-ported to illustrate the effectiveness of our new algorithms.This is joint work with Jingjun Han and Zhi Jin at PekingUniversity.

Bican XiaPeking [email protected]

MS22

Exact Safety Verification of Interval Hybrid Sys-tems Based on Symbolic-Numeric Computation

In this talk, we address the problem of safety verificationof interval hybrid systems in which the coefficients are in-tervals instead of explicit numbers. A hybrid symbolic-numeric method, based on SOS relaxation and intervalarithmetic certification, is proposed to generate exact in-equality invariants for safety verification of interval hybridsystems. As an application, an approach is provided toverify safety properties of non-polynomial hybrid systems.Experiments on the benchmark hybrid systems are givento illustrate the efficiency of our method. This is joint workwith Min Wu and Wang Lin.

Zhengfeng YangShanghai Key Laboratory of Trustworthy Computing

[email protected]

Min Wu, Wang LinShanghai Key Laboratory of Trustworthy ComputingEast China Normal [email protected], [email protected]

MS22

Computing Rational Solutions of Linear Matrix In-equalities

Consider a (D×D) symmetric matrix A whose entries arelinear forms in Q[X1, . . . ,Xk] with coefficients of bit size≤ τ . We provide an algorithm which decides the existenceof rational solutions to the linear matrix inequality A � 0and outputs such a rational solution if it exists. This prob-lem is of first importance: it can be used to compute alge-braic certificates of positivity for multivariate polynomials.

Our algorithm runs within (kτ )O(1)2O(min(k,D)D2)DO(D2)

bit operations; the bit size of the output solution is dom-

inated by τO(1)2O(min(k,D)D2). These results are obtainedby designing algorithmic variants of constructions intro-duced by Klep and Schweighofer. This leads to the bestcomplexity bounds for deciding the existence of sums ofsquares with rational coefficients of a given polynomial.We have implemented the algorithm; it has been able totackle Scheiderer’s example of a multivariate polynomialthat is a sum of squares over the reals but not over therationals; providing the first computer validation of thiscounter-example to Sturmfels’ conjecture.

Lihong Zhi, Qingdong GuoAcademia [email protected], [email protected]

Mohab Safey El DinUPMC, Univ Paris [email protected]

MS23

Computing Persistent Homology in Chunks

We present a parallel algorithm for computing the persis-tent homology of a filtered chain complex. Our approachdiffers from the commonly used reduction algorithm by firstcomputing persistence pairs within local chunks, then sim-plifying the unpaired columns, and finally applying stan-dard reduction on the simplified matrix. The approachgeneralizes a technique by Gunther et al., which uses dis-crete Morse Theory to compute persistence; we derive thesame worst-case complexity bound in a more general con-text. The algorithm employs several practical optimizationtechniques which are of independent interest. Our sequen-tial implementation of the algorithm is competitive withstate-of-the-art methods, and we improve the performancethrough parallelized computation.

Ulrich BauerInstitute of Science and Technology (Austria)[email protected]

Michael KerberStanford [email protected]

Jan ReininghausIST [email protected]

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MS23

Computational (co)Homology in ElectromagneticModelling and Material Analysis

In this talk we show engineering and material science ap-plications of topology. With so called Discrete GeometricApproach discrete counterparts of Maxwell’s laws will beprovided. We show how cohomology generators make themconsistent. Some cohomology algorithms will be stressed.Later we will show efficient ways of computing homologyof nodal domains. We will motivate their importance inanalyzing so called spinodal decomposition in alloys. Thisis joint work with Ruben Specogna and Thomas Wanner.

Pawel DlotkoInstitute of Computer ScienceJagiellonian [email protected]

MS23

One the Persistent Homology of Time-Delay Em-beddings

We present in this talk a theoretical framework for thestudy of the persistent homology of time-delay embeddings.In particular, we propose maximum persistence as a mea-sure of periodicity at the signal level, present structuraltheorems for the resulting diagrams, and derive estimatesfor their dependency on window size and embedding di-mension. We apply this methodology to quantifying peri-odicity in synthetic signals, and present comparisons withstate-of-the-art methods in gene expression analysis.

Jose Perea, John HarerDuke [email protected], [email protected]

MS23

New Topological Methods for Robotic Graspingand Machine Learning

This talk focusses on our recent efforts in adapting ideasfrom topology for applications in robotic grasping and ma-chine learning. We will present some of the research ques-tions and challenges that one faces in these fields and wewill discuss how techniques from topology might be usedthere. In particular, we will discuss an application of ap-proximately shortest homology generators for grasping andwe will describe how topological constraints can be used indensity estimation.

Florian T. PokornyKTH Royal Institute of [email protected]

MS23

A Categorical Approach to Multipersistent Homol-ogy

Multipersistent homology represents an interesting chal-lenge both from an applied and a theoretical point of view.Introduced by G.Carlsson and A.Zomorodian, this methodanalyzes a point cloud (noisy data set in a metric space)through two or more parameters. Classically, it is studiedin the category of multigraded modules over the polynomialring. We will rephrase multipersistence in the categoryof functors from Nr to vector spaces, give a constructivemethod to build free minimal resolutions and characterize

Betti numbers.

Martina ScolamieroPolitecnico di [email protected]

Wojciech ChacholskiKTH Royal Institue of [email protected]

Francesco VaccarinoPolitecnico di [email protected]

MS24

Exotic Cluster Structure in Gln

Abstract not available at time of publication.

Michael GekhtmanUniversity of Notre [email protected]

MS24

Tropical Curves in the Planar Dimer Model

Associated to a bipartite, periodic, edge-weighted planargraph is a bivariate polynomial P (z, w), which is the deter-minant of a certain adjacency matrix arising in the dimermodel. In an appropriate ”low temperature” limit thecurve P = 0 becomes a tropical curve. Various featuresof the tropical curve have corresponding probabilistic in-terpretations for the resulting dimer model.

Richard KenyonBrown [email protected]

MS24

Generalized Discrete Toda Lattices

I will discuss a discrete-time dynamical system which gen-eralizes the discrete Toda lattice. In a continuous limit thisdynamical system becomes a variation of the Toda lattice.In the tropical limit this dynamical system gives a varia-tion of the box-ball system. There are also connections toa combinatorial recurrence known as the octahedron recur-rence, and to the study of graphs and networks embeddedin a torus. This is joint work with Rei Inoue and PavloPylyavskyy.

Thomas LamUniversity of [email protected]

MS24

Higher-Dimensional Analogues of Tropical ClusterCombinatorics

Various tropical features of cluster structures (exchange re-lations, g-vectors) generalize beyond the setting of clusteralgebras. I will recall the tropical structures of interest,specialized to the case of a polygon, and discuss what hap-pens when the polygon is replaced by a higher-dimensionalcyclic polytope. This is based on arXiv:1001.5437 and sub-sequent work.

Steffen Oppermann

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Norwegian University of Science and [email protected]

Hugh ThomasUniversity of New [email protected]

MS25

Identifiability of Linear Structural Equation Mod-els

This talk presents combinatorial conditions for genericidentifiability of linear structural equation models. Thesemodels relate random variables of interest via a linear equa-tion system and can be represented by a graph with twotypes of edges that correspond to non-zero coefficients inthe linear equations and correlations among noise terms.Identifiability holds if the coefficients and correlations as-sociated with the edges of the graph can be uniquely re-covered from the covariance matrix they define.

Mathias DrtonDepartment of StatisticsUniversity of [email protected]

Rina FoygelDepartment of StatisticsUniversity of [email protected]

Jan DraismaEindhoven University of Technology, The NetherlandsDepartment of Mathematics and Computer [email protected]

MS25

Identifiability of Structural Equation Models on 6Random Variables

Structural equation models (SEMs) are used to formalizea variety of causal queries as certain types of probabilitydistributions. A central problem in SEMs is the analysisof identification. A model is identified if it only admitsa unique parametrization to be compatible with a givendata set. Here, we present the results of a computationalexperiment to find all the algebraically d-identified SEMmodels on at most six random variables.

Luis D. Garcia-PuenteSam Houston State [email protected]

MS25

Algebraic Theory for Discrete Models in SystemsBiology

Systems biology aims to explain how a biological systemfunctions by investigating the interactions of its individualcomponents from a systems perspective. Modeling is a vi-tal tool as it helps to elucidate the underlying mechanismsof the system. Many discrete model types can be trans-lated into the framework of polynomial dynamical systems(PDS), that is, time- and state-discrete dynamical systemsover a finite field where the transition function for eachvariable is given as a polynomial. This allows for usinga range of theoretical and computational tools from com-puter algebra, which results in a powerful computational

engine for model construction, parameter estimation, andanalysis methods.

Franziska B. HinkelmannICAMVirginia [email protected]

MS25

Scaling Invariants and Symmetry Reduction of Dy-namical Systems

Scalings form a class of group actions that have theoreticaland practical importance. A scaling is accurately describedby a matrix of integers. Tools from linear algebra over theintegers are exploited to compute their invariants, rationalsections (a.k.a. global cross-sections), and offer an algo-rithmic scheme for the symmetry reduction of dynamicalsystems. A special case of the symmetry reduction algo-rithm applies to reduce the number of parameters in phys-ical, chemical or biological models: we provide a monomialreparametrisation of the variables that leads to a modelthat depends on less parameters. This can be understoodas an (algorithmic) non-dimensionalisation. We illustratethe technique on models from the mathematical biologyliterature.

Evelyne HubertINRIA [email protected]

George LabahnUniversity of [email protected]

MS26

An Incremental Algorithm for Computing Cylin-drical Algebraic Decomposition and Its Applicationto Quantifier Elimination

In this talk, we present a new approach for computingcylindrical algebraic decomposition of real space via trian-gular decomposition of polynomial systems. Based on it,a general quantifier elimination method is proposed. Theusage of our method is illustrated by several applicationexamples.

Changbo ChenUniversity of Western [email protected]

Marc Moreno MazaComputer Science DepartmentUniversity of Western [email protected]

MS26

Turning CAD Upside Down

We present a new decision procedure for solving the exis-tential fragment of non-linear arithmetic. The classic ap-proach to the problem is to project the problem polynomi-als, followed by model construction (lifting). The new pro-cedure performs the lifting optimistically, and performs fo-cused model-based projection only on the polynomials thatare relevant in inconsistencies. The new approach, and theleaner projection operator, are effective in practice, which

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we support with an extensive experimental evaluation.

Dejan JovanovicNew York [email protected]

Leonardo de MouraMicrosoft [email protected]

MS26

An Application of Quantifier Elimination to Auto-matic Parallelization of Computer Programs

In automatic parallelization of computer programs, the so-called polyhedron model is a powerful geometrical tool foranalyzing the relation between iterations of nested loops.To be practically efficient, one should avoid a too fine-grained parallelization. It is also desirable for the gen-erated code to depend on parameters such as number ofprocessors, cache sizes, etc. These extensions of the poly-hedron model lead to the manipulation of system of non-linear polynomial equations and the use of techniques likequantifier elimination (QE), Unfortunately, classical algo-rithms for QE are not suitable, since they do not alwaysproduce conjunctions of atomic formulas, while this formatis required in order to generate code automatically. As weshall demonstrate in this talk, this issue is addressed byour recent algorithm for computing cylindrical algebraicdecomposition (C. Chen and M. Moreno Maza, 2012). In-deed, this algorithm supports QE in a way that the outputof a QE problem has the form of a case discussion: this isappropriate for code generation.

Marc Moreno MazaComputer Science DepartmentUniversity of Western [email protected]

Changbo ChenUniversity of Western [email protected]

MS26

Relative Equilibria in the Four-Vortex Problemwith Two Pairs of Equal Vorticities

The N-vortex problem concerns the dynamics of N pointvortices moving in the plane. Of particular interest in thisproblem are solutions that appear fixed when viewed in anuniformly rotating frame. Such solutions are called relativeequilibria. In our work we gave a complete classificationof the relative equilibria in the four vortex problem withtwo pairs of equal vorticities. In this talk I will outlinethe problem and describe some of the methods used to ob-tain our results, with particular emphasis to the triangulardecomposition of semi-algebraic systems.

Manuele SantopreteWilfrid Laurier [email protected]

Marshall HamptonUniversity of Minnesota, [email protected]

Gareth RobertsCollege of the Holy [email protected]

MS27

Projective Path Tracking for Homotopy Continua-tion Methods

The homotopy continuation methods is a large class ofreliable and efficient numerical methods for solving sys-tems of polynomial equations. An essential component insuch methods is the path tracking algorithm for trackingsmooth paths of one real dimension. “Divergent paths’pose a tough challenge to these algorithms. A well knownremedy is to operate inside the complex projective spaceinstead. This talk focuses on the path tracking algorithmsin the complex projective space.

Tianran ChenMichigan State [email protected]

Tien-Yien LiDepartment of MathematicsMichigan State [email protected]

MS27

Numerically Computing Polynomial Images of Al-gebraic Sets with Applications

Given a polynomial map π defined on an algebraic set V , acommon goal is to compute the irreducible decomposition

of π(V ). In this talk, we will discuss numerical methodsfor determining the dimension, degree, and other invariantsfor each component of this decomposition. The talk willconclude by applying these methods to systems arising inphysics.

Noah Daleo, Jonathan HauensteinNorth Carolina State [email protected], [email protected]

MS27

Applications of Numerical Elimination Theory

An extensive collection of problems in algebraic geome-try, algebraic statistics, biology, geophysics, linear algebra,and physics can be formulated in terms of computing theirreducible decomposition of the closure of the polynomialimage of an algebraic set. Often in these applications, thepolynomial map is a nonlinear mapping from hidden statesto observable states. This talk will explore using variousnumerical algebraic geometric algorithms to solve problemsarising in several of these applications.

Jonathan HauensteinNorth Carolina State [email protected]

MS27

On Massively Parallel Algorithms to Track OnePath of a Polynomial Homotopy

The latest generation of graphics processors delivers oneTflop of peak performance but requires massively parallelalgorithms occupying thousands of threads. In previouswork we obtained good speedups for two building blocksto run Newton’s method: the evaluation and differentiationof polynomial systems, and the solving of linear systems inthe least squares sense, using double double and quad dou-ble arithmetic. This talk will present a massively parallel

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algorithm to track one solution path of a homotopy definedby a polynomial system.

Jan VerscheldeDepartment of Mathematics, Statistics and ComputerScienceUniversity of Illinois at [email protected]

Genady YoffeDept. of Mathematics, Statistics, and CSUniversity of Illinois, [email protected]

Xiangcheng YuDepartment of Mathematics, Statistics, and ComputerScienceUniversity of Illinois at [email protected]

MS27

Numerical Algebraic Intersection Using Regenera-tion

In numerical algebraic geometry, algebraic sets are repre-sented by witness sets. This paper presents an algorithm,based on the regeneration technique, that solves the follow-ing problem: given a witness set for pure-dimensional alge-braic set Z, along with a system of polynomial equationsf : Z → Cn, compute a numerical irreducible decomposi-tion of Z ∩ V (f). Also treated is the case where Z is thecross product of two or more pure-dimensional sets, eachgiven in terms of a witness set. Two existing algorithms,diagonal intersection and the homotopy membership test,can be seen as special cases of the new algorithm. In addi-tion to this unification, the method also extends the rangeof problems that can be solved using numerical algebraicgeometry.

Charles WamplerGeneral Motors Research Laboratories, EnterpriseSystems Lab30500 Mound Road, Warren, MI 48090-9055, [email protected]

Jonathan HauensteinNorth Carolina State [email protected]

MS28

K-Theory of Toric Varieties Revisited

Algebraic K-theory is a synthetic subject, drawing ideasfrom various mathematical disciplines. In this talk we willreview algebraic K-theory of monoid rings and toric vari-eties, starting from the 1980s and including very recent de-velopments. The classical part of this theory is algebraic inthe true meaning of the word, allowing very meaningful ap-plication: algorithms for solving systems of linear equationswith multivariate polynomial coefficients, factoring invert-ible matrices into elementary ones (and such algorithmshave been implemented in the past). However, as we moveto higher K-groups, the nature of the subject changes fromalgebraic to topological and even categorial. Correspond-ingly, the applied element quickly becomes ephemeral - butnot in the sense of applications to other areas of mathemat-ics.

Joseph Gubeladze

San Francisco State [email protected]

MS28

Frobenius Splitting and Toric Varieties

Frobenius splittings provide some very useful techniquesfor the study of algebraic varieties. After a quick reviewof Frobenius splittings I will present some results on theF -splitting ratio of a toric ring, an invariant closely relatedto the F -signature that measures the singularities of thevariety.

Milena HeringUniversity of [email protected]

Kevin TuckerPrinceton [email protected]

MS28

Equivariant Vector Bundles on T-Varieties

Klyachko has shown that there is an equivalence of cate-gories between equivariant vector bundles on a toric varietyX and collections of filtered vector spaces satisfying somecompatibility conditions. I will discuss joint work with H.Suß which generalizes this equivalence to the setting of T -equivariant vector bundles on a normal variety X endowedwith an effective action of an algebraic torus T . Indeed, T -equivariant vector bundles on X correspond to collectionsof filtered vector bundles on a suitable quotient of X. Thiscorrespondence can be applied to show that T -equivariantbundles of low rank on projective space split, as well as toeasily compute the space of global vector fields on rationalcomplexity-one T -varieties.

Nathan IltenUniversity of California, [email protected]

MS28

The Hodge Theory of Hypersurfaces

Geometric properties of generic hypersurfaces in projectivetoric varieties are often determined by their correspondinglattice polytopes. Pioneering work of Danilov-Khovanskiigave combinatorial descriptions for their Euler character-istic and χy-characteristic in terms of combinatorics. Injoint work with Alan Stapledon, we outline an alternativeapproach to hypersurfaces. Here, we degenerate the hyper-surface into a union of linear subspaces and use the limitmixed Hodge structure to understand the cohomology. Wemay also explain some applications to Ehrhart theory.

Eric KatzDepartment of Combinatorics and OptimizationUniversity of [email protected]

MS28

Syzygies and Singularities of Tensor Product Sur-faces of Bidegree (2,1)

Let {f1, . . . , f4} be four bihomogeneous polynomials ofbidegree (2,1), having no common basepoints on X =P 1 × P 1. The fi define a regular map from X to P 3.

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We study the associated bigraded ideal I of the fi fromthe standpoint of commutative algebra, proving that thereare exactly six numerical types of possible bigraded min-imal free resolution. These resolutions play a key role indetermining the implicit equation of the image, and weshow that there is a pleasing dictionary which relates thesyzygies on I to the singularities of the image.

Hal SchenckMathematics DepartmentUniversity of [email protected]

A. SeceleanuUniversity of [email protected]

J. ValidashtiUniveristy of [email protected]

MS29

Degree of Regularity of Hfe Family of Cryptosys-tems

In this talk, we will present the recent results on the de-gree of regularity of Hidden Field Equation (HFE) familyof multivariate public key cryprosystems, including, HFE,HFE-, HFEv and HFEv-, in particular, the bound formu-las for the degree of regularity of these systems . Further-more we will discuss the implication of these results on thesecurity of those multivariate pubic key cryptosystems un-der direct algebraic attack using multivariate polynomialsolvers.

Jintai DingUniversity of [email protected]

MS29

Degree of Regularity of Generalized HFE Cryp-tosystems

Ding and Hodges recently showed that the algebraic attackon Patarin’s HFE system is quasi-polynomial over any basefield. We extend this result to more generalized systemswhere the central polynomial is not necessarily quadratic(i.e., is of higher degree) over the base field. The core ofthe proof lies in some interesting combinatorial propertiesof generalized lacunary binomial coefficients.

Timothy HodgesUniversity of Cincinnati, [email protected]

MS29

Overview of Post-Quantum Cryptography

This talk gives an overview of post-quantum cryptogra-phy to set the stage for the mini symposium. It motivatesthe need for cryptography resisting attacks using quantumcomputers and briefly presents the most relevant 4 groupsof systems: code-based cryptography, lattice-based cryp-tography, multivariate signatures, and hash-based signa-tures.

Tanja LangeTechnische Universiteit Eindhoven

Eindhoven Institute for the Protection of Systems [email protected]

MS29

On the Practical and Asymptotic Complexity ofSolving Generic Systems of Equations

Existing literature overwhelmingly list F4 /F5 algorithmsas optimal general solution to MQ, or solving a system ofm quadratic equations in n variables in Fq. We show thatthe complexity of system-solving should be estimated dif-ferently in the following cases: 1. For random or generic F2

systems, brute-force is often better. 2. For many randomsystems (small-field, high-degree, or somewhat overdeter-mined with m ∝ n) XL and F4/F5 operates at degreesmax 1 apart, and here Sparse XL variants are likely prag-matically or asymptotically better.

Bo-Yin YangAcademia [email protected]

MS30

Persistence Barcode Signatures for Image Classifi-cation

Persistent topology provides a method for understandingdata sets in general, and also specifically for data sets inwhich the points are ”shapes” of some kind. We will discussthis methodology, with some examples.

Gunnar E. CarlssonStanford [email protected]

MS30

Image Segmentation with Topological Information– Yet Another Application of Persistent Homology

Topological information could be very useful for the taskof image segmentation. However, most existing works havetrouble incorporating such combinatorial information intothe optimization framework. Persistent homology, on theother hand, is a natural tool to quantitatively measuretopological features, in the view of a given scalar func-tion. In this talk, I will introduce how to use persistenthomology in image segmentation, as well as how topologi-cal information could improve segmentation results.

Chao ChenRutgers [email protected]

MS30

Persistence Simplification with Iterated MorseComplex Decomposition

In this talk we will combine ideas of discrete Morse the-ory and (persistent) homology. A finite sequence of Morsecomplexes each of which preserves (persistent) homologyof the initial object will be constructed. The final complexin the sequence is a minimal complex with the same (per-sistent) homology as the initial object. Later we will showhow to continue the construction to obtain barcodes of agiven object. This is joint work with Hubert Wagner.

Pawel Dlotko

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Institute of Computer ScienceJagiellonian [email protected]

MS30

Stability of Persistence Spaces for Vector-ValuedFunctions

In shape analysis it is common to deal with data carry-ing multi-parameter information. Multidimensional persis-tence allows for a topological analysis of these data. Unfor-tunately, multidimensional persistent modules do not ad-mit concise representations analogous to persistence dia-grams in the one-parameter setting. However, there is noobstruction for multidimensional persistent Betti numbersto admit one. In this talk, the persistence space of a multi-filtration is shown to generalize the concept of persistencediagram in this sense.

Claudia LandiUniversity of Modena and Reggio [email protected]

Andrea [email protected]

MS30

Metric Geometry and Persistent Homology

We show how metric measure spaces – a mathematicalmodel for datasets – induce certain stable persistence ho-mology invariants whose study provides some insights intothe problem of computing statistical summaries of thepersistent homology of large datasets. These invariantsare a family of probability distributions over persistencediagrams which turn out to be stable in the Gromov-Wasserstein sense.

Facundo MemoliStanford University, Mathematics DepartmentThe University of [email protected]

MS31

Extremal Betti Tables

We will discuss extremal Betti tables of resolutions in threedifferent contexts. We begin over the graded polynomialring, where extremal Betti tables correspond to pure reso-lutions and can be realized via so-called tensor complexes.We then contrast this behavior with that of extremal Bettitables over regular local rings and over a bigraded ring.

Christine BerkeschDuke [email protected]

Daniel ErmanUniversity of [email protected]

Manoj KumminiChennai Mathematical [email protected]

MS31

A Set-theoretic Proof of the Salmon Conjecture

In 2007, motivated by applications to molecular phyloge-netics, Elizabeth Allman offered the challenge of determin-ing the polynomials that vanish on the irreducible varietyof 4 x 4 x 4 complex valued tensors of border rank at most4. In this talk, we will show that this variety is the zero setof polynomials of degrees 5, 6, and 9. We will focus mainlyon the degree 6 polynomials.

Elizabeth GrossUniversity of Illinois at [email protected]

Shmuel FriedlandDepartment of MathematicsUniversity of Illinois, [email protected]

MS31

Tangential Varieties of Segre-Veronese Varieties

I will report on joint work with Luke Oeding, involving thedetermination of the homogeneous coordinate ring, and ofthe minimal generators of the defining ideal of the tan-gential variety of a Segre-Veronese variety. In the specialcase of a Segre variety, our results confirm a conjecture ofLandsberg and Weyman.

Claudiu RaicuPrinceton [email protected]

Luke OedingUC [email protected]

MS32

Lozenge Tilings and the Weak Lefschetz Property

Let I be a monomial ideal in K[x, y, z]. We describe aconnection between I and a sequence of planar regions ofthe triangular grid. We use this connection to determinethe presence (or absence) of the weak Lefschetz propertyfor I . This is joint work with Uwe Nagel at the Universityof Kentucky.

David Cook IIUniversity of Notre [email protected]

MS32

Computer-aided Unirationality Proofs

An algebraic variety is called unirational if there is domi-nant map from some projective space to it. Of particularinterest is the case of unirational moduli spaces of certainobjects since this translates to a parametrization of theobjects in free parameters in terms of the dominant map.In this talk we present a recent unirationality results forHurwitz spaces (i.e. parameter spaces of simply branchedcoverings of the projective line by smooth curves) and showhow computer-algebra can be used to substantially ease thestep of showing that the parametrizing map is dominantby using computer algebra. The implementation of sucha parametrization can be useful for various applications.We present how this method enters in existence results for

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certain vector bundles on cubic hypersurface threefolds.

Florian Gei, Frank-Olaf SchreyerUniversitat des [email protected], [email protected]

MS32

Inferring Biologically Relevant Models: NestedCanalyzing Functions

Inferring dynamic biochemical networks is one of the mainchallenges in systems biology. Given experimental data,the objective is to identify the rules of interaction amongthe different entities of the network. However, the numberof possible models fitting the available data is huge, andidentifying a biologically relevant model is of great interest.Nested canalyzing functions, where variables in a given or-der dominate the function, have recently been proposed asa framework for modeling gene regulatory networks. Pre-viously, we described this class of functions as an algebraictoric variety. In this talk , we present an algorithm thatidentifies all nested canalyzing models that fit the givendata.

Franziska HinkelmannOhio State [email protected]

MS32

Groebner-free Computations with Binomial Ideals

The membership problem for a pure difference binomialideal I is graph connectivity: A binomial is contained in Iiff there exists a path connecting its two monomials usingbinomials in I. This idea is applicable even if Grobner basismethods are infeasible. We present a C++ implementationwhich computed the first counterexample to radicality ofa global Markov ideal. Based on joint work with JohannesRauh and Seth Sullivant.

Thomas [email protected]

MS32

Algebraic Statistics and Macaulay2: RunningMarkov Chains on Network Fibers

Sampling from a fiber of a (social) network is a crucial stepin goodness-of-fit testing for network models in statistics.In order to apply recent theoretical results, one needs tobe able to explore a fiber of an observed network. Thistalk will focus on an algorithm for p1 random graph mod-els. Macaulay2 plays a crucial role in the development andexperimentation.

Elizabeth GrossUniversity of Illinois at [email protected]

Sonja PetrovicDepartment of StatisticsPennsylvania State [email protected]

Despina StasiPennsylvania State [email protected]

MS33

Polytopes with Minimal Semidefinite Representa-tions

Finding good semidefinite representations of polytopes is avery relevant problem for combinatorial optimization andmany methodologies for doing so have been proposed. Theproblem of finding the smallest possible representation ofa polytope has been recently recast as a problem of ma-trix factorization. In this talk, we will give an overview ofrecent developments in this area, with special focus on acharacterization of polytopes which have the smallest pos-sible representations.

Joao GouveiaUniversidade de [email protected]

Richard Robinson, Rekha ThomasUniversity of [email protected], [email protected]

MS33

Realizing Hyperbolicity Cones As Spectrahedraand Their Projections

Spectrahedral cones are the feasible sets of semidefiniteprogramming. Hyperbolicity cones form a potentiallylarger class of convex cones. Trying to realize hyperbol-icity cones as spectrahedral cones or projections thereofis an interesting and important task. Important, since ithelps classifying the sets on which semidefinite program-ming can be performed. Interesting, since methods andresults from different areas, such as algebraic geometry,positivity and sums of squares, non-commutative geome-try, and more, are involved when solving such problems. Iwill report on recent methods and results concerning thesequestions.

Tim NetzerUniversity of [email protected]

MS33

On Elliptesque and Hyperbolesque Curves

Any five points in the plane (no four on a line) determinea unique conic section. What can be said about a curveC with the property that any five points chosen from Ceither always determine an ellipse or always determine ahyperbola?

Bruce ReznickUniversity of Illinois, [email protected]

MS33

Polynomial-Sized Semidefinite Representations ofDerivative Relaxations of Spectrahedral Cones

The hyperbolicity cones associated with the elementarysymmetric polynomials provide an intriguing family of non-polyhedral relaxations of the non-negative orthant whichpreserve its low-dimensional faces and successively discardhigher dimensional facial structure. We show by an explicitconstruction that this family of convex cones (as well astheir analogues for symmetric matrices) have polynomial-sized representations as projections of slices of the PSD

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cone. This, for example, allows us to solve the associatedlinear cone program using semidefinite programming.

James Saunderson, Pablo A. ParriloMassachusetts Institute of [email protected], [email protected]

MS34

Recursive Interpolation of a Sparse PolynomialGiven by a Straight-Line Program

We consider the problem of interpolating a sparse univari-ate polynomial given by a straight-line program. Supposewe are given a univariate polynomial f(z) represented by astraight-line program, and that we have bounds T and Don the sparsity and degree of f(z) respectively. We presenta recursive, randomized algorithm for interpolating f, withcomplexity quasi-linear with respect to T log3(D). Thisalgorithm builds previous work by Giesbrecht and Roche,and Garg and Schost.

Andrew ArnoldCheriton School of Computer ScienceUniversity of [email protected]

Mark GiesbrechtSchool of Computer ScienceUniversity of [email protected]

Dan RocheUnited States Naval [email protected]

MS34

New Approaches to Sparse Interpolation and Sig-nal Reconstruction

Abstract not available at time of publication.

Mark GiesbrechtSchool of Computer ScienceUniversity of [email protected]

MS34

Numerical Issues with Sparse Interpolation

Abstract not available at time of publication.

George LabahnUniversity of [email protected]

MS34

Sparse Interpolation with Noise and Outliers

We address the problem of interpolation of sparse poly-nomials in the presence of noise and errors in the evalua-tions. Our approach is based on Ben-Or & Tiwari’s andBerlekamp/Massey algorithms with early termination. Weshow a lower bound on the number of terms required in asequence in order to recover its minimal linear generator.We then provide an algorithm that finds this generator. Aduality with Reed-Solomon decoding will be shown. Lastly,we will address the more general problem of rational func-

tion and number reconstruction with errors.

Clement PernetUniiversite de [email protected]

MS34

Combining Tricks for Exact Sparse Interpolation

Abstract not available at time of publication.

Daniel S. RocheDept. of Computer ScienceUS Naval [email protected]

MS35

A Numerical Algorithm for Zero Counting

In memoriam of our dear friend Mario In a series of threearticles we produce and analyze a numerical (iterative) al-gorithm for computing the exact number of real projectivezeros of a system of n real homogeneous polynomial equa-tions in n + 1 variables. In the first one, we show thatthe number of iterations, and the cost of each iteration,depends on a condition number of the system, in additionto other usual parameters. The algorithm can be imple-mented with finite precision as long as the round-off unitsatisfies some bound depending on the same parameters. Inthe second one, we show that the condition number that weintroduced satisfies an Eckard-Young theorem, as it repre-sents the inverse of the distance of the input system to theill-posed systems. We derive from this smoothed analysisbounds for it, applying other general results. Finally, in thethird one, we use specific probability techniques as a Ricetheorem to obtain more precise bounds for the tail and theexpected value of the condition number, considered as arandom variable when imposing a probability measure onthe space of polynomial systems.

Felipe CuckerCity University, Hong [email protected]

Teresa KrickUniversidad de Buenos [email protected]

Gregorio MalajovichUFRJ, [email protected]

Mario WscheborUniversidad de la Republica, [email protected]

MS35

How Far Are Archimedean Tropical Varieties fromAmoebae?

There is a simple, and arguably natural way to asso-ciate a polyhedral approximation to any complex hyper-surface Z(f): we call this approximation the Archimedeantropical variety ArchTrop(f). The univariate casedates back to work of Ostrowski around 1940, and thehigher-dimensional case has appeared in work of Passare,Mikalkin, and other authors starting around 2000. How-ever, the distance between Z(f) and ArchTrop(f) has not

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been made studied in general. We give a simple and ex-plicit upper bound for the Hausdorff distance between Z(f)and ArchTrop(f). As an application, we show how to ef-ficiently decide whether a given ball contains any roots ofa given polynomial system.

J. Maurice Rojas, Martin AvendanoTexas A&M [email protected], [email protected]

MS35

On the Complexity of Computing the Dimensionof Semi-algebaic Sets

Let S be a semi-algebraic set defined by a system of spolynomial inequalities in n variables of degree ≤ D. Wegive an algorithm that computes the real dimension of thesemi-algebraic set S in (sD)O(n) arithmetic operations.

Mohab Safey El DinUPMC, Univ Paris [email protected]

Elias TsigaridasINRIA/Univ. Pierre et Marie Curie, [email protected]

MS35

On the Complexity of Solving Bivariate PolynomialSystems

We give an algorithm for solving bivariate polynomial sys-tems over either k(T )[X,Y ] or Q[X,Y ] using a combinationof lifting and modular composition techniques.

Esmaeil MehrabiORCCAWestern [email protected]

Eric SchostComputer Science DepartmentUniv. of Western [email protected]

Romain LebretonLIRMMUniversite Montpellier [email protected]

MS36

Computing Hypergeometric Solutions of SecondOrder Differential Equations with Five Singulari-ties

If a generating function of an integer or a near-integer se-quence is convergent and holonomic (satisfies a linear dif-ferential equation), then we have observed that such differ-ential equation of order 2 or 3 has either algebraic solutionsor logarithmic singularities. In the later case we can ex-press its solution in terms of 2F1-hypergeometric functions.That gives us a way to prove whether a sequence is an inte-ger or a near-integer sequence. In this talk we will discussabout computing 2F1-type solutions of a second order lin-ear differential equation with five singularities.

Vijay KunwarFlorida State University

[email protected]

Mark van HoeijFlorida State UniversityTallahassee, [email protected]

MS36

Computing Decompositions of HypergeometricTerms

Two algorithms will be described in this talk. The first de-composes a univariate hypergeometric term into a sum of ahypergeometric-summable term and a non-summable one.It is an improvement of Abramov-Petkovsek’s algorithmfor additive decompositions of hypergeometric terms. Thesecond computes a multiplicative decomposition of a mul-tivariate hypergeometric term. It is based on the Ore-Satotheorem, which states that such a term is the product of arational function and several factorials.

Ziming LiKey Laboratory of Mathematics MechanizationAMSS, Chinese Academy of [email protected]

MS36

A Combinatorial Approach to Lattice Path Asymp-totics

Several approaches for finding asymptotic formulas for lat-tice path models restricted to various regions have re-cently been proposed. This talk will describe some recentprogress, including algorithmic and probabilistic strategies,and then focus on our new, combinatorial point of view.Our approach uses only elementary combinatorial argu-ments; eliminates the case analysis that was previously nec-essary; and can be extended to a much larger framework,for example, models in higher dimensions, and cones otherthan the quarter plane. Work in progress with SamuelJohnson and Karen Yeats (Simon Fraser).

Marni MishnaSimon Fraser [email protected]

MS36

Holonomic Integer Sequences and TranscendentalNumbers

Let L be a recurrence relation, with polynomial coefficientsover the rational numbers. If the quotient u(n)/v(n) con-verges, for every pair of non-zero sequences u, v that (a)satisfy L, (b) have rational values, and (c) have non-zerotails, then we say that L is a single-growth recurrence. Suchrecurrences are quite common (e.g. Apery’s recurrence). Ifthe tails of u, v, w are linearly independent then the limitsof u(n)/w(n), v(n)/w(n) and 1 must be linearly indepen-dent over the rational numbers. This can be used to provevarious irrationality results, including Baker’s formulationof the Lindemann-Weierstrass theorem.

Mark van HoeijFlorida State [email protected]

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MS37

Automatic Natural Language Mathematical Prob-lem Solving Using Real Quantifier Elimination

We present a logic-based architecture for solving mathe-matical problems written in natural language. At the coreof the system is a computer algebra solver that handles anyformula in the language of real closed field. The naturallanguage processing (NLP) module translates the problemtext into a logical form on the basis of linguistic analysisof the text. We aim at an end-to-end problem solving sys-tem through the synthesis of the advances in linguistics,NLP, and computer mathematics. We show some empir-ical results on Japanese university entrance examinationproblems.

Hirokazu AnaiKyushu [email protected]

Hidenao IwaneFUJITSU LABORATORIES LTD / Kyushu University4-1-1 Kamikodanaka, Nakahara-ku, Kawasaki 211-8588,[email protected]

Takuya Matsuzaki, Norico AraiNational Institute of [email protected], [email protected]

MS37

Constructing a Single Cell in a Cylindrical Alge-braic Decomposition

In a 2012 paper, Jovanovic and de Moura present NLSAT,an novel algorithm that uses Conflict-Driven Clause Learn-ing (CDCL)-style search to decide the satisfiability of sys-tems of polynomial equalities and inequalities over the realnumbers. An essential step in this algorithm is to takean assignment of real values that do not satisfy the orig-inal system, and generalize it to a larger region in whichall points fail to satisfy the original system. They do thisby constructing a single cell in a Cylindrical Algebraic De-composition (CAD) defined from a set of polynomials —namely the cell that contains the unsatisfying assignment.This motivates the consideration of a new problem: Givena set P of polynomials in n variables and a point α in n-dimensional real space, how can we efficiently construct arepresentation of the cell containing α from the CAD de-fined (with respect to some projection operator) by the setP . This talk will present an algorithm for this problem.

Christopher BrownUnited States Naval [email protected]

MS37

Sparse Interpolation and Error-Correcting Coding

Interpolation of univariate polynomials from values,with several of the values erroneous, is used to con-struct the Reed-Solomon error correcting code. TheBerlekamp/Welch decoder recovers a polynomial of degree< d from values at d + 2E distinct points, where k ≤ Evalues are faulty, that for unkown k at unknown locations.With Matthew Comer and Clement Pernet [ISSAC 2012]we have considered recovery of a t-sparse polynomial ofunknown sparsity t, with a given bound T ≥ t. Our decod-ing procedure requires (2E + 1)2T values, while an error-

correcting code is possible for evaluations at 2T + 2E dis-tinct positive real points. We use our decoding procedurefor numeric sparse interpolation with outlier errors. In mytalk I will discuss the hardness of the Ω(ET ) lower boundof evaluations for the task of correcting errors in exact andnumeric sparse interpolation.

Erich KaltofenNorth Carolina State UniversityMathematics [email protected]

MS37

Hybrid Methods for Exact Geometric Computa-tion

Classical methods for exact computation with geometricobjects such as algebraic curves and surfaces mainly suf-fer from the high computational cost for the consideredsymbolic operations. For instance, most of the existingmethods for computing the topology of a planar algebraiccurve C = V (f), f ∈ Z[x, y], need to compute the en-tire subresultant sequence of the defining polynomial fand its derivative fy, and the latter computation has tobe considered even in the case, where the given curve hasa very simple geometry, or where we are only interestedin the topology of the curve restricted to a certain localneighborhood. This non-adaptive behavior partially alsoexplains why such exact methods based on a heavy alge-braic machinery have nearly not been used in algorithmsfrom Computational Geometry. In this talk, we summa-rize some recent results on the development of symbolic-numerical algorithms to compute arrangements of planaralgebraic curves. The novel algorithms need a consider-ably less amount of purely symbolic operations, whereasmost computations are based on approximate but certifiedarithmetic. In addition, the remaining symbolic computa-tions are outsourced to a highly efficient implementation ongraphics card. Exhaustive benchmarks show that the cor-responding implementations do not only outperform exist-ing methods by magnitudes, but they are even comparableto purely numerical approaches with respect to runningtime. Furthermore, a theoretical analysis gives evidencethat the novel algorithms achieve record bounds for thebit complexity of the considered problem which seems tobe almost optimal. Finally, we outline how to combine thelatter mentioned methods with a purely numerical subdivi-sion approach. We expect that such a hybrid method willeventually yield excellent local behavior as well.

Michael SagraloffMPII, [email protected]

MS37

New Algorithms for Computing Roadmaps inSmooth Bounded Real Algebraic Sets

We consider the problem of constructing roadmaps of realalgebraic sets. This problem was introduced by Canny toanswer connectivity questions and solve motion planningproblems. Canny’s algorithm is recursive, but decreasesthe dimension only by one through each recursive call.In this talk, we present a divide-and-conquer version ofCanny’s algorithm, where the dimension is halved througheach recursive call.

Mohab Safey El DinUPMC, Univ Paris 6

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[email protected]

Eric SchostComputer Science DepartmentUniv. of Western [email protected]

MS38

Nontrivial Bounds for Steady State Concentrations

We develop tools from computational algebraic geometryfor the study of biochemical reaction networks with mass-action kinetics. If there exists a positive conservation re-lation involving a given species, one gets a trivial upperbound for its concentration all along the trajectory. Undergeneral conditions, we get better nontrivial upper boundsat steady state, regardless of explicit analytic expressionsor the occurrence of multistationarity.

Alicia Dickenstein, Mercedes Perez MillanUniversidad de Buenos [email protected], [email protected]

MS38

Zero-Eigenvalue Turing Instability in GeneralChemical Reaction Networks

We describe a necessary condition for zero-eigenvalue Tur-ing instability, i.e., Turing instability arising from a realeigenvalue changing sign from negative to positive, for gen-eral chemical reaction networks modeled with mass-actionkinetics. The reaction mechanisms are represented by thespecies-reaction graph (SR graph). If the SR graph satis-fies certain conditions, similar to the conditions for rulingout multiple equilibria in spatially homogeneous differen-tial equations systems, then the corresponding mass-actionreaction-diffusion system cannot exhibit zero-eigenvalueTuring instability for any parameter values.

Maya MinchevaNorthern Illinois [email protected]

Gheorghe CraciunDepartment of Mathematics, University [email protected]

MS38

Ruling out Hopf Bifurcations in Systems of Inter-acting Elements

We describe a new graphical approach to the questionof whether dynamical systems modeling systems of inter-acting elements admit Hopf bifurcations. The techniquesmake use of the spectral properties of additive compoundmatrices: in particular, we show that a condition on thecycles of a labelled digraph (called the DSR[2] graph) rulesout the possibility of nonreal eigenvalues of the Jacobianmatrix passing through the imaginary axis. The connec-tion with the DSR graph and relevant biological exampleswill also be discussed.

Casian PanteaImperial College [email protected]

Murad Banaji

University of [email protected]

David AngeliImperial [email protected]

MS38

Enzymatic Networks and Toric Steady States

A chemical reaction system has toric steady states if itssteady state ideal is binomial. In these systems, somequestions can be answered easily (for example, the exis-tence of positive steady states and multistationarity). Itwas shown in Perez Millan et al. (2012) that the networkfor multisite phosphorylation in a sequential and distribu-tive mechanism has toric steady states. We show here abroader family of enzymatic networks with this property.

Mercedes Perez Millan, Alicia DickensteinUniversidad de Buenos [email protected], [email protected]

MS39

A-Hypergeometric Systems and Estimation Prob-lems in Statistics

Nakayama et al. (2011) proposed the holonomic gradientdescent (HGD) which is a new method in statistics to makea maximal likelihood estimate for a given unnormalizedprobability distribution. We give new algorithms of trans-forming a given A-hypergeometric system into a Pfaffiansystem which the HGD requires as an input. We expectthat our algorithms yield a new class of an exponentialfamily of probability distributions for which we can applythe HGD efficiently.

Takayuki HibiOsaka University / JST [email protected]

Kenta NishiyamaShizuoka Prefecture [email protected]

Nobuki TakayamaKobe [email protected]

MS39

Holonomic Gradient Descent in Directional Statis-tics

Abstract not available at time of publication.

Tomonari SeiKeio [email protected]

MS39

Holonomic Gradient Method for Multivariate Nor-mal Distribution Theory

Abstract not available at time of publication.

Akimichi TakemuraTokyo University

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[email protected]

MS39

A-Hypergeometric Systems

Abstract not available at time of publication.

Uli WaltherDepartment of MathematicsPurdue [email protected]

MS40

Solving Polynomial Systems in Noether Positionwith Puiseux Series

A system of polynomials is said to be in Noether positionwhen we may set the first d coordinates to zero and obtaina smaller system, which defines a zero dimensional solutionset. Such smaller systems are called initial form systems.We obtain the initial form systems by identifying specialexponent vectors, called pretropisms. Pretropisms are theanalogues to the points in the tropical prevariety of the sys-tem. Each d dimensional cone of pretropisms we find, leadsto an initial form system and identifies which coordinatesmay be set to zero. The d dimensional cone of pretropismsmay become the leading exponents of the Puiseux series ofa d dimensional solution set and the regular solutions of theinitial form system the leading coefficients, provided thatwe can find a second term in the Puiseux series. In thistalk we show how to obtain the second terms in the series.We first show how to identify the exponents of the secondterms and then we show how the coefficients of the secondterms can be obtained by solving smaller linear systems.Once the second terms are obtained, more terms can becomputed by iterating the same method that was used forthe second terms.

Danko Adrovic, Jan VerscheldeDepartment of Mathematics, Statistics and ComputerScienceUniversity of Illinois at [email protected], [email protected]

MS40

Application of Numerical Algebraic Geometry toGeometric Data Analysis

It is desirable to represent large data sets with smaller ones.One naive approach is to take the arithmetic mean of thedata set, though this is not always sufficient. In this talk,I will define a subspace mean that is unitarily invariant.Computing this mean comes with a price; it is the solutionto a challenging optimization problem. I will reformulatethis as a polynomial system and discuss solution strategiesvia numerical algebraic geometry.

Brent R. DavisColorado State [email protected]

Daniel J. BatesColorado State UniversityDepartment of [email protected]

Chris Peterson, Michael Kirby, Justin MarksColorado State University

[email protected], [email protected],[email protected]

MS40

Bifurcation Analysis and Continuation Methods inCelestial Mechanics

The computation at the heart of many mathematical mod-els of physical systems is the solution of a parameter-dependent system of nonlinear equations. Many such prob-lems give rise to families of solutions. The computation ofsuch solution families and their singularities, e.g., folds,bifurcation points, etc., is the domain of numerical param-eter continuation algorithms. In this talk we will applysuch methods to periodic trajectories in the Circular Re-stricted Three Body Problem (CR3BP) from celestial me-chanics. In particular, we will focus on computing familiesof periodic orbits arising from Hopf bifurcations, commonlyreferred to as Libration points. We will show how to com-pute families of periodic solutions of conservative systemswith two-point boundary value problem continuation soft-ware, including detection of bifurcations and correspondingbranch switching.

Randy [email protected]

MS40

Polynomial Systems and Algebraic Number Fields

Let F(z) be a polynomial system with coefficients in an al-gebraic number field A. In this talk we discuss work withChris Peterson (Colorado State Univ.) and Tim McCoy(Notre Dame Univ.) to numerically compute the irre-ducible decomposition of the solution set V(F) over A.

Andrew SommeseDept. of Applied & Computational Mathematics &StatisticsUniversity of Notre [email protected]

MS41

Arithmetic of Jacobians over Function Fields

By work of Shioda and Katsura, a Fermat variety admits adominant rational map from a product of lower dimensionalFermat varieties. A generalization of their work allows oneto explicitly construct surfaces, dominated by products ofcurves, with this property retained under extension of thebase field. We describe this generalization and applicationsto the arithmetic of Jacobians over function fields.

Lisa BergerSUNY Stony [email protected]

MS41

The Local-global Principle for Divisibility in theCohomology of Elliptic Curves

The Grunwald-Wang theorem implies that a rational num-ber is an n-th power if and only if it is an n-th power every-where locally. When the multiplicative group is replacedby a general commutative algebraic group this local-globalprinciple for divisibility by n can fail. I will discuss recentwork in which we show that for elliptic curves the local-

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global principle for divisibility by n can fail if and only ifn is divisible by 4 or 9. Time permitting I will also discussanalogous results for divisibility in the Weil-Chatelet groupand other higher cohomology groups of elliptic curves.

Brendan CreutzUniversity of SydneySchool of Mathematics and [email protected]

MS41

Effective One-Dimensional Infrastructure in Func-tion Fields of Arbitrary Degree

The distinguished principal ideals in the maximal order ofa global field with two infinite places form the infrastruc-ture of the field. This is “almost” a group under idealcomposition, only barely failing associativity. The infras-tructure can be used to significantly speed up computationof the regulator, the ideal class number and (in the case offunction fields) the degree zero divisor class number of thefield. This is joint work with Adrian Tang.

Renate ScheidlerUniversity of [email protected]

MS41

Canonical Heights in Arithmetic Geometry andArithmetic Dynamics

Let f : X → X be a dominant rational self-map of asmooth projective variety defined over Q. The dynam-ical degree δf of f measures the dynamical complexityof the iterates fn of f . The arithmetic degree αf (x) =

lim hX

(fn(x)

)1/nof a point x ∈ X(Q) similarly mea-

sures the arithmetic complexity of the f -orbit O{(§) of x.I will discuss the inequality αf (x) ≤ δf , the conjecturethat αf (x) is an algebraic integer, with a proof if f isa morphism, and the conjecture that if O{(§) is Zariskidense, then αf (x) = δf , with a proof when X is an abelianvariety. (Joint work with Shu Kawaguchi)

Joseph H. SilvermanBrown [email protected]

MS41

Computing L-Series of Low Genus Curves

David Harvey recently proposed a new algorithm that com-putes the zeta function of a hyperelliptic curve of genus gat all primes p ≤ N , with an average running time (perprime) is polynomial in both g and log p. I will describejoint work with Harvey on the practical implementation ofthis algorithm for g ≤ 3, and also discuss extensions of thealgorithm to handle some non-hyperelliptic cases.

Andrew V. [email protected]

David HarveyUniversity of New South [email protected]

MS42

On Polar Coding with Algebraic Geometric Ker-nels

Arikan developed polar coding as a breakthrough methodfor an explicit construction of symmetric capacity achiev-ing codes for binary DMCs with low encoding and decodingcomplexity. Recently, Reed-Solomon and BCH codes havebeen considered as kernels of polar codes. We implementmore general constructions with algebraic geometry codesas kernels, specifically codes from maximal and optimalfunction fields. In addition, we demonstrate that multi-point code kernels arise naturally from shortening thoseassociated with one-point codes.

Sarah AndersonClemson [email protected]

MS42

Smooth Models for the Deligne-Lusztig Curves

The curves of Deligne-Lusztig type come in three fami-lies: The Hermitian curves, the Suzuki curves, and the Reecurves. They are defined over a finite field and have manyrational points and a large automorphism group, makingthem attractive for various applications. The curves havenatural abstract definitions and they have known singularmodels that yield descriptions of their function field. Someapplications and further questions about the curves requirea nonsingular model. We provide natural explicit smoothembeddings for the Suzuki curves in P 4 and for the Reecurves in P 13. We use these to describe new properties ofthe curves.

Iwan DuursmaUniversity of Illinois at Urbana-ChampaignDepartment of [email protected]

Abdulla EidUniversity of [email protected]

MS42

On q-Ary Polar Coding

We discuss polar coding for q-ary channels, where q is apower of a prime. In polar coding, independent copies of adiscrete memoryless channel are synthesized into a secondset of channels so that asymptotically (in block length) allchannels, except for a vanishing fraction, are either noise-less or almost pure noise. In this talk, we address the useof algebraic geometry codes in this setting.

Gretchen L. MatthewsClemson [email protected]

MS42

List Decoding of Subspace Codes

Subspace codes are defined to be subsets of the projectivegeometry over a finite field. If all elements of such a codehave the same dimension, such a code is called a constantdimension code. These codes are useful for different appli-cations, e.g. random linear network coding. List decodingis the concept of finding not only the closest codeword to

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a received word but the list of codewords inside a ball ofa given radius around the received word. In this talk weshow how a list decoding algorithm of constant dimensioncodes can be derived in the well-known Plcker embeddingof the Grassmannian.

Anna-Lena TrautmannUniversity of [email protected]

Joachim RosenthalInstitut fur MathematikUniversitat [email protected]

MS42

The Minimum Distance and the Index of Nilpo-tency

Let C be a [n, k, d]−linear code. Let A be a k × n gener-ating matrix for C. Assuming that A has no proportionalcolumns, nor zero columns, we think of the columns ofA as the homogeneous coordinates of n distinct points inP k−1. Under the extra condition that any k − 1 of thesepoints span a hyperplane, we find a formula that relatesthe minimum distance d to the index of nilpotency of theideal of a fat-points scheme with support the coatoms ofthe arrangement of dual hyperplanes to these points.

Stefan TohaneanuUniversity of Western [email protected]

MS43

Object Image Correspondence for AlgebraicCurves under Projections.

We present a novel algorithm for deciding whether a givenplanar curve is an image of a given spatial curve, obtainedby a central or a parallel projection. The motivation comesfrom the problem of establishing a correspondence betweenan object and an image, taken by a camera with unknownposition and parameters. The computational advantage ofour algorithm, in comparison to algorithms based on thestraightforward approach, lies in a significant reduction ofa number of real parameters that need to be eliminated tosolve this problem.

Joseph Burdis, Irina Kogan, Hoon HongNorth Carolina State [email protected], [email protected], [email protected]

MS43

Simplified Morse Skeletons from Digital Images

The closely related techniques of Morse theory and persis-tent homology provide a rigorous framework for general-izing and unifying various tools from image analysis thataddress the problems of segmentation and shape analy-sis (e.g. skeletonization, watersheds, and ad hoc simplifica-tion). We present the latest foundations and efficient im-plementations of discrete morse theory based algorithmsfor building simplified skeletons from real x-ray micro-CTdata of rock cores and other samples with intricate three-dimensional structure.

Vanessa Robins, Olaf Delgado-Friedrichs, AdrianSheppard

Australian National [email protected],[email protected],[email protected]

MS43

Object/Image Equations for Object Recognition,Shape Analysis, and Statistics.

We will discuss two fundamental problems related to objectrecognition. Our focus will be on point features under weakperspective projection, since this is a case with a completesolution. The first problem involves the geometric con-straints (object-image equations) that must hold betweena set of object feature points (object configuration) and anyimage of those points under a weak perspective projection.These constraints are formulated in an invariant way, sothat object pose, image orientation, or the choice of coor-dinates used to express the feature point locations eitheron the object or in the image are irrelevant. These con-straints turn out to be expressions in the so-called shapecoordinates on the moduli space of configurations. Thesecond problem concerns the notion of a shape space andshape statistics. These spaces have certain natural metricsthat allow for statistical analysis when faced with noisyfeature data.

Peter F. StillerTexas A&M [email protected]

MS43

Using Gaussian-Weighted Graph Laplacian in Ge-ometric Shape Processing

The Laplace-Beltrami operator of a given manifold is afundamental object encoding the intrinsic geometry of theunderlying manifold, and has been widely used in graph-ics and machine learning applications. In practice, theshape of interest is often available only through discreteapproximations, such as a set of points sampled from it.The Gaussian-weighted graph Laplacian is a popular dis-cretization of the Laplace-Beltrami operator for such pointclouds data. In this talk, I will talk about our recent workwhere we use the Gaussian-weighted graph Lapacian fortwo shape analysis applications. The first one is to recon-struct a singular surface (a collection of possibly intersect-ing surface patches with non-smooth creases) from a setof points sampled from it. The second one is about tore-distribute points on a hidden domain to achieve a cer-tain prescribed target probabilistic distribution (such as tomake the points distribution uniform). The development ofthese applications leverages some theoretical understand-ing of the Gaussian-weighted graph Laplace operator.

Yusu WangThe Ohio State University, Columbus, [email protected]

MS44

On Waring’s Problem for Systems of Skew-Symmetric Forms

This talk explores a problem of finding the smallest pos-itive integer s(m,n, k) such that (m + 1) generic skew-symmetric (k + 1)-forms in (n + 1) variables as linearcombinations of the same s(m,n, k) decomposable skew-symmetric (k+1)-forms. In this talk, we will describe how

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objects in algebraic geometry can be associated to systemsof skew-symmetric forms and discuss algebraic and geomet-ric approaches to establish the existence of triples (m,n, k),where s(m,n, k) is more than expected.

Hirotachi Abo, Jia WanUniversity of [email protected], [email protected]

MS44

Ranks and Nuclear Norms of Tensors

Unfortunately, it is often difficult to determine the rank ofa higher order tensors. A common simplification is convexrelaxation: instead of the rank of a tensor, we may con-sider its nuclear norm. For many tensors, for which we donot know the rank, we can determine the nuclear norm.Examples are: the matrix multiplication tensor, the deter-minant, permanent, and the multiplication tensor in groupalgebras. We also will generalize the notion of the SingularValue Decomposition (at least for some tensors) and findthe singular values of some tensors of interest.

Harm DerksenUniversity of [email protected]

MS44

Higher Secants of Sato’s Grassmannian

Sato’s Grassmannian from integrable systems is an inverselimit of all finite-dimensional Grassmannians. I will dis-cuss progress on a project to determine whether its highersecant varieties are defined in bounded degree.

Jan DraismaEindhoven University of Technology, The NetherlandsDepartment of Mathematics and Computer [email protected]

MS44

Equations of Secant Varieties

It is a classical result of Mumford that in a sufficientlyample embedding, any variety is defined by determinantalequations. There is a lot of interest in the correspondingquestions for secant varieties. Namely, if X ⊂ Pn and ifwe are given an integer r0, can we find another integer d0such for all d ≥ d0 and for all r ≤ r0 we have that σr(Xd)is determinantal. We use the notation Xd to denote νd(X),the image of X under the d-th Veronese re-embedding. Wewill talk about results of Buczynski-Ginsensky-Landsbergand Buszynska-Buczynski which reduce the problem to thecase of X = Pn but show that the problem for X = Pn ismore complicated than perhaps was thought.

Adam GinenskyUniversity of [email protected]

MS44

Writing Forms as Sums of Higher Powers of LowerDegree Forms

Complex forms in n variables of degree kd can always bewritten as a sum of k-th powers of forms of degree d. Wediscuss various questions related to minimal representa-tions of this kind, especially when n = 2 and when k ≥ 3

and d ≥ 2.

Bruce ReznickUniversity of Illinois, [email protected]

MS45

Combinatorial Degree Bound for Toric Ideals ofHypergraphs

Associated to any hypergraph is a toric ideal encoding thealgebraic relations among its edges. We study these ide-als and the combinatorics of their minimal generators, andderive general degree bounds for both uniform and non-uniform hypergraphs. As an application, we show thatthe defining ideal of the tangential variety is generated byquadratics and cubics in cumulant coordinates.

Elizabeth GrossUniversity of Illinois at [email protected]

Sonja PetrovicDepartment of StatisticsPennsylvania State [email protected]

MS45

Numerical Computations and Galois Groups inSchubert Calculus

Schubert calculus is an important class of geometric prob-lems involving linear spaces meeting other fixed but gen-eral linear spaces. Problems in Schubert calculus can bemodeled by systems of polynomial equations. Thus, wecan use numerical methods to find the solutions to thesegeometrical problems. We present a Macaulay2 implemen-tation of numerical algorithms that solve Schubert prob-lems. These algorithms are based on the geometric Pieriand Littlewood-Richardson homotopies. We use our im-plementation to study Galois groups of Schubert problems.This work is partially joint with Anton Leykin and FrankSottile.

Abraham Martin del CampoIST [email protected]

MS45

Comparison of Symbolic and Ordinary Powers ofIdeals

Given an ideal I in a ring R, we can define the t-th symbolicpowers I(t) by evaluating elements of R to the order t ateach associated prime of I . Comparison symbolic and or-dinary powers has attracted considerable attention in thepast decades. We will show some uniform bounds, com-putable examples and conjectures in this area.

Wenbo NiuPurdue [email protected]

MS45

Hyperdeterminants of Polynomials

Hyperdeterminants were brought into a modern light byGel’fand, Kapranov, and Zelevinsky. Inspired by their

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84 AG13 Abstracts

work, we study the hyperdeterminant applied to a poly-nomial (interpreted as a symmetric tensor). The hyperde-terminant of a polynomial factors into several irreduciblefactors with multiplicities. These factors along with theirdegrees and their multiplicities are identified geometrically(via projective duals of Chow and Segre-Veronese varieties)and have a nice combinatorial interpretation.

Luke OedingUniversity of California BerkeleyUniversita degli Studi di [email protected]

MS45

Software for Computing Multiplier Ideals

Multiplier ideals are interesting for a range of applicationsin birational geometry, commutative algebra, and algebraicstatistics. Unfortunately they are usually hard to compute.Shibuta’s algorithm, implemented in a Macaulay2 pack-age by Christine Berkesch and Anton Leykin, can computemultiplier ideals of modestly sized examples. I describea Macaulay2 package being developed that uses combina-torial algorithms for particular classes of ideals includingmonomial ideals, monomial curves, and determinantal ide-als, allowing computations of somewhat larger examples.

Zach TeitlerBoise State [email protected]

MS46

Toward a Rigorous Variation of Coppersmith’s Al-gorithm on Three Variables

Coppersmith’s technique for finding small roots on a poly-nomial p1(x, y) works by constructing p2(x, y) independentfrom p1, sharing the same root and then solving the sys-tem {p1 = 0, p2 = 0}. For p1(x, y, z), the generalization,which finds p2 and p3 independent from p1, is heuristicsince the system {p1 = 0, p2 = 0, p3 = 0} can sometimesnot be solved. In this talk, we propose a rigorous variationof Coppersmith’s algorithm providing algebraically inde-pendent polynomials that makes use of Grobner bases andwhich can be generalized for multivariate polynomials.

Aurelie BauerANSSI, [email protected]

MS46

Polynomial Analogues of Coppersmith’s Method,with Applications to List-decoding of Error-correcting Codes

I will show how the analogue of Coppersmith’s methodover the ring of polynomials gives a “lattice-based’ algo-rithm for list-decoding several families of important error-correcting codes, including Reed-Solomon codes in the uni-variate case and Parvaresh-Vardy codes in the multivariatecase. This approach yields algorithms that are extremelyfast both in theory and in practice. Interestingly, in thecoding-theory application it is possible to construct codesso that the heuristic assumptions commonly used in mul-tivariate Coppersmith-type methods over the integers arenot necessary. I will discuss joint work with Henry Cohnand with Casey Devet and Ian Goldberg.

Nadia Heninger

Princeton [email protected]

MS46

The Heuristic Coppersmith Technique from a Com-puter Algebra Point of View

In this talk we introduce objects and algorithms comingfrom computational commutative algebra that let us de-scribe precisely why the Coppersmith’s technique becomesheuristic in more than two variables. We then present how,in some cases, one can circumvent this heuristic.

Guenael RenaultUniversite Pierre et Marie Curie, [email protected]

MS46

An Introduction to Coppersmith’s Theorem and itsApplications

In this talk, we give an overview of how Don Coppersmithuses lattice reduction techniques to compute small roots ofunivariate polynomials in Z/nZ efficiently even when thefactorization of n is unknown, and mention some applica-tions to number theoretic problems and cryptanalysis. Asan introduction to the other talks of the minisymposium,we also briefly discuss the difficulties that arise when tryingto directly apply similar techniques in higher dimensions.

Mehdi TibouchiNTT, [email protected]

MS47

Implicitization by Interpolation: Symbolic and Nu-merical Methods

We apply symbolic or numerical methods in order to com-pute a matrix kernel and determine the coefficients of theimplicit equation knowing its support. We propose solu-tions for the multidimensional kernel case, that occurs ifthe predicted support is a superset of the actual. We com-pare different approaches for constructing a real or com-plex matrix in Maple and SAGE. In our experiments wehave used classical algebraic curves and surfaces as well asNURBS.

Tatjana KalinkaNational and Kapodistrian University of [email protected]

MS47

Coping with Singular Isolated Zeros of PolynomialSystems Using Symbolic perturbations

We present techniques for extracting information from anisolated singular zero of a polynomial ideal. We work onthe dual polynomial ring and derive a description of themultiplicity structure in terms of local differential condi-tions. The algorithm is based on matrix-kernel computa-tions, which can be carried out numerically, starting froman approximation of the zero in question. Based on thisalgorithm, we derive a deflation technique, by augmentingthe original system using a number of new parameters, sothat the multiplicity is obviated: the deflated system inter-polates the initial root with multiplicity one; consequently,it can be used in Newton-based iterative schemes in order

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to refine the approximate root with quadratic convergence.

Angelos MantzaflarisRICAMAustrian Academy of [email protected]

Bernard MourrainINRIA Sophia [email protected]

MS47

Vanishing Ideals of Limited Precision Points

In this talk we address the problem of characterizing aset X of limited precision points of Rn from a symbolic-numerical point of view. In contrast to other approaches ofthe recent literature we define the numerical vanishing idealassociated to X as a zero-dimensional complete intersectiongenerated by polynomials of low degree and whose exactzero-set contains a small perturbation of the input points.

Maria-Laura TorrenteUniversity of [email protected]

MS47

Interpolation and Ehrhart Theory

We explore the interplay between interpolation andEhrhart theory. The computation of Ehrhart polynomi-als is a difficult problem with many applications. Theuse of interpolation techniques for the computation ofEhrhart polynomials seems natural and potentially impor-tant in practice. Changing the point of view, we investigatehow certain interpolation problems can be interpreted inEhrhart theory. Although, in general, Ehrhart polynomi-als cannot be computed efficiently, such a connection wouldprovide insight and possible algorithmic advantages.

Zafeirakis ZafeirakopoulosJohannes Kepler [email protected]

MS48

Sheaves, Cosheaves and Applications

This talk outlines a unified approach to persistence, net-work coding, and sensor networks. In all of these appli-cations, data is parameterized over a cell complex. Theextraction and analysis of qualitative features of data ismediated through the use of constructible (co)sheaf the-ory. A generalized version of a barcode is introduced toaid in visualization and interpretation. Finally, a new ho-mology theory that is sensitive to the embedding of datais presented.

Justin CurryUniversity of [email protected]

MS48

Generalized Interleavings and Weak Laws of LargeNumbers for 2-D Persistent Homology

Implicit in the results of the 2009 paper “Persistence-Based

Clustering in Riemannian Manifolds” by Chazal, Guibas,Oudot, and Skraba is a (loose) analogue of the weak lawof large numbers for 1-D persistent homology. I’ll intro-duce (J1, J2)-interleavings, which are generalized interleav-ings defined on multidimensional filtrations and persistencemodules. I’ll then explain how (J1, J2)-interleavings can beused to formulate and prove 2-D adaptations of the “weaklaw of large numbers for persistent homology.”

Michael LesnickStanford [email protected]

MS48

A Continuous Mean for Sets of Persistence Dia-grams

Recent progress has shown that the abstract space of per-sistence diagrams is Polish, and found necessary and suf-ficient conditions for a set to be compact. To understandthe average of a set of diagrams, Frechet means were used,a construction which is possible for any metric space. TheFrechet mean is a set, not an element. Mileyko et al showedthat for well behaved distributions, the Frechet mean isnon-empty. However, there are simple counterexamplesshowing that the mean is non-unique, which creates is-sues when we are interested in continuous, time-varyingsystems. In order to solve this problem, we present a gen-eralization of the Frechet mean which is continuous for setsof continuously varying persistence diagrams.

Elizabeth MunchDuke [email protected]

MS48

Measuring the Stability of Intersections to C0 Per-turbations

Consider two smooth submanifolds M and N of X. Thetwo intersect transversally if, at each point x in the inter-section M∩N , the tangent space of M at x plus the tangentspace of N at x is the tangent space of X at x. Transverseintersections are interesting because they are stable to in-finitesimally small or, in other words, C∞ perturbations.Motivated by the philosophy of science, we would like toquantify the stability of the intersection. In this talk, I willpresent the relatively new idea of the well group. The wellgroup measures the stability of the homology of M ∩ Nwith respect to C0 perturbations of the embedding of Minto X. I will show that the well group is the abstractionof the persistent homology group.

Amit PatelRutgers [email protected]

MS48

Kernel Distance for Geometric Inference

We study the geometric inference from a noisy point cloudusing the kernel distance. Recently Chazal, Cohen-Steiner,and Mergot have introduced distance to a measure, whichis a distance-like function that is robust to perturbationsand noise on the data. Their work and its extensions rep-resents the state-of-the-art in geometric inference on noisypoint cloud data. The point of this talk is to show that thekernel distance can be used in place of the distance to ameasure. We prove that the kernel distance possesses very

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similar properties to the distance to a measure of Chazalet.al., but also has several advantages. (i) Its inference hasa very naturally interpretation as the superlevel set of a ker-nel density estimate. (ii) The kernel distance is Lipschitzwith respect to the outlier parameter s. Similar propertiesare not know for the distance to a measure. (iii) The kerneldistance has a small coreset representation, which permitsefficient inferences on tens of millions of points.

Bei Wang, Jeff PhillipsUniversity of [email protected], [email protected]

MS49

Discriminants and Applications

Discriminant is a key tool when examining well-constrainedsystems, including the case of one univariate polynomial.We present some useful properties and applications in orderto motivate their study. We focus on exploiting the sparse-ness of polynomials via the theory of Newton polytopesand sparse elimination. We also discuss a multiplicativityformula of the discriminant in the case of two polynomialswith fixed supports when one of them factors.

Alicia DickensteinUniversidad de Buenos [email protected]

Ioannis Z. EmirisNational and Kapodistrian University of [email protected]

Anna KarasoulouUniversity of [email protected]

MS49

Khovanskii-Rolle Continuation for Real Solutions

Over 30 years ago, Askold Khovanskii used a multivari-ate generalization of Rolle’s Theorem to give a bound onthe number of positive solutions to a system of polynomialequations that depends only on the number of terms ap-pearing in the equations, and not on their degree. Reflect-ing this dependence, this class of bounds are called fewno-mial bounds. Khovanskii’s proof was revisited by Bihanand Sottile, who gave a refined bound that is asymptoti-cally sharp, in a certain sense. Subsequently, we observedthat Khovanskii’s use of Rolle’s Theorem can form the basisof a continuation algorithm for numerically computing thereal solutions to a system of fewnomials. This Khovanskii-Rolle algorithm finds the real solutions by following realarcs, and avoids computing complex solutions to the fewno-mial system. Consequently its complexity depends uponthe fewnomial bound and not on the number of complexsolutions. This talk will explain this Khovanskii-Rolle al-gorithm and describe some technical challenges that areencountered in its implementation. This represents jointwork with Dan Bates.

Frank SottileTexas A&M [email protected]

Daniel J. BatesColorado State UniversityDepartment of [email protected]

MS49

Computing Critical Points with Grobner Bases:Complexity and Applications

Computing critical points of a polynomial map g restrictedto an algebraic variety V is a topical issue in several algo-rithms in real algebraic geometry and in global minimiza-tion problems. In this talk, we present new complexitybounds for computing such critical points with symbolicalgorithms. Let f1, . . . , fp be polynomials in Q[x1, . . . ,xn]defining a variety V and I be the ideal vanishing on thecritical points of g restricted to V . The ideal I is generatedby f1, . . . , fp and by the maximal minors of the Jacobianmatrix of (f1, . . . , fp, g). Under genericity assumptions onthe coefficients of f1, . . . , fp, g, we show new upper boundson the arithmetic complexity of computing a Grobner ba-sis of I , which reflects the behavior observed in practice.These bounds are obtained by taking into account struc-tural properties of determinantal ideals generated by max-imal minors. In particular, these bounds are polynomial inthe number of critical points for several families of param-eters.

Pierre-Jean SpaenlehauerUPMC, Univ. Paris [email protected]

Jean-Charles FaugereSalsa project, INRIA [email protected]

Mohab Safey El DinUPMC, Univ Paris [email protected]

MS49

An Algorithm to Compute the Dimension of RealAlgebraic Sets

We present a new probabilistic algorithm for computing thereal dimension of a real solution set S ⊂n of s polynomialsof degree bounded by D in n variables.

Mohab Safey El DinUPMC, Univ Paris [email protected]

Elias TsigaridasInria [email protected]

MS50

Introduction and Overview of the Topic Area ’For-mulas in Interpolation’

This workshop will explore the use of interpolation in bothSymbolic and Numerical Algebraic Geometry. From closedformulas for both Cauchy and osculatory rational interpo-lation to more sophisticated applications including Birkhoffinterpolation and its multivariate versions. We will presentsome of recent results and challenges.

Carlos D’AndreaUniversitat de [email protected]

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MS50

Polynomial Algebra for Birkhoff Interpolants

We introduce a unifying formulation of a number of re-lated problems which can all be solved using a contourintegral formula. Each of these problems requires find-ing a non-trivial linear combination of some of the val-ues of a function f , and its first and higher derivatives,at a number of data points. This linear combination isre- quired to have zero value when f is a polynomial ofup to a specific degree p. Examples of this type of prob-lem include Lagrange, Hermite and Hermite-Birkhoff in-terpolation; fixed-denominator rational interpolation; andvarious numerical quadrature and differentiation formulae.Other applications include the estimation of missing dataand root-finding.

John C. ButcherDepartment of MathematicsThe University of [email protected]

Rob M. CorlessDepartment of Applied MathematicsUniversity of Western [email protected]

Laureano Gonzalez-VegaUniversidad de CantabriaDpto. [email protected]

Azar ShakooriUniversidad de Cantabria, [email protected]

MS50

An Algorithm for Evaluation and Interpolation inHigher Dimensions

We propose efficient new algorithms for multi-dimensionalmulti-point evaluation and interpolation on certain subsetsof tensor product grids. These point-sets naturally occur inthe design of efficient multiplication algorithms for finite-dimensional algebras of the form K[x1, . . . , xn]/I , where Iis generated by monomials.

Joris van der HoevenLIX, Ecole [email protected]

Eric SchostComputer Science DepartmentUniv. of Western [email protected]

MS50

Sylvester Double Sums and Divided Differences

Abstract not available at time of publication.

Aviva SzpirglasUniversite de [email protected]

MS51

A Bayesian Information Criterion for Singular

Models

For singular models, such as reduced rank regression ormixture models, the Bayesian Information Criterion (BIC)may fail to reflect the large-sample behavior of the marginallikelihood. This talk will suggest a practical extension ofBIC that uses recently developed large-sample theory forsingular marginal likelihood integrals to resolve this prob-lem.

Mathias DrtonDepartment of StatisticsUniversity of [email protected]

Martyn PlummerInternational Agency for Research on [email protected]

MS51

Singular Learning Theory and Causal Inference

Many algorithms for inferring causality are based on partialcorrelation testing. Partial correlations define hypersur-faces in the parameter space of a directed Gaussian graph-ical model. The volumes obtained by bounding partialcorrelations play an important role for the performance ofcausal inference algorithms and the quantification of biasin causal inference. We develop an asymptotic theory forcomputing these volumes. Our analysis involves comput-ing the singular loci of the partial correlation hypersurfacesand using the method of real log canonical thresholds.

Shaowei LinA*[email protected]

Caroline UhlerInstitute of Science and Technology [email protected]

Bernd SturmfelsUniversity of California , [email protected]

Peter BuehlmannETH [email protected]

MS51

Resolution of Singularities and Statistical ModelEvaluation

It has been difficult to evaluate statistical models whichdo not satisfy regularity condition, since asymptotic the-ory of likelihood function near singularities has been leftunknown. In this presentation, we show that resolutionof singularities provides us the standard representation ofthe parameter space, by which we can make the universalstatistical estimation theorem. As a result, statistical con-cepts, AIC and BIC in regular models, are generalized ontosingular models.

Sumio WatanabeDepartment of Computational Intelligence and SystemsScienceTokyo Institute of Technology, [email protected]

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88 AG13 Abstracts

MS51

Asymptotic Inference for Gaussian Hidden TreeModels

Hidden tree models are Bayesian networks on rooted treeswith all inner nodes representing data which are unob-served. Models of this type can be for example found inphylogenetics and they are complicated both from alge-braic and statistical point of view. The likelihood functionis multimodal, estimates of the parameters lie often on theboundary of the parameter space. The fact that the modelis not identifiable rises many additional issues. In this talkwe show how singular learning theory can be used to get abetter insight into asymptotic analysis of these models inthe case when studied system is multivariate Gaussian.

Piotr ZwiernikMittag-Leffler InstituteStockholm, [email protected]

MS52

Numerical Methods for Highly Structured Polyno-mial Systems Coming from Magnetism

Abstract not available at time of publication.

Daniel J. BatesColorado State UniversityDepartment of [email protected]

MS52

Applying Fiber Products to Polynomial Maps andthe Planar Pentad

Fiber product techniques in numerical algebraic geometrycan be used to determine the parameters over which a poly-nomial system has solution set with dimension greater thanthe expected dimension. For parameterized polynomialsystems, the dimension of the (complex) solution set is thesame for almost all choices of parameters. Fiber productscan identify exceptional sets of parameter values where thedimension of the solution set is greater than expected. Anapplication of these techniques arises in Kinematics, wereparameters in polynomial systems correspond to physicalparameters of mechanisms. An exceptional set of param-eters corresponds to mechanism configurations that havemore mobility than is expected. Fiber product techniquesrequire finding solutions to polynomial systems of increas-ing size. We will discuss algorithms for avoiding or limitingthe growth of these systems, in particular we will considerapplying these techniques to the Planar Pentad.

Eric HansonDepartment of MathematicsColorado State [email protected]

Dan BatesColorado State [email protected]

Jon HauensteinNorth Carolina State [email protected]

Charles WamplerGeneral Motors Research Laboratories, Enterprise

Systems Lab30500 Mound Road, Warren, MI 48090-9055, [email protected]

MS52

Numerical Determination of Witness Points onReal Solution Components of Polynomial and Dif-ferential Polynomial Equations

In this talk we extend complex homotopy methods to find-ing witness points on the irreducible components of realvarieties. In particular we compute such witness pointsas the isolated real solutions of a constrained optimiza-tion problem by a homotopy mixed volume based solverover C. First a random hyperplane characterized by itsrandom normal vector is chosen. Witness points are com-puted which are at the intersection of this hyperplane withcomponents. Other witness points are the regular localcritical points of the distance from the plane to compo-nents. A method is given for constructing regular witnesspoints on components, in the case that the critical pointsare singular. The method is applicable to systems satisfy-ing certain regularity conditions. Illustrative examples aregiven. We show that the method can be used in the consis-tent initialization phase of a popular method due to Pryceand Pantelides for preprocessing such differential algebraicequations for numerical solution.

Wenyuan WuChongqing Institute of Green and Intelligent TechnologyChinese Academy of [email protected]

Greg ReidApplied Math DepartmentUniversity of Western [email protected]

MS53

Shellability and Freeness of Continuous Splines

We provide an example of a shellable polyhedral complex

P in R2 such that the module of continuous splines C0(P)is not a free module over the polynomial ring in three vari-ables, answering a question raised in [Schenck, EquivariantChow Cohomology of Non-Simplicial Toric Varieties, 2012].

Michael DiPasqualeUniversity of [email protected]

MS53

Computational Topology and Visualization

In this talk, we outline various challenges and problemsthat arise in high performance scientific visualization. Inparticular, we concern ourselves with the fidelity of theoutput. Inspired by applications, we discuss methods ofensuring topological fidelity for curves and surfaces withcurvature adaptive sampling.

Lance E. MillerUniversity of [email protected]

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MS53

Splines on Polyhedral Complexes

We discuss recent results on splines on polyhedral com-plexes, including results on freeness of the spline module,combinatorial and geometric aspects, and bounds for de-grees of generators. The polyhedral case is quite differentfrom the simplicial case, and I will illustrate with severalexamples the subtle points that arise.

Hal SchenckMathematics DepartmentUniversity of [email protected]

MS53

Syzygies and Singularities of Tensor Product Sur-faces

This project is motivated by questions in geometric model-ing, where one would like to understand the implicit equa-tion and singular locus of a parametric surface. Considerthe polynomial ring R = k[s, t, u, v] over an algebraicallyclosed field k. Regard R as a bigraded k-algebra, in whichs, t have degree (1, 0) and u, v have degree (0, 1). We con-sider a class of surfaces S in P 3 parametrized by four bi-homogeneous polynomials in R. Such surfaces are calledtensor product surfaces. We classify all possible minimalfree resolutions of the ideal given by the parametrizationfunctions and we relate the syzygies to the singularities ofthe projective surface S. These resolutions play a key rolein determining the implicit equation for S.

Alexandra SeceleanuUniversity of [email protected]

Hal SchenckMathematics DepartmentUniversity of [email protected]

Javid ValidashtiUniversity of Illinois at [email protected]

MS53

The Schenck-Stiller Conjecture on the Dimensionof Splines

Let Δ be a triangulation of an open topological disk inR2. Let r ≥ 0 be an integer. Let Cr

d(Δ) be the spaceof polynomial splines on Δ of smoothness r and degree d.If d ≥ 3r + 2 (or d ≥ 3r + 1 for generic Δ) Alfeld andSchumaker give a formula for the dimension of this vectorspace that depends on the combinatorics and the local ge-ometric data of Δ. Schenck and Stiller conjectured thatthis formula holds for d ≥ 2r + 1, for any d and r. Therehas been little progress into proving this conjecture, and Iwill talk about an example that verifies the conjecture andalso that makes it tight. We also investigate the validityof the conjecture when pseudo edges (edges that touch theboundary) are added to any triangulation known to satisfySchenck-Stiller conjecture.

Stefan TohaneanuUniversity of Western [email protected]

MS54

Quantum Algorithms for the Subset-sum Problem

This talk introduces a subset-sum algorithm with heuris-tic asymptotic cost exponent below 0.25. The new algo-rithm combines the 2010 Howgrave-Graham–Joux subset-sum algorithm with a new streamlined approach to quan-tum walks.

Daniel BernsteinUniversity of Illinois at [email protected]

MS54

McBits: fast constant-time code-based cryptogra-phy

This talk presents extremely fast algorithms for code-basedpublic-key cryptogra- phy, including full protection againsttiming attacks. For example, our software achieves a recip-rocal throughput of just 36615 cycles per decryption at a80-bit security level on a single Ivy Bridge core. These algo-rithms rely on an additive FFT for fast root computation,a transposed additive FFT for fast syndrome computation,and a sorting network to avoid cache-timing attacks.

Tung ChouTechnische Universiteit EindhovenThe [email protected]

Daniel BernsteinUniversity of Illinois at [email protected]

Peter SchwabeRadboud University NijmegenThe [email protected]

MS54

Solving the Shortest Vector Problem in LatticesFaster Using Quantum Search

By applying Grovers quantum search algorithm to the lat-tice algorithms of Micciancio and Voulgaris, Nguyen andVidick, Wang et al., and Pujol and Stehle, we obtain im-proved asymptotic quantum results for solving the shortestvector problem. With quantum computers we can provablyfind a shortest vector in time 21.799n+o(n), improving uponthe classical time complexity of 22.465n+o(n) of Pujol andStehle and the 22n+o(n) of Micciancio and Voulgaris, whileheuristically we expect to find a shortest vector in time20.312n+o(n), improving upon the classical time complexityof 20.384n+o(n) of Wang et al. These quantum complexitieswill be an important guide for the selection of parametersfor post-quantum cryptosystems based on the hardness ofthe shortest vector problem.

Thijs LaarhovenTechnische Universiteit [email protected]

Michele MoscaUniversity of [email protected]

Joop van de Pol

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90 AG13 Abstracts

University of [email protected]

MS55

The Geometry of Lightlike Surfaces in MinkowskiSpace

We investigate the geometric properties of lightlike sur-faces in the Minkowski space R2,1, using Cartan’s methodof moving frames to compute a complete set of local in-variants for such surfaces. Using these invariants, we give acomplete local classification of lightlike surfaces of constanttype in R2,1 and construct new examples of such surfaces.

Jeanne ClellandUniversity of [email protected]

Brian CarlsenUniversity of [email protected]

MS55

Moving Frames and Flows for Curves in Cen-troaffine Space

Centroaffine geometry in 3 is the study of invariant proper-ties of objects with respect to the action of SL(3) withouttranslations. Along curves free of inflection points, onecan construct a unique SL(3)-valued moving frame, an in-variant arclength, and fundamental differential invariants(the analogues of Euclidean curvature and torsion). Givena non-stretching invariant curve flow, the passage fromthe velocity components (relative to the moving frame)to the time derivatives of the curvatures gives a skew ad-joint differential operator P . We use this to construct adouble hierarchy of mutually commuting bi-Hamiltoniancurve flows, whose curvature evolution equations turn outto be a version of the Boussinesq hierarchy. A subsequenceof linear combinations of flows in this hierarchy preservecurves lying along a cone through the origin, and inducethe Kaup-Kuperschmidt hierarchy at the level of curvature.

Thomas IveyCollege of [email protected]

Annalisa M. CaliniCollege of CharlestonDepartment of [email protected]

Gloria Mari BeffaUniversity of Wisconsin, [email protected]

MS55

Cohomology of Variational Bicomplexes Invariantunder a Pseudo-Group Action

I will describe some new techniques based on the mov-ing frames method for computing the cohomology of varia-tional bicomplexes invariant under a pseudo-group action.Moving frames can be used to produce complete sets of dif-ferential invariants and invariant coframes on jet bundles

and to analyze their algebraic structure, thus providing abasic tool for studying invariant bicomplexes. Surprisingly,the moving frames method often allows one to computetheir local cohomology by purely algebraic means.

Juha PohjanpeltoOregon State [email protected]

MS55

Symbols, Tableaux, and Pseudogroups (with Sage)

A century ago, the most profound discovery in both algebraand geometry was the classification of Lie groups via rep-resentation theory. Lie pseudogroups generalize Lie groupsand lie behind all moving-frame and equivalence problems;however, there has been no real progress in their classifi-cation. In this talk, I will discuss a program-in-progressfor using refined knowledge of the characteristic variety to(I hope) unveil some structure within the category of Liepseudogroups. An important aspect of this project is thedevelopment of effective, open-source computational toolsto study examples, so I will discuss how Lie pseudogroups,symbols, tableaux, and characteristic varieties can be stud-ied with a new Sage package.

Abraham D. SmithFordham [email protected]

MS55

Recursive Moving Frames

In the standard implementation of the equivariant mov-ing frame method, differential invariants of a Lie pseudo-group action are constructed by first prolonging the actionto the infinite submanifold jet space and then normalizingthe pseudo-group parameters. Typically, this procedureyields unwieldy expressions that limit the method’s practi-cal scope. In the recursive approach the prolonged action iscomputed incrementally, and pseudo-group parameters arenormalized as they appear to control, as much as possible,the expression swell.

Francis ValiquetteDalhousie [email protected]

MS56

Rank of Tensors via Secant Varieties and Fat Points

This talk explores the problem of finding the dimensionof the s-secant variety of X, where X is a Segre varietyor a Segre-Veronese variety. The method for computingsuch a dimension is essentially based on Terracini’s Lemma.Starting from that viewpoint, a projective scheme Z thatis the union of fat points (and in some cases fat linearspaces) can be built. The scheme Z has the property thatits Hilbert function in a suitable degree gives the dimensionof the secant variety.

Maria Virginia CatalisanoUniversity of [email protected]

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MS56

The Common Lines Variety

A key step in the reconstruction of 3D molecular structuresin cryo-EM microscopy is recovering the original imagingdirections from noisy 2D cryo-EM images. In 2009 Shkol-nisky and Singer discovered an algorithm that can success-fully recover cryo-EM imaging directions even from im-ages corrupted with substantial error. In 2011, Hadaniand Singer provided a rigorous mathematical explanationfor both the algorithm’s correctness and its robustness tonoise. This algorithm centers on constructing a certainlinear operator, the common lines operator. We study thealgebraic variety of all possible common lines operators,provide defining equations, and explore applications of thiscommon lines variety to denoising of experimental cryo-EMdata.

David M. DynermanUniversity of [email protected]

MS56

The Waring Rank of the Vandermonde Determi-nant

The Waring rank for a homogeneous polynomial is thesmallest number of linear forms needed to write the polyno-mial as a sum of pure powers of linear forms. We computethe rank of the Vandermonde determinant using the lowerbound of Ranestad and Schreyer and classical facts frominvariant theory. Our approach generalizes to the defin-ing polynomial for any real reflection arrangement and thereflection multi-arrangements for some complex reflectiongroups. In particular we recover some but not all of theknown results on the Waring rank of monomials.

Alexander WooUniversity of [email protected]

Zach TeitlerBoise State [email protected]

MS57

Merging Divisorial with Colored Fans

There are two generalizations of toric varieties: First, em-beddings of spherical homogeneous spaces can be repre-sented by colored fans. Second, lower-dimensional torusactions on algebraic varieties can be described by so-calledpolyhedral divisors. They are hybrid objects between alge-braic and discrete geometry. We are going to merge bothconcepts by using the Tits fibration and a special torus ac-tion on spherical varieties. This will be based on the won-derful compactification of associated sober spherical vari-eties.

Klaus AltmannFU [email protected]

Valentina KiritchenkoHSE [email protected]

Lars PetersenDB Frankfurt

[email protected]

MS57

Computations with Mori Dream Spaces

The concept of a Mori dream space is a natural generaliza-tion of that of a toric variety. Linking suitable combinato-rial data with the Cox ring leads to a complete encoding ofMori dream spaces. This can be used for explicit compu-tations, e.g. of geometric invariants. In the talk we give asurvey on current results and present a software package.

Jurgen HausenUniversitat [email protected]

MS57

Using Polyhedral Divisors in Algebraic Geometry

Polyhedral divisors generalize the concept of toric varietiesin order to understand varieties with the action of a lowerdimensional torus. This talk will give a brief introductioninto the concept of polyhedral divisors as well as elaborateon the broad field of applications. Special emphasis will beput on the implementation in Singular and polymake.

Lars KastnerFU [email protected]

MS57

Computing Cox Rings

We present a technique to compute Cox rings using toricambient modifications. We apply this method to com-pute the Cox rings of certain (singular) del Pezzo surfaces,Gorenstein log del Pezzo surfaces of Picard number one andblow ups of projective space. The algorithms are availableas software packages.

Simon KeicherUniversitat [email protected]

MS57

New Developments in Singular and Application tothe Computation of the Git Fan

This talk provides a quick overview over newly introducedfeatures in Singular on one running example. The fea-tures include:

• a toolbox for parallelizing interpreter level code

• custom data types and overload of existing operatorson interpreter level

• an interface for integrating custom C++ code

• an interface to polymake for convex geometry

This talk reflects the work of multiple authors in and out-side the Singular team. No prior knowledge is necessaryfor understanding it.

Yue RenTU [email protected]

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92 AG13 Abstracts

MS58

Lie Markov Models with Prescribed Symmetry

Recent work has discussed the importance of multiplicativeclosure for the Markov models used in phylogenetics. A suf-ficient condition for multiplicative closure is ensured if theset of rate-matrices of the model form a Lie algebra. Thesemodels are Lie Markov models and they include group-based and equivariant models. We discuss how to gener-ate Lie Markov models by demanding certain symmetriesunder nucleotide permutations and explore the geometricembedding of the cone of stochastic rate matrices withinthe ambient space.

Jess Fernandez-SanchezUniversitat Politecnica de [email protected]

Peter D. Jarvis, Jeremy SumnerUniversity of [email protected], [email protected]

MS58

Tensor Rank and Toric Structure

Markov process depends on a number of unknown param-eters. By varying the parameters we obtain different prob-ability distributions. This gives us a parametrisation ofan algebraic variety. I will present relations between toricgeometry and phylogenetic models. In particular, I willfocus on the general Markov model, represented by the se-cant variety of the Segre embedding. This relates to theproblem of determining the rank of a tensor.

Mateusz MichalekMax Planck [email protected]

MS58

Invariant Based Phylogenetic Reconstruction: Op-portunities and Obstacles

Invariant based algorithms for phylogenetic reconstructionwere widely dismissed by practicing biologists because in-variants were perceived to have limited accuracy in con-structing trees. Advances in algebraic geometry have pro-vided a deeper understanding of phylogenetic invariants.These advances have led to a theoretical framework for un-derstanding phylogenetic reconstruction. Here we describean effort to translate these theoretical advances to effectivephylogenetic reconstruction software usable by practicingbiologists.

Joseph P. RusinkoWinthrop [email protected]

MS58

Nonparametric Estimation of Phylogenetic TreeDistributions

As population-based models of gene trees such as coales-cent have been developed to more accurately model distri-butions of gene trees across genomes, meanwhile detectionof horizontal gene transfers and discordances among genetrees have become important problems in phylogenetics.Here we focus on the problem of discordance among genetrees, and the distribution on gene trees as a whole. We

view “typical’ gene trees as samples from some distributionf that generates gene trees as independent samples. Wealso suppose there may be rare outlier gene trees whichare not “typical,’ and are samples from some other dis-tribution f ′ very different from f . Given the tremendousamount of ongoing effort to develop better parametric mod-els for gene tree distributions, here we take a nonparametricapproach. One advantage of nonparametric estimation isthat modeling decisions and assumptions are avoided. Incontrast to parametric models such as coalescents, using anonparametric approach avoids issues such as model mis-specification which might potentially confound detectionof outlier trees. While most of methods to detect outliersapply statistical methods over an Euclidean (vector) space,using a Markov Chain Monte Carlo technique, we proposeto develop statistical methods to estimate the distributionof trees over the space of trees which is not an Euclideanspace.

Grady WeyenbergUniversity of [email protected]

Peter [email protected]

Christopher Schardl, Daniel Howe, Ruriko YoshidaUniversity of [email protected], [email protected],[email protected]

MS59

Recovering a Sparse Polynomial Model from Datawith Noise and Outliers

We consider black-boxes that produce values with possiblynoise and outliers. We aim to recover a sparse model froma small number of probes from this black-box. Such prob-lems include recovering a sparse polynomial, possibly in ashifted bases, or producing a sparse multiple of the inter-polating polynomial or rational fraction. We used minimi-sation techniques from linear algebra or numerical analysis(e.g. compressed sensing).

Brice B. BoyerNorth Carolina State [email protected]

Erich L Kaltofen, Matthew [email protected], [email protected]

MS59

Numerical Reconstruction of Polytopes from Di-rectional Moments

In the shape-from-moments problem, we are interested inretrieving the vertices of polytopes in any dimension. Pre-vious algorithms used formulas in the complex plane withProny or pencil methods to recover only polygons. A newformula allows us to link directional moments and pro-jected vertices. With the help of algebraic and numericalmethods, we develop an algorithm to recover the whole setof vertices.

Mathieu CollowaldInria Sophia-AntipolisFrance

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[email protected]

Annie CuytUniversity of AntwerpDept of Mathematics and Computer [email protected]

Evelyne HubertINRIA [email protected]

Wen-shin LeeAntwerp [email protected]

MS59

Sparse Models, Interpolation and Polynomials – ASummary

In my presentation, I will summarize what I have observedand learned from the talks in our minisymposium. I willalso give my own current view of the subject of sparsemodel construction and fitting and approximate interpola-tion.

Erich KaltofenNorth Carolina State UniversityMathematics [email protected]

MS59

Sparse Interpolation and Signal Processing

In signal processing the interest in sparsity has exploded.The fact that signals can be reconstructed from undersam-pled data opens up a whole new range of possibilities. Atthe same time, research in sparse models has existed fordecades. In computer algebra, sparse representations aredeveloped to improve computational performance. Recentachievements allow input and output representations in fi-nite precision. Based on these, we develop new techniquesfor signal processing. Selected applications are presented.

Wen-shin LeeUniversity of [email protected]

Annie CuytUniversity of AntwerpDept of Mathematics and Computer [email protected]

MS59

Sparse Multivariate Function Recovery From Val-ues with Noise and Outlier Errors

Error-correcting decoding is generalized to multivariatesparse rational function recovery from evaluations that canbe numerically inaccurate and where several evaluationscan have severe errors (“outliers’). The generalization ofthe Berlekamp-Welch decoder to exact Cauchy interpola-tion of univariate rational functions from values with faultsis by Kaltofen and Pernet in 2012. We give a differentunivariate solution based on structured linear algebra thatyields a stable decoder with floating point arithmetic. Ourmultivariate polynomial and rational function interpola-tion algorithm combines Zippel’s symbolic sparse polyno-mial interpolation technique [Ph.D. Thesis MIT 1979] with

the numeric algorithm by Kaltofen, Yang, and Zhi [Proc.SNC 2007], and removes outliers (“cleans up data’) throughtechniques from error correcting codes. Our multivariatealgorithm can build a sparse model from a number of eval-uations that is linear in the sparsity of the model.

Erich KaltofenNorth Carolina State UniversityMathematics [email protected]

Zhengfeng YangShanghai Key Laboratory of Trustworthy Computing, SEIEast China Normal University, Shanghai 200062, [email protected]

MS60

Statistics for Points on Curves over Finite Fields

We will look at the distribution of the zeroes of the zetafunctions of Artin-Schreier covers over a fixed finite field ofcharacteristic p as the genus grows. We will focus on twocases: the p-rank zero locus and the ordinary locus.

Alina BucurUniversity of California at San [email protected]

Chantal DavidConcordia [email protected]

Brooke FeigonThe City College of New [email protected]

Matilde LalinUniversity of [email protected]

MS60

Random Matrices and Cohen-Lenstra Distribu-tions in Function Fields

The Cohen-Lenstra-Martinet heuristics predict the fre-quency with which a fixed finite abelian group appears asa class group of certain extensions of a given number field.Recently, Malle found numerical data suggesting that theseheuristics fail when the base field contains unexpected rootsof unity. He then proposed a modified frequency, correct-ing this failure. I will explain a random matrix heuristicthat leads to a function field version of Malle’s conjecture(and generalizations of it).

Derek GartonNorthwestern [email protected]

MS60

Degenerations and Non-algebraically Closed Fields

Abstract not available at time of publication.

Brian OssermanUniversity of California, [email protected]¡mailto:[email protected]

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94 AG13 Abstracts

MS60

The Ekedahl-Oort type of Supersingular Curves

A curve X defined over a finite field is supersingular if itsJacobian is isogenous to a product of supersingular ellipticcurves. Equivalently, the Newton polygon of the L-functionhas all slopes equal to 1/2. The Ekedahl-Oort type is an-other invariant of a curve defined in positive characteristicp. It characterizes the p-torsion group scheme of the Jaco-bian of X or, equivalently, the structure of the Dieudonnemodule or the de Rham cohomology. In this talk, I dis-cuss results about the Ekedahl-Oort type of supersingularcurves.

Rachel PriesColorado State [email protected]

MS60

Counting Dihedral Function Fields

Research into the construction of certain low degree func-tion fields has surged in recent years, in part due tothe cryptographic significance of elliptic and hyperellipticcurves. However there is comparatively little data availablefor higher degree function fields, leaving open many ques-tions about the number of function fields of fixed degreeand given discriminant. We will present a Kummer the-oretic algorithm for tabulating a complete list of dihedralfunction fields of bounded genus over a fixed finite field.This method also reveals how one can (heuristically) com-pute the expected number of such fields. We will presentthis heuristic and show how it compares to the actual data.

Colin J. WeirUniversity of [email protected]

MS61

Order-Degree Bounds for Recurrence and Differ-ential Operators

Desingularization is the problem of finding a left multipleof a given Ore operator in which some factor of the leadingcoefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the(r, d)-plane such that for all points (r, d) above this curve,there exists a left multiple of order r and degree d of thegiven operator. We give a new algorithm for a desingu-larization result by Abramov and van Hoeij for the shiftcase, and show how desingularization implies order-degreecurves which are extremely accurate in examples.

Shaoshi ChenDept. of Mathematics / [email protected]

Maximilian JaroschekRISC / Johannes Kepler UniversityLinz, [email protected]

Manuel KauersRISC, JKU [email protected]

Michael SingerNorth Carolina State University

[email protected]

MS61

Valuations of Sequences: Examples in Search of aTheory

Let p be a prime and x ∈ N. The p-adic valuation of x, de-noted by ˚νp(x), is the highest power of x that divides x. Aclassical theorem of Legendre shows that the p-adic valua-tion of factorials, ˚νp(n!), can be expressed in terms of thedigits of n in base p. From here it is simple to recover basicresults on arithmetic properties of factorials and binomialcoefficients. The talk describes a variety of examples ofsequences ˚νp(xn), where {xn : n ∈ N} has some combi-natorial or number-theoretical interest. These sequences ofvaluations are represented by a binary tree structure and(conjectured) properties of these trees will be presented.The examples include Stirling numbers of the second kind,the ASM sequence counting Alternating Sign Matrices andalso partial sums of elementary functions.

VIctor MollTulane [email protected]

MS61

Symbolic Summation for Combinatorial and Phys-ical Problems

We illustrate a symbolic summation toolbox based on anenhanced difference field theory of Karr’s ΠΣ fields. Inthis setting we can apply Z’s creative telescoping to com-pute recurrence relations for definite sums whose sum-mands are given in terms of indefinite nested sums andproducts. Furthermore, given such a recurrence, one cancompute all solutions that are expressible in terms of in-definite nested sums and products (such solutions are alsocalled d’Alembertian solutions, a subclass of Liouvillian so-lutions). With this machinery we will discover (and prove)new identities related to combinatorics and number theory.In addition, we report on a cooperation with J. Blumlein(DESY, German Electron-Synchrotron) how these summa-tion tools can be utilized to simplify multi-sums, comingfrom massive 3-loop Feynman integrals, to expressions interms of indefinite nested sums and products.

Carsten SchneiderRISC / Johannes Kepler UniversityLinz, [email protected]

MS61

Open Combinatorial Problems Arising from NewSequences in the OEIS

The On-Line Encyclopedia of Integer Sequences is adatabase of over 200,000 number sequences. In the past49 years it has evolved from punched cards to a wiki. Iwill describe the present version, illustrating it with a num-ber of unsolved combinatorial problems arising from recentsubmissions.

Neil SloaneAT&T Shannon LabsRoom [email protected]

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AG13 Abstracts 95

MS62

Refined Bounds on Connected Components of SignConditions on a Variety II

Let R be a real closed field and Q1, . . . , Q� ∈ R[X1, . . . , Xk]such that for each i, 1 ≤ i ≤ , deg(Qi) = di. For 1 ≤ i ≤ ,denote by Q〉 = {Q∞, . . . ,Q〉}, Vi the real variety definedby Q〉, and ki the real dimension of Vi (by convention

V0 = Rk and k0 = k). Suppose also that for 2 ≤ i ≤ ,di ≥ (k − ki−2 + 1)di−1, and that ≤ k. We prove thatthe number of semi-algebraically connected components of

V� is bounded by O(k)2k(∏

1≤j<�dkj−1−kj

j

)dk�−1

� . Addi-

tionally, if P ⊂ R[X1, . . . ,Xk] is a finite family of polynomi-als with deg(P ) ≤ d for all P ∈ P , cardP = ∫ , and d ≥ (k−k�−1+1)d�, we prove that the number of semi-algebraicallyconnected components of the realizations of all realizablesign conditions of the family P restricted to V� is bounded

by O(k)2k(sd)k�dk−k11 dk1−k2

2 · · · dk�−2−k�−1

�−1 dk�−1−k�

� .

Sal P. BaronePurdue University, [email protected]

Saugata BasuPurdue [email protected]

MS62

The geometry of the TDOAbased localization

The localization of a radiant source is a common issue tomany scientific and technological research areas. We solvedthe existence and uniqueness problem for planar source lo-calization based on the measurements of the time differ-ences of arrival (TDOAs) of the source signal to three dis-tinct receivers. In geometric terms, one has to characterizea particular algebraic double cover of the real plane, thatwe studied using multilinear algebra and (real) algebraicgeometry techniques.

Marco CompagnoniDipartimento di Matematica ”Francesco Brioschi”Politecnico di [email protected]

Roberto NotariDipartimento di MatematicaPolitecnico di [email protected]

Fabio Antonacci, Augusto SartiDipartimento di Elettronica ed InformazionePolitecnico di [email protected], [email protected]

MS62

Applications of Real Numerical Algebraic Geome-try

Large-scale polynomial systems arise in many applicationsincluding biology, chemistry, engineering, and physics. Of-ten in these applications, only a few real isolated solutionsand real connected components are of interest. Given awitness set for an irreducible algebraic set X, this talk willexplore using various numerical algebraic geometric algo-rithms to compute the real connected components of thereal points of X. This talk will conclude by demonstrating

the algorithms on several of these applications.

Jonathan HauensteinNorth Carolina State [email protected]

Charles WamplerGeneral Motors Research Laboratories, EnterpriseSystems Lab30500 Mound Road, Warren, MI 48090-9055, [email protected]

MS62

Some Applications of Cylindrical Algebraic Decom-position

Cylindrical Algebraic Decomposition (CAD) can be ap-plied to determine the truth value of a given system ofquantified polynomial inequalities. Gerhold and Kauersintroduced a method based on CAD to prove inequalitieson expressions that are defined by (linear or nonlinear) sys-tems of recurrences with polynomial coefficients. Specialfunctions as well as many combinatorial objects are withinthis class. In this talk we present some of our recent appli-cations of the method.

Veronika PillweinResearch Institute for Symbolic Computation (RISC)Johannes Kepler Univ. [email protected]

MS62

Safety Verification of Cyber-Physical Systems Us-ing the Theory of Reals

We describe computational techniques for verification andsynthesis of cyber-physical systems. These techniques in-variably rely on reasoning over the theory of reals. Re-cently, the formal methods community has made someprogress in integrating and improving algorithms basedon cylindrical algebraic decomposition for reasoning aboutsemialgebraic sets within the larger verification context.We will survey these recent advances in scaling reasoningover reals to large formulas.

Ashish TiwariSRI International, [email protected]

MS63

Subresultants, Sylvester Sums and the Rational In-terpolation Problem

This talk presents a solution for the classical univariaterational interpolation problem by means of (univariate)subresultants. In the case of Cauchy interpolation (inter-polation without multiplicities), explicit formulas for thesolution in terms of symmetric functions of the input dataare recovered, generalizing the well-known formulas for La-grange interpolation. In the case of the osculatory rationalinterpolation (interpolation with multiplicities), determi-nantal expressions in terms of the input data are presented.

Carlos D’AndreaUniversitat de [email protected]

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Teresa KrickUniversidad de Buenos [email protected]

Agnes SzantoNorth Carolina State [email protected]

MS63

Divided Differences and Combinatorial Nullstellen-satz

Alon’s combinatorial Nullstellensatz, and in particular theresulting nonvanishing criterion is one of the most power-ful algebraic tools in combinatorics, with a diverse arrayof applications. In this paper we extend the nonvanishingtheorem in two directions. We prove a version allowingmultiple points. Also, we establish a variant which is validover arbitrary commutative rings, not merely over subringsof fields. For the proofs we use the technique of divideddifferences to finite fields. We extend some of the knownapplications of the original nonvanishing theorem to a set-ting allowing multiplicities, including the theorem of Alonand Furedi on the hyperplane coverings of discrete cubes.

Geza Kos, Lajos RonyaiComputer and Automation Research InstituteHungarian Academy of [email protected], [email protected]

Tamas MeszarosDept. of Mathematics, Central European [email protected]

MS63

Fraction-Free Polynomial Arithmetic with Interpo-lation Bases

In this talk we consider the problem of fraction free com-putation working with polynomials represented in non-standard bases. Examples of these non-standard basesinclude Lagrange, Newton, and various orthogonal poly-nomial bases. Operations covered include rational interpo-lation, polynomial gcd and matrix gcd problems.

George LabahnSchool of Computer ScienceUniversity of [email protected]

MS63

Interpolation and Walks on the Hilbert Scheme

The interpolation problem at a given set of points can beinterpreted as the construction of an explicit representa-tion of idempotents in some quotient algebra. A naturalquestion is how this construction changes when the pointsare moving. We will first describe a generalization of theunivariate Weierstrass method which explicitly relates theperturbation of a complete intersection system defining acertain point set to the perturbation of the point set. Nextwe will consider the description of the variety of polyno-mial systems defining a given number of points countedwith multiplicities, called the punctual Hilbert Scheme. Wewill give equations of degree 2 related to the commutationrelations of border bases, describe its tangent space at agiven border basis and discuss some limit problems in this

interpolation process.

Bernard MourrainGALAADINRIA Sophia [email protected]

MS64

Calabi-Yau 3-folds. Collect them all!

Viable models of particle physics from string theory requirecompactification of 6 spatial dimensions onto a manifoldwith specific properties put forward by Calabi and Yau,including a complex structure and vanishing first Chernclass. These conditions suggest that we might find a large,and more importantly, calculable class of such manifoldsexisting as hypersurfaces in a 4d complex toric variety -and consequently encoded in a class of 4d reflexive latticepolytopes. Kreuzer and Skarke have painstakingly enumer-ated all such reflexive polytopes, and have made strides inextracting the desired Calabi-Yau 3-fold geometries. I dis-cuss these results, a few ensuing difficulties, and my owncurrent efforts to obtain a more complete exposition usingthe computer package SAGE.

Ross AltmanNortheastern [email protected]

MS64

Numerical Algebraic Geometry and Potential En-ergy Landscapes

The surface drawn by a potential energy function, whichis usually a multivariate nonlinear function, is called thepotential energy landscape (PEL) of the given Physi-cal/Chemical system. The stationary points of the PEL,where the gradient of the potential vanishes, are used toexplore many important Physical and Chemical propertiesof the system. Recently, we have employed the numericalalgebraic geometry (NAG) method to study the stationarypoints of the PELs of various models arising from Physicsand Chemistry and have discovered their many interestingcharacteristics. In this talk, I will mention some of theseresults after giving a very brief introduction to the NAGmethod. I will then go on discussing our latest adventure:exploring the PELs of random potentials with NAG, whichwill address not only one of the classic problems in Alge-braic Geometry but will also find numerous applications indifferent areas such as String Theory, Statistical Physics,Neural Networks, etc.

Dhagash MehtaDepartment of PhysicsSyracuse [email protected]

MS64

Algebraic Geometry and the Search for Calabi-YauManifolds with Large Volume Vacua

We describe an efficient, construction independent, algo-rithmic test to determine whether Calabi–Yau threefoldsadmit a structure compatible with the Large Volume mod-uli stabilization scenario of type IIB superstring theory.Using the algorithm, we scan complete intersection andtoric hypersurface Calabi-Yau threefolds with small Picardnumbers, and deduce that 418 among 4434 manifolds havea Large Volume Limit with a single large four-cycle. We de-

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scribe major extensions to this survey, which are currentlyunderway.

Brent NelsonDept of Physics, Northeastern [email protected]

MS64

Supersymmetric Hidden Sectors for HeteroticStandard Models

Abstract not available at time of publication.

Burt OvrutDept of Physics, University of [email protected]

MS64

Integrand Reduction of High-loop Scattering Am-plitudes via Computational Algebraic Geometry

Integrand reduction is a powerful method for loop scat-tering amplitudes computation. We present an algebraicgeometry approach for Integrand-level reduction, which isvalid for any renormalizable theories and arbitrary loop or-ders, in principle. The integrand form of amplitudes canbe determined by Groebner basis method and the unitar-ity structure is systematically studied by algebraic geom-etry. We also show some explicit two-loop and three-loopN=0,1,2,4 (super)-Yang-Mills amplitudes examples, basedon our methods.

Yang ZhangNiels Bohr Institute, [email protected]

MS65

Paramotopy: Parallel Parameter Homotopy Soft-ware Through Bertini

Many real-world problems involve parameterized systemsof polynomials. The quick solution of such systems forlarge sets of parameter samples will enable the scientistto scan for regions of interest in parameter space, collectdata for conjectures, and other applications. Paramotopyis a program which enables efficient solution of parameter-ized systems via parameter homotopy, leaning heavily onBertini as the numerical solver. It uses parallel computingtechniques, provides easy back-end access to processing ofdata, and can automatically re-solve the system at anytrouble points. This talk will discuss some features andapplications of Paramotopy.

Daniel A. BrakeColorado State [email protected]

MS65

Applications of Homotopy Method to NonlinearPdes

This talk will cover some recent progress on numericalmethods to solve systems of nonlinear partial differentialequations (PDEs) arising from biology and physics. Thisnew approach, which is used to compute multiple solutionsand singularity of nonlinear PDEs, makes use of polyno-mial systems (with thousands of variables) arising by dis-cretization. Examples from hyperbolic systems will be used

to demonstrate the ideas.

Wenrui HaoDept. of Applied and Comp. Mathematics and StatisticsUniversity of Notre [email protected]

MS65

Chebyshev Method for a Free Boundary ProblemModeling Tumor Growth

Abstract not available at time of publication.

Oliver KernellUniversity of Notre [email protected]

MS65

Numerical Algebraic Geometry in Algebraic Statis-tics

Maximum likelihood estimation is a fundamental compu-tational task in statistics and involves beautiful geometry.We discuss this task for determinantal varieties (matriceswith rank constraints) and show how numerical algebraicgeometry can be used to maximize the likelihood function.Our computational results with the software Bertini ledto surprising conjectures and duality theorems. This isjoint work with Jan Draisma, Jon Hauenstein, and BerndSturmfels.

Jose RodriguezDepartment of MathematicsUniversity of California, [email protected]

MS65

A Web Interface for PHCpack

Our web interface for PHCpack allows users to submitpolynomial systems through a browser, after signing upvia an automated email process. Through a password pro-tected account, the user has access the solutions of the sub-mitted systems stored on our server. Solved problems mayserve as start systems for similar problems. The web serveris connected to a remote computational server which man-ages the job queue for the distributed solving of polynomialsystems. The user administration is done with MySQL andthe servers are implemented in Python.

Jan VerscheldeDepartment of Mathematics, Statistics and ComputerScienceUniversity of Illinois at [email protected]

Xiangcheng YuDepartment of Mathematics, Statistics, and ComputerScienceUniversity of Illinois at [email protected]

MS66

Kolmogorov’s Problem on the Class of MultiplyMonotone Functions

In this talk we give necessary and sufficient conditions forthe system of positive numbers Mk1 ,Mk2 , ...,Mkd

, 0 ≤

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k1 < ... < kd≤r, to guarantee the existence of an r-monotone function defined on the negative half-line − andsuch that ‖x(ki)‖∞ = Mki , i = 1, 2, ..., d.

Yuliya BabenkoKennesaw State [email protected]

Vladislav Babenko, Oleg KovalenkoDnepropetrovsk National UniversityDnepropetrovsk, [email protected],[email protected]

MS66

Toric Degenerations of (irrational) Bezier Patches

A Bezier surface is intuitively governed by control points;in particular, the surface lies within the convex hull of itscontrol points. This convex hull is often indicated by draw-ing some edges between the control points, the resultingstructure is called a control mesh. In this talk, we givea precise mathematical description of how these meshes”control” the Bezier surface. We will present our results inthe general context of (irrational) toric Bezier patches.

Luis D. Garcia-PuenteSam Houston State [email protected]

Frank SottileTexas A&M [email protected]

Chungang ZhuDalian University of Science and [email protected]

Elisa PostinghelInstitute of Mathematics of the Polish Academy [email protected]

Nelly VillamizarJohann Radon Institute for Computational and [email protected]

MS66

Towards an Algebra for Rational Curves and Sur-faces in Two and Three Dimensions

The complex numbers allow us to multiply complex val-ued rational curves and surface patches in the plane. Byspherical inversion (perspective projection) this approachgenerates an algebra for curves and surface patches on thesphere. Similarly, the quaternions allow us to multiply(with some restrictions) rational curves and surfaces in 3-dimensions. The goal of this talk is to explore the geometryaccompanying this algebra for rational curves and surfacesin two and three dimensions.

Ron GoldmanRice UniversityComputer [email protected]

MS66

Wachspress Varieties

Wachspress Varieties Eugene Wachspress introduced ra-tional barycentric coordinates for convex polygons in R2,which Warren generalized to all convex polytopes in Rd.These Wachspress barycentric coordinates are the uniquerational barycentric coordinates of minimal degree. TheWachspress coordinates of a polytope P in Rd define a mapof projective d-space to a projective space spanned by thevertices of P whose image is a Wachspress variety. In thistalk, I will describe the Wachspress variety and its defin-ing ideal, which is the ideal of algebraic relations amongthe Wachspress coordinates. This is joint work with CoreyIrving, Hal Schenck, and Greg Smith.

Frank SottileTexas A&M [email protected]

Corey IrvingSanta Clara [email protected]

Henry SchenckUniversity of [email protected]

Gregory G. SmithQueen’s [email protected]

MS66

Representation of Surface Pencil with A CommonLine of Curvature

Developable surface and line of curvature play importantroles in geometric modeling and pratical applications. Byutilizing the Frenet frame, we derive the necessary and suf-ficient condition for the parametric surface pencils possess-ing the given curve as the line of curvature, and presentthe conditions for the resulting surfaces are developable.

Chungang ZhuDalian University of Science and [email protected]

Caiyun Li, Renhong WangDalian University of [email protected], [email protected]

MS67

Graph Based Codes for Flash Memories

Finite geometries have been useful in several coding ap-plications, from early Write-Once-Memory (WOM) codesfor data storage to the design of LDPC error-correctingcodes. However, modern data storage techniques imposeseveral conflicting design constraints that make the directuse of finite geometry LDPC codes harder to apply. In thistalk we will discuss how graph based codes can be used incoding for non-volatile memories, and suggest how to mod-ify algebraic codes (such as finite geometry codes) for thisapplication.

Christine A. KelleyUniversity of [email protected]

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Kathryn HaymakerUniversity of [email protected]

MS67

GF (q)-Linear Codes over GF (qt)

We will examine the classical theorems for linear codesand see if and how they apply to GF (q)-linear codesover GF (qt), We will discuss some of the following: theMacWilliams Identities, the Gleason polynomials, theGleason-Pierce Theorem, Mass Formulas, the BalancePrinciple, the Singleton Bound, and MDS codes.

Cary HuffmanLoyola University [email protected]

MS67

Geometric Perspective on Alternant Codes

Alternant codes are a large class of codes containing manyof the most well-known constructions, such as BCH-codesand Goppa codes. Many other constructions containingbest known codes exist, but there is no known way to guar-antee optimal constructions. We will present a geometricdescription of alternant codes and show how this descrip-tion can lead to some new bounds and ideas for construct-ing good codes.

Kyle MarshallUniversity of [email protected]

MS67

On the Number of Constacyclic Codes on a Classof Local Finite Frobenius Rings

The study of cyclic and constacyclic codes over various al-phabets has been of considerable interest in Coding Theory.These alphabets include finite fields, Galois rings, and finitechain rings. In this talk, the number of constacyclic codesthe alphabet of which is a finite local non-chain Frobeniusring with a maximal ideal of nilpotency index 3, is deter-mined, where the length of the codes is relatively prime tothe characteristic of the residue field of the ring. Resultsfrom Commutative Algebra including the length of a mod-ule, Nakayama’s Lemma, and Galois extension of a finitelocal ring, are helpful in this discussion.

Horacio Tapia-RecillasUAM [email protected]

MS68

Calculations in the Lie Invariant Calculus of Vari-ations - the General Case

In recent works, the authors considered Lagrangians invari-ant under a Lie group action, in the case where the inde-pendent variables are themselves invariant. Using a movingframe for the Lie group action, we showed how to calcu-late the Euler Lagrange equations for the invariants and toobtain the space of conservation laws in terms of a vectorof invariants and the adjoint representation of the frame.In this talk, we show how our calculations extend to caseswhere the independent variables participate in the action.We take for our main expository example the standard lin-

ear action of SL(2) on the two independent variables. Thischoice is motivated by potential applications to variationalfluid problems which conserve potential vorticity.

Elizabeth MansfieldUniversity of [email protected]

Tania GoncalvesUniversidade Federal de So [email protected]

MS68

Symbolic Computation of Lax Pairs of Systemsof Partial Difference Equations Using ConsistencyAround the Cube

A three-step method due to Bobenko & Suris and Nijhoffto derive Lax pairs for scalar partial difference equationsis extended to systems which are defined on a quadrilat-eral and consistent around the cube. Lax pairs will bepresented for several systems including the integrable 2-component potential Korteweg-de Vries lattice system, aswell as nonlinear Schrodinger and Boussinesq-type latticesystems. Previously unknown Lax pairs will be presentedfor systems of partial difference equations recently derivedby Hietarinta. The method is algorithmic and is being im-plemented in Mathematica.

Willy A. HeremanDepartment of Applied Mathematics and StatisticsColorado School of [email protected]

Terry BridgmanDept Appl Mathematics & Statistics, Colorado School [email protected]

MS68

Induced Curvature Flow and Integrability

The operator providing the induced curvature flow for a ge-ometric flow can be computed algorithmically. The methodrelies on the construction of a complete set of syzygies for agenerating set of differential invariants of the geometry anddifferential elimination. With the Maple packages AIDA[Algebraic Invariants and their Differential Algebras] anddiffalg to re-examine geometries in the plane. By removingthe non-stretching condition, an exhaustive search providesnew generalisations of the usual integrable equations as in-duced curvature flow. This is work in progress.

Evelyne HubertINRIA [email protected]

Peter van der KampUniversity of La [email protected]

MS68

Pseudo-spherical Twisted Columns

The aim of this presentation is to explain the global geo-metric properties of singular helicoids with Gaussian curva-ture K=-1. We will show that, for every positive integer nthere is a one-parameter family of pseudo-spherical singular

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helicoids which are topologically embedded cylinders withthe shape of twisted columns. Numerical routines to de-tect and visualize the pseudo-spherical twisted columns aregiven. The geometrical problem while being rather simplefrom the conceptual point of view, presents a certain com-putational complexity which is caused by the presence ofsingularities. The theoretical framework of our approach isbased on the method of moving frame while the problemsof computational and numerical nature have been dealtwith the use of the software Mathematica 9. The interestof the result lies in the fact that the known examples ofpseudo-spherical surfaces that are topologically embeddedcylinders are very sporadic and substantially boil down tothe Beltrami pseudo-sphere and the pseudo-spherical sur-faces of rotation of cylindrical type, whose origin dates backto the early days of the theory. Another reason that givesmeaning to our result must be sought in possible appli-cations to the study of the ”modulated wave solutions”of certain mathematical models which have been recentlyused in the theory of non-linear elasticity.

Emilio MussoDipartimento di [email protected]

MS68

Holomorphic Differentials and Laguerre deforma-tion of surfaces

We present a local characterization of L-minimal surfacesand Laguerre deformations (T-transforms) of L-minimalisothermic surfaces in terms of the holomorphicity of aquartic and a quadratic differential form. This result isthen discussed in relation with the differential geometry ofthe Laguerre Gauss map of surfaces, viewed as a space-like immersion in Minkowski spacetime. As an applica-tion, we will show that the Lawson correspondence be-tween certain constant mean curvature surfaces in differentspace forms and the analogous correspondence between cer-tain constant mean curvature spacelike surfaces in differentLorentzian space forms can be viewed as special cases ofthe Laguerre deformation of L-isothermic surfaces. Ourapproach uses the method of moving frames. This is basedon joint work with E. Musso.

Lorenzo NicolodiUniversita degli Studi di [email protected]

MS69

Real Rank of Real Symmetric Tensors

I will explain the ways in which symmetric real tensor de-composition differs from the complex case and present re-cent results. Open question abound in the real case.

Greg BlekhermanGeorgia Institute of [email protected]

MS69

Tensor Network Decompositions

Abstract not available at time of publication.

Jason MortonPennsylvania State [email protected]

MS69

Eigenvectors of Tensors and Waring Decomposi-tion.

In recent work with Ottaviani, we have developed algo-rithms for symmetric tensor decomposition. I will explainthese algorithms, their connection to algebraic geometry,and possible applications.

Luke OedingUniversity of California BerkeleyUniversita degli Studi di [email protected]

Giorgio OttavianiUniversity of [email protected]

MS69

Algorithms for Tensor Decomposition via Numeri-cal Homotopy and Optimization

This talk will present consider several numerical algorithmsfor tensor decompositions and tensor rank. The algorithmsare based in numerical homotopy and in semi-definite pro-gramming but arise from the point of view of algebraicgeometry.

Chris PetersonColorado State [email protected]

MS69

Computational Complexity and Linear Preservers

We study the relation between computational complexityand linear preservers of various problems. We first inves-tigate this relation for determinant, immanants, and α-permenant. Then we consider the same question for rankand non-negative rank, p-norms for different values of p,and positive definiteness and complete positivity.

Ke Ye, Lek-Heng LimUniversity of [email protected], [email protected]

MS70

Bernstein-Sato Ideals

For any topological space, one can define the cohomologjump loci. These are natural strata in the space of lo-cal systems of rank one on the space. When the spaceis given by polynomials, i.e. a quasi-projective variety,the cohomology jump loci have rigid arithmetic structure.This is a result obtained together with Botong Wang. Wewill address the practical question of how to compute thecohomology jump loci, the conjectural answer being: viaBernstein-Sato ideals. This generalizes the computation ofMilnor monodromy eigenvalues of a hypersurface.

Nero BudurNotre [email protected]

MS70

Modularity and the Reciprocal Plane

Let A be a collection of n linear hyperplanes in K�, where

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K is an algebraically closed field. The Orlik-Terao alge-bra of A is the subalgebra R(A) of the rational functionsgenerated by reciprocals of linear forms vanishing on hy-perplanes of A. It determines an irreducible projectivevariety Y (A), the reciprocal plane. We show that a flat Xof A is modular if and only if R(A) is a split extension ofthe Orlik-Terao algebra of the subarrangement AX . Thisprovides another refinement of Stanley’s modular factor-ization theorem and a new characterization of modularity.We note that if A is supersolvable, then R(A) is Koszul,and characterize when R(A) is a complete intersection.

Graham DenhamUniversity of Western [email protected]

MS70

F-jumping Numbers of Homogeneous Polynomials

This is joint work with Luis Nunez Betancourt, EmilyWitt, and Wenliang Zhang. F -jumping numbers are im-portant invariants of singularities in characteristic p > 0,and may be thought of as positive characteristic analogsof jumping exponents in characteristic zero. The aim ofthis work is to investigate certain arithmetic properties ofthese invariants; in particular, we will present a descriptionof F -pure thresholds of homogeneous polynomials with anisolated singluarity.

Daniel HernandezUniversity of [email protected]

Luis C. Nunez-BetancourtUniversity of [email protected]

Emily E. WittUniversity of [email protected]

Wenliang ZhangUniversity of [email protected]

MS70

A Stopping Condition to Compute Test Ideals andF-Jumping Numbers

We will discuss a stopping condition that gives algorithmsto compute test ideals and F-jumping numbers, which playan important role in the study of singularities in positivecharacteristic. In addition, we will see properties of F-purethresholds that are a consequence of this condition.

Luis C. Nunez-BetancourtUniversity of [email protected]

Daniel Hernandez, Emily E. WittUniversity of [email protected], [email protected]

MS70

An Algorithm to Find Annihilators of ArtinianModules Compatible with a Frobenius Map

I will discuss an algorithm to find radical annihilators of

submodules, compatible with a Frobenius map, of an Ar-tinian module over a ring of formal power series over a fieldof characteristic p. This is a joint work with MordechaiKatzman.

Wenliang ZhangUniversity of [email protected]

Mordechai KatzmanUniversity of Sheffieldm.katzman@@sheffield.ac.uk

MS71

Boolean Matrices with Prescribed Row and Col-umn Sums, Associated Partition Functions and Hy-perbolic Polynomials

Hyperbolic (and their generalizations) polynomials were re-cently used to improve several lower bounds in combina-torics and geometry. These new lower bounds can be usedto obtain poly-time algorithms to approximate many hardinteresting quantities, notably the mixed volume, withinsimply exponential multiplicative error. The talk will de-scribe one refinement of the lower bounds and its appli-cation to the approximation of partition functions associ-ated with Boolean matrices with prescribed row and col-umn sums. In a way, we will describe a natural class ofn × n nonnegative matrices which allows a poly-time de-terministic algorithm to approximate the permanent with

the factor const√

n. Besides, we will explain how the cele-brated Gale-Ryzer Theorem follows immediately from thehyperbolic framework.

Leonid GurvitsLos Alamos National [email protected]

MS71

Norm-constrained Determinantal Representationsof Multivariable Polynomials

For every multivariable polynomial p, with p(0) = 1, weconstruct a determinantal representation

p = det(I −KZ),

where Z is a diagonal matrix with coordinate variables onthe diagonal and K is a complex square matrix. Whennorm constraints on K are imposed, we give connectionsto the multivariable von Neumann inequality, Agler de-nominators, and stability. We show that if a multivariablepolynomial q, q(0) = 0, satisfies the von Neumann inequal-ity, then 1 − q admits a determinantal representation withK a contraction. On the other hand, every determinantalrepresentation with a contractive K gives rise to a rationalinner function in the Schur–Agler class.

Dmitry Kaliuzhnyi-VerbovetskyiDrexel [email protected]

MS71

Determinantal Representations of Singular Hyper-surfaces in Pn

A (global) determinantal representation of projective hy-persurface X in Pn is a matrix whose entries are lin-ear forms in homogeneous coordinates and whose deter-

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minant defines the hypersurface. We study the propertiesof such representations for singular (possibly reducible ornon-reduced) hypersurfaces. In particular, we obtain thedecomposability criteria for determinantal representationsof globally reducible hypersurfaces. Further, we classify thedeterminantal representations in terms of the correspond-ing kernel sheaves on X. Finally, we extend the resultsto the case of symmetric/self-adjoint representations, withimplications to hyperbolic polynomials and generalized Laxconjecture.

Dmitry KernerBen Gurion [email protected]

MS71

Primal-Dual Algorithms for Optimization over Hy-perbolicity Cones

Hyperbolicity cones are rich in structure, yet it remainsunclear whether the structure can be utilized to designoptimization algorithms that generally are more efficientthan interior-point methods. However, as we will discuss,when a warm-start is available – that is, when an approxi-mate optimal solution is available (by, say, having alreadysolved a problem differing only slightly in the data) – thenthe boundary structure of the hyperbolicity cone most cer-tainly can be efficiently harnessed. Unsurprisingly, themost effective harnessing makes use of duality, but only du-ality structure arising from the hyperbolicity cone’s bound-ary in a neighborhood of the warm-start. Far more chal-lenging is understanding the global structure of the dualcones of (general) hyperbolicity cones. We present stepsin this direction, and discuss their relevance for designingprimal-dual algorithms beyond the warm-start setting.

James M. RenegarCornell UniversitySchool of [email protected]

MS71

Hyperbolic Cone Programming: Structure andInterior-Point Algorithms

I will first discuss some structural and geometric proper-ties of convex cones which are hyperbolicity cones of somehyperbolic polynomial. Then, I will present some interior-point algorithms and their theoretical features on the hy-perbolic cone programming problems. This talk is basedon joint work with Tor Myklebust.

Levent TuncelUniversity of WaterlooDept. of Combinatorics and [email protected]

MS72

Linear Algebra with Errors: On the Complexity ofthe Learning with Errors Problem

The Learning With Errors problem (LWE) - a general-isation of the LPN problem to larger moduli - enjoyswidespread appreciation from the cryptographic commu-nity because it allows for the construction of a wide rangeof cryptographic primitives and comes with convincing se-curity arguments. However, the concrete hardness of con-crete LWE instances is not that well understood. The bestknown algorithm for LWE is an adaptation of the BKW

algorithm to the larger moduli case. In this work we in-vestigate this adaptation in detail, refine it and provideconcrete costs for solving LWE instances from the litera-ture.

Martin AlbrechtUniversite Pierre et Marie CurieParis, [email protected]

MS72

Fast Matrix Decomposition in F2

An efficient algorithm to perform a block decomposition(and so to compute the rank) of large dense rectangularmatrices with entries in F2 is presented. Depending on theway the matrix is stored, the operations acting on rows orblock of consecutive columns (stored as one integer) shouldbe preferred. In this paper, an algorithm that completelyavoids the column permutations is given. In particular, ablock decomposition is presented and its running times arecompared with the ones adopted into SAGE.

Enrico BertolazziDipartimento di Ingegneria IndustrialeFacolta di Ingegneria - Universita di Trento, [email protected]

Anna RimoldiDipartimento di MatematicaUniversita di [email protected]

MS72

Relaxed Hensel Lifting for Dense, Sparse andStructured Linear System Solving

This talk is about linear system solving on integers or poly-nomials via lifting techniques. We study the Hensel liftingtechnique using relaxed algorithms and take a careful lookat the adaptation of this method to different matrix repre-sentations (dense, structured, sparse...). We compare ourresults to the concurrent lifting technique (Newton itera-tion) and show improvements both in theory and practice.

Romain LebretonLIRMMUniv. Montpellier 2, [email protected]

MS72

Rational Linear Solvers and Local Smith Forms andHow They Apply to Homology Computation

Simplicial Homology is determined by the integer Smithnormal forms of boundary matrices, which are (1,0,-1) ma-trices that are most often very sparse. It is important to beefficient in both time and space on Smith form computa-tion. I will report on some recent developments concerningspace efficiency and show what remains open. I will discussthe use of block blackbox methods, important for effectiveparallel implementations.

David SaundersDept of Computer and Information SciencesUniversity of Delaware, [email protected]

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MS73

p-Adic Height Pairings and Integral Points on Hy-perelliptic Curves

We give a formula for the component at p of the p-adicheight pairing of a divisor of degree 0 on a hyperellipticcurve in terms of iterated p-adic integrals. We use thisto give a Chabauty-like method for computing p-adic ap-proximations to integral points on such curves when theMordell-Weil rank of the Jacobian equals the genus.

Jennifer BalakrishnanHarvard [email protected]

Amnon BesserUniversity of [email protected]

J. Steffen MuellerUniversitaet [email protected]

MS73

Real Hyperelliptic Curves of Genus 2

We will present optimized explicit formulas for the arith-metic of real hyperelliptic curves of genus 2 in both affineand projective coordinates. We will compare the results toimaginary curves and timings of cryptographic protocols inboth cases.

Stefan EricksonColorado [email protected]

MS73

A Lutz-Nagell Theorem for Hyperelliptic Curves

We’ll present a version of the Lutz-Nagell Theorem forcurves C defined by y2 = f(x), where f is a monic polyno-mial of degree 2g + 1 defined over the ring of integers of anumber field or its non-archimedean completions. Namely,if J is the Jacobian of C, and φ is the Abel-Jacobi map ofC into J sending the point at infinity on C to the origin ofJ , then if P = (a, b) is a rational point of C such that φ(P )is torsion in J , then a and b are integral if the order n of Pis not a prime power, and we can bound the denominatorsof a and b if n is. When a and b are integral, we’ll discusscriteria for when b2 divides the discriminant of f .

David GrantUniversity of [email protected]

MS73

Improved Scalar Multiplication on HyperellipticCurves

Scalar multiplication, adding a divisor class to itself ntimes, is a central operation of elliptic and hyperellipticcurve cryptosystems. In this talk, we describe recent re-sults and on-going efforts to improve this operation usingideas involving double-base expansions of the scalar, in-cluding an algorithm for anomalous binary hyperellipticcurves that requires a sublinear number of divisor addi-tions in the size of the scalar. This is joint work with H.

Labrande and S. Lindner.

Michael JacobsonUniversity of [email protected]

MS73

Effective Chabauty for symmetric Powers ofCurves

While we know by Faltings’ theorem that curves of genusat least 2 have finitely many rational points, his theorem isnot effective. In 1985, R. Coleman showed that Chabauty’smethod, which works when the Mordell-Weil rank of theJacobian of the curve is small, can be used to give a goodeffective bound on the number of rational points of curvesof genus g > 1. We draw ideas from tropical geometry toshow that we can also give an effective bound on the size ofthe finite component of SymdX, where X is a curve thatsatisfies certain Chabauty-type condition, hence showingthat Chabauty’s method can be used for certain higher-dimensional varieties.

Jennifer ParkMassachusetts Institute of [email protected]

MS74

Evasion Paths in Mobile Sensor Networks

Suppose disk-shaped sensors wander in a planar domain.A sensor doesn’t know its location but does know whichsensors it overlaps. We say that an evasion path exists inthis sensor network if a moving evader can avoid detection.Vin de Silva and Robert Ghrist give a necessary condition,depending only on the time-varying connectivity graph ofthe sensors, for an evasion path to exist. Can we sharpenthis result? We consider an example where the existence ofan evasion path depends not only on the network’s connec-tivity data but also on its embedding. We also study thespace of evasion paths using a generalization of the unsta-ble J. F. Adams spectral sequence to diagrams of spaces.

Henry Adams, Gunnar E. CarlssonStanford [email protected], [email protected]

MS74

Spaces of Shapes: Creating Moduli Spaces ofChemical Compounds for Drug Discovery

Abstract not available at time of publication.

Anthony BakStanford [email protected]

MS74

Homological Algebra over Semirings for Optimiza-tion

Some optimization dualities (e.g. MFMC) generalize toa Poincare Duality for sheaves of semimodules. We de-tail how a generalized homological algebra for semimod-ules over general semirings yields new methods for solv-ing stochastic optimization problems. No familiarity with

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semimodule theory will be assumed.

Sanjeevi KrishnanUniversity of [email protected]

MS74

Multicore Homology

We design and implement a framework for parallel compu-tation of homology of cellular spaces over field coefficients,by decomposing the space. Theoretically, we show thatoptimal decomposition into local pieces is NP-Hard. Inpractice, we achieve roughly an 8x speedup of homologycomputation on a 3-dimensional complex with about 10million simplices using 11 cores.

Ryan LewisStanford [email protected]

MS74

Inferring Dynamics with Perisistence

Dynamical systems are often easier to analyze if we have agood parametrization. For periodic or recurrent systems,this is the map to a circle. The first part of the talk willdescribe how persistent cohomology can help construct anatural parametrization of certain dynamical systems. Thesecond part will go into specific cases how this informationaugments traditional invariants such as the Conley indexand allows us to infer properties of a dynamical system.

Primoz SkrabaJozef Stefan InstituteLjubljana, [email protected]

MS75

Algebraic Geometry of Partially Nested Canalyz-ing Functions

A Boolean function is canalyzing if some variable has theproperty that taking a specified input determines the out-put. If this variable does not take that input, then theoutput is a function on n − 1 variables. We may ask if ittoo is canalyzing, and so on. In this talk I will discuss thenested canalyzing depth of a function, which measures howmany times we can recursively pick off canalyzing variablesin this manner.

Qijun HeClemson [email protected]

MS75

Hodge-Kodaira Decomposition of Evolving NeuralNetworks

Although the network topology of the brain can be very im-portant, the conventional network analyses are often lim-ited to locally defined variables such as degrees. Thoseintrinsically local variables cannot capture global recur-rent structures, which are frequently observed in neuralnetworks. For example, a network that alters its cou-pling strengths under STDP learning rule has tendencyto make paths and loops of sequential firings. We appliedthe Hodge-Kodaira decomposition of graph flows to evolv-

ing neural network models in order to count the number ofglobal loops as a topological measure of network structures.The dimension of each flow not only reflected the known bi-furcation diagrams but also discovered the inhomogeneityof the chaotic region.

Keiji MiuraTohoku [email protected]

Takaaki AokiKagawa [email protected]

MS75

Algebraic Geometry in the Life and Physical Sci-ences: Past, Present, and Future

We will give an overview of how algebraic geometry hasimpacted the life and physical sciences in recent years. Wewill also discuss highlights from a workshop at the Math-ematical Biosciences Institute that immediately precededthis conference on novel ways to integrate discrete and alge-braic methods into undergraduate education. We will con-clude by sharing some surprising and upcoming researchareas.

Brandilyn StiglerSouthern Methodist [email protected]

Matthew MacauleyMathematical SciencesClemson [email protected]

MS75

Reverse Engineering Functional Networks of theHuman Brain within the Polynomial DynamicalSystems Framework

In this talk we will present an application example of areverse engineering method within the framework of poly-nomial dynamical systems. The objective is to unravel theinteraction patters among functional brain regions identi-fied by task-based functional magnetic resonance imaging(fMRI) time series data. These interaction patterns areused to construct functional networks of the human brainto enable a comparative study across groups of individuals.

Paola Vera-LiconaInstitut [email protected]

MS75

Detection of De Novo Copy Number Variants fromWhole Exome Sequencing Data

Exome sequencing is becoming a standard tool for mappingMendelian disease-causing variations, yet the exome dataare often more noisy. Here we describe a novel method forthe detection of de novo copy number variants from suchdata. We developed a likelihood model combined with ahidden Markov model to detect the variants jointly fromtrio samples. Our model outperforms other methods thatcall variants separately.

Mingfu Zhu, Yongzhuang Liu

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Duke [email protected], [email protected]

MS76

Reducing Implicitization to Interpolation via Sup-port Prediction

We reduce implicitization of parametric curves and (hy-per)surfaces to computing the nullspace of a numeric ma-trix, given a superset of the support of the implicit equa-tion. For predicting the support we compute the Newtonpolytope of the implicit polynomial, via sparse resultanttheory. We shall discuss both the support prediction andthe interpolation steps of the method, highlight their con-nections and describe an application of the method to com-puting discriminants.

Christos KonaxisACMAC, University of [email protected]

Ioannis Z. EmirisNational and Kapodistrian University of [email protected]

Tatjana KalinkaUniversity of [email protected]

Thang Luu BaHanoi National Univ. of [email protected]

MS76

Remarks on Nagata’s Conjecture

Abstract not available at time of publication.

Rick MirandaColorado State [email protected]

MS76

Polynomial Interpolation and Sums of Powers

The affine space of polynomials of degree ≤ d in n vari-ables, with assigned values of any number of general linearcombinations of first partial derivatives, has the expecteddimension if d = 2 with five exceptional cases. Four amongthese cases appear in Alexander-Hirschowitz theorem, andcorrespond to the assignment of all the first partial deriva-tives. If d = 2 the exceptional cases are described. Thisproblem is related to the decomposition of a polynomial assum of powers of linear forms.

Giorgio OttavianiUniversity of [email protected]

Chiara BrambillaPolitecnico delle Marche, [email protected]

MS76

Subresultants in Multiple Roots and Connections

to Hermite-Birkhoff Interpolation

This is a joint work with Carlos D’Andrea and TeresaKrick. The topic is a continuation of our previous workon Poisson-like formulas for univariate and multivariatesubresultants in terms of the roots of a polynomial sys-tem, as well as our work on the relationship between uni-variate subresultants and Sylvester double sums, providedthat all these roots were simple. As it was pointed out byan anonymous referee, it would be interesting to work outthese results for the case of multiple roots. This work is afirst attempt in that direction. We successfully extendedPoisson-like formulas for univariate and multivariate subre-sultants in the presence of multiple roots. We also obtainedclosed formulas for subresultants in roots in the univariatesetting for some non-trivial extremal cases. However, it isstill open how to generalize Sylvester double sums in themultiple roots case. In the talk we will also emphasize therelationship of this problem to Hermite-Birkhoff interpola-tion formulas.

Carlos D’AndreaUniversitat de [email protected]

Teresa KrickUniversidad de Buenos [email protected]

Agnes SzantoNorth Carolina State [email protected]

MS77

Invariant Theory for Matrix Product States

Abstract not available at time of publication.

Jacob BiamonteISI [email protected]

MS77

Tensor Networks

Abstract not available at time of publication.

Jason MortonPennsylvania State [email protected]

MS77

Contracting Tensor Networks

Abstract not available at time of publication.

Jacob TurnerPennsylvania State [email protected]

MS78

Title Not Available at Time of Publication

Abstract not available at time of publication.

Amihay HananyTheoretical Physics Division, Imperial College [email protected]

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MS78

Vector Bundles, Calabi-Yau Threefolds and theHeterotic String

We review some recent progress in explicitly constructingstable vector bundles, inspired by the search for the stan-dard model within the context of heterotic string compact-ifications, on Calabi-Yau threefolds. In due course, we willsee the necessity of computational and algorithmic geome-try on a large scale.

Yang-Hui HeUniversity of OxfordCity University of [email protected]

MS78

Generalized T-duality, String Theory and the RealWorld’

We examine the notion of T-duality in string theory from aphase space point of view and then emphasize the conceptof Born’s reciprocity, a natural generalization of T-duality,which posits the more general duality between spacetimeand momentum space. We propose that Born’s reciprocitydemands a dynamical momentum space, and thus a dy-namical phase space, opening up an exciting possibility fornew dualities in string theory and new connections to thereal world. [This work is done together with Laurent Frei-del (Perimeter Institute) and Rob Leigh (Urbana).

Djordje MinicVirginia [email protected]

MS78

New Moduli Spaces of Brane Tilings on RiemannSurfaces

Brane tilings are bipartite periodic graphs on the toruswhich represent one of the largest known classes of 4d N=1supersymmetric gauge theories. A brief review of branetilings will lead to a discussion on recent developments ofthe subject. These include the addition of boundaries aswell as the variation of the genus of the Riemann surface ofthe bipartite graphs. The talk will give an overview of thephysical significance of these extensions to brane tilings.

Rak-Kyeong SeongTheoretical Physics Division, Imperial College [email protected]

MS79

Linear Obstructions for Linear Systems in Pn

The linear system L of degree-d hypersurfaces of Pn withprescribed general multiple points is said to be special if theconditions imposed by the multiple points are not linearlyindependent. This phenomenon occurs when the multi-plicities force L to contain in its base locus, besides themultiple points, also higher dimensional cycles. We willdiscuss the case when such cycles are linear and introducethe notion of linear speciality. It takes into account theobstructions given by multiple linear base cycles and pro-vides a geometric interpretation of algebraic conjecturesby Froberg and Iarrobino on the Hilbert function of idealsgenerated by powers of general linear forms.

Elisa Postinghel

Institute of Mathematics of the Polish Academy [email protected]

Chiara BrambillaPolitecnico delle Marche, [email protected]

Olivia DumitrescuDepartment of MathematicsUniversity of California, [email protected]

MS79

Some Algebraic Problems in Polynomial Interpola-tion

We will discuss some problems in multivariate polynomialinterpolation and their relationship to questions in alge-braic geometry; in particular to questions regarding sub-space arrangements and irreducibility of border schemes.

Boris ShekhtmanDepartment of Mathematics and StatisticsUniversity of South [email protected]

MS79

Special Positions of Body-and-cad Frameworks

A body-and-cad framework consists of a finite set of full-dimensional rigid bodies and geometric constraints amongthem. These frameworks are the combinatorial models ofstructures arising in computer-aided design software. Be-ing able to detect constraints that specifiy a minimallyrigid body-and-cad framework has the potential to improveuser feedback. Following work of White and Whiteleyfor body-and-bar frameworks, it is possible to character-ize generic minimal infinitesimal rigidity by the nonvanish-ing of a polynomial, the ”pure condition,” which can bewritten as a bracket polynomial in the coordinate ring of aGrassmannian. This talk will focus on how understandingthe relationship between the combinatorics of the frame-work, the irreducible factors of the pure condition, and theGrassmann-Cayley factorization of the bracket polynomialcan yield insight into the identification and behavior ofspecial positions.

Jessica Sidman, Ruimin CaiMount Holyoke [email protected], [email protected]

James FarreUniversity of Texas, [email protected]

Audrey Lee-St.JohnMount Holyoke [email protected]

Louis TheranFreie Universitat [email protected]

MS79

Classification of Planar Pythagorean Hodograph

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Quintics

We describe PH quintic curves using two parameters, whichdetermine their global shape up to Euclidean similarities.Based on this description we give examples of all possiblekinds of PH quintics with respect to the distribution oftheir real singularities.

Zbynek SirCharles University in PragueCzech [email protected]

MS80

Partitions of Frobenius Rings Induced by the Ho-mogeneous Weight

The values of the homogeneous weight on a finite commu-tative Frobenius ring are determined. This is used to in-vestigate the partition induced by this weight and its dualpartition under character-theoretic dualization. A charac-terization of the rings is given for which the induced par-tition is reflexive (i.e., coincides with its bidual) or evenself-dual.

Heide Gluesing-LuerssenUniversity of [email protected]

MS80

Efficient Representation for the Trace Zero Sub-group via Rational Functions

The trace zero subgroup is a subgroup of the divisor classgroup of a (hyper)elliptic curve, defined over a finite field.Being able to represent the elements of this group as com-pactly (i.e., using as few bits) as possible is relevant forthe cryptographic applications. In this talk, I will discussa new method for representing the elements of the tracezero subgroup. The method uses as representation the co-efficients of certain rational functions.

Elisa GorlaUniversity of [email protected]

Maike MassiererUniversity of Basel and University of [email protected]

MS80

Partial Spreads in Network Coding

As in the novel approach by R. Kotter and F. R. Kschis-chang, we study network codes as families of k-dimensionallinear spaces over a finite field. Following an idea in finiteprojective geometry, we introduce a class of network codeswhich we call partial spread codes. Partial spread codesnaturally generalize the known family of spread codes. Weprovide an easy description of such codes, discuss theirmaximality, and explain how to decode them efficiently.

Alberto Ravagnani, Elisa GorlaUniversity of [email protected], [email protected]

MS80

An MQ/Code Cryptosystem Proposal

We describe a new trap-door (and PKC) proposal. Theproposal is “multivariate quadratic’ (relies on the hardnessof solving systems of quadratic equations); it is also code-based, and uses the code-scrambling technique of McEliece(1978). However, in the new proposal, the error-correctingcode is not revealed in the public key, which protectsagainst the leading attacks on McEliece’s method. Theproposal motivates an open question in coding theory.

Leonard [email protected]

Jonathan HallMichigan State [email protected]

MS80

Colorability of Hypergraphs Using CommutativeAlgebra

A hypergraph is properly k-colorable if each vertex can becolored by one of k colors such that no edge is monochro-matic. The talk will present a complete algebraic char-acterization of the 2-colorability of uniform hypergraphsand discuss a decomposition result for hypergraph 2-colorings. We will also discuss extensions of our resultsto k-colorability of uniform hypergraphs.

Lubos Thoma, Michael KrulDepartment of MathematicsUniversity of Rhode [email protected], [email protected]

MS82

Moment Matrices and Applications

Moment matrices play an important role in inverse prob-lems. Examples are sparse interpolation, pole detection,frequency estimation, etc. In many applications, the tar-geted problem is associated with a structured moment ma-trix. We employ tools from numerical linear algebra toinvestigate the numerical behavior and conditioning of thetargeted problem. We present some selected applicationsand demonstrate how the techniques can be generalized toa multivariate problem statement.

Annie CuytUniversity of AntwerpDept of Mathematics and Computer [email protected]

Wen-shin LeeUniversity of [email protected]

MS82

Truncated Moment Problems, Extensions, andPositivity

We discuss two interrelated abstract solutions to the mul-tivariable Truncated Moment Problem (TMP). One solu-

tion shows that an n-dimensional sequence y(2d) of degree2d admits a representing measure if and only if the mo-ment matrix Md(y) has a positive flat extension Md+k for

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some k ≥ 0. A second solution shows that y has a rep-resenting measure if and only if the corresponding Rieszfunctional Ly : R[x]2d =⇒ R admits a positive extension

L : R[x]2d+2 =⇒ R. The difficulty in both approaches isto develop concrete conditions for the desired extensions.In the case n = 2, d = 3, TMP for y(6) remains largely un-solved, but we show that if M3 � 0 and rank M3 ≤ 6, thenLy is positive; consequently, y(5) has a representing mea-

sure and y(6) admits approximate representing measures.

Lawrence A. FialkowSUNY New [email protected]

MS82

A Semidefinite Approach to the Ki Cover Problem

We apply theta body relaxations to the Ki cover problemand use this to show polynomial time solvability for certainclasses of graphs. In particular, we give an effective relax-ation where all Ki-p-hole facets are valid, addressing anopen question of Conforti, Corneil and Mahjoub. For thetriangle free problem, we also provide an integrality gapfor the first relaxation step as well as negative convergenceresults for the complete graph.

Joao GouveiaUniversidade de [email protected]

James PfeifferUniversity of [email protected]

MS82

Quadratic Forms, Flat Extensions and Applications

We describe the properties of a quadratic form associatedto a linear form over a polynomial ring and show how torecover its decomposition in terms of evaluations and dif-ferentials. Using a general flat extension property, we de-scribe a method for computing such decompositions frommoment matrices. Applications illustrate this approach,including tensor decomposition problems, the solution ofexponential polynomials or the construction of cubaturerules.

Bernard MourrainINRIA Sophia [email protected]

MS82

Shape from Moments

Abstract not available at time of publication.

Dmitrii PasechnikNTU [email protected]

MS83

Torus Invariants and Binomial D-Modules

Binomial D-modules are given by a binomial ideal and ho-mogeneity operators. Combinatorial tools from toric ge-ometry have been successful at analyzing many aspects of

binomial D-modules, which carry a torus action. We willconsider how to interpret taking invariants of D-moduleswith torus actions, with the goal of gaining a new under-standing of the classical hypergeometric systems studiedby, among others, Gauss, Appell, and Lauricella.

Christine BerkeschDuke [email protected]

Laura Felicia MatusevichTexas A&M [email protected]

Uli WaltherDepartment of MathematicsPurdue [email protected]

MS83

Transformations of Hypergeometric Functions

We show how some classical transformation identities in-volving hypergeometric functions arise from symmetries ofthe underlying polytope. This provides us with an algo-rithm to search for such identities in more general situa-tions.

Laura MatusevichTexas A&M [email protected]

Jens ForsgaardStockholm [email protected]

MS83

A Charaterization of F-jumping Numbers

In their study of test ideals and F-jumping numbers,Blickle, Mustata and Smith introduce the D-modules Mα.We prove that the simplicity of these modules determineif α is an F-jumping number and describe an algorithm todecide whether α is an F-jumping number. This is jointwork with Luis Nunez-Betancourt.

Felipe PerezUniversity of [email protected]

MS83

Differential Operators and Invariant Theory

I will describe the role played by differential operators inclassical invariant theory.

Will TravesUS Naval [email protected]

MS83

Explicit formulas for F -pure Thresholds and Rootsof Bernstein-Sato Polynomials

The F -pure threshold is an invariant of singularities incharacteristic p > 0. Mustata, Takagi, and Watanabe

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proved that if a special type of formula for an F -purethreshold can be found, roots of the Bernstein-Sato poly-nomial (in characteristic zero) can be recovered; Budur,Mustata, and Saito found such formulas for monomial ide-als. We find (quite subtle) explicit formulas for F -purethresholds of certain classes of polynomials, and investi-gate their implications. This is joint work with DanielHernandez.

Emily E. WittUniversity of [email protected]

Daniel HernandezMathematical Sciences Research [email protected]

MS84

Combinatorics of Hyperbolic Polynomials

I will talk about hyperbolic polynomials in combinatorics.In particular their relevance to the (a)symmetric simpleexclusion process, and its rich combinatorial structure.

Petter BrandenKTH [email protected]

MS84

Hyperbolicity Cones and Projections of Spectrahe-dra

The Generalized Lax Conjecture states that every hyper-bolicity cone is a section of the cone of positive semidefinitematrices. This central open question has been verified forvarious classes of hyperbolic polynomials and, keeping thebalance, has been disproved for various strengthenings. Inthis talk I will discuss a weaker version, the Projected LaxConjecture: Every hyperbolicity cone is the projection of aspectrahedral cone. While this is certainly weaker than theGeneralized Lax Conjecture, I will show that it holds forthe large (dense) class of smooth hyperbolic polynomials.This is joint work with Tim Netzer.

Raman SanyalFreie Universitat [email protected]

MS84

Determinantal Representations of Projective Hy-perbolic Curves

A real projective curve C in the d-dimensional projectivespace is called hyperbolic with respect to a real d − 2-dimensional projective subspace V (we assume that C andV do not intersect) if for every real hyperplane H throughV, C intersects H in real points only. In this talk wedescribe a Livsic-type determinantal representations of acomplex projective curve, which is given by an element

of∧2

Cd+1 ⊗ Cn×n, and show that similarly to the wellknown case of plane curves (d = 2), a hyperbolic curvealways admits a real symmetric Livsic-type determinantalrepresentation satisfying a positivity condition that givesa witness to hyperbolicity.

Eli ShamovichDepartment of MathematicsBen Gurion University of the Negev

[email protected]

MS84

Hyperbolic Polynomials, Interlacers, and Sums ofSquares

Hyperbolic polynomials are real polynomials whose realhypersurfaces are nested ovals, the inner most of which isconvex. These polynomials appear in many areas of math-ematics, including optimization, combinatorics and differ-ential equations. I’ll give an introduction to this topic anddiscuss the special connection between hyperbolic polyno-mials and their interlacing polynomials (whose real ovalsinterlace the those of the original). This will let us relatethe inner oval of a hyperbolic hypersurface to the coneof nonnegative polynomials and, sometimes, to sums ofsquares. An important example will be the basis gener-ating polynomial of a matroid. This is joint work withMario Kummer and Daniel Plaumann.

Cynthia VinzantUniversity of MichiganDept. of [email protected]

MS84

Stable Polynomials and Sums of Squares in Ma-troid Theory

A multivariate complex polynomial is “stable’ providedthat it never vanishes when all variables take values withpositive imaginary part. Early last decade the relevanceof stable polynomials to matroid theory (and conversely)came to light. (This comes from an abstraction of electri-cal network theory.) Branden subsequently characterizedreal multiaffine stable polynomials by means of a systemof parameterized inequalities, forging another connectionwith semidefinite optimization. Wei and I supplementedBranden’s criterion with another one that is easier to check,and used it to verify the stable (HPP) property for sevensmall matroids, including the Vamos matroid. Brandenthen used this to settle a conjecture of Helton and Vin-nikov. I will sketch these developments, concentrating onmy work with Wei and my plans for extending it.

David WagnerUniversity of Waterloo, CanadaDept. of Combin. and [email protected]

MS85

Accelerating Block Wiedemann Implementation inLinBox

Block Wiedemann algorithm allows to solve sparse linearsystem Ax = b in quadratic time. The key ingredient isto compute the minimal matrix generator of the sequenceS = {UTAiV }i=1..l with matrices U, V having much lesscolumns then A. In this talk, I will discuss the practical ef-ficiency of such approach within the LinBox library. I willshow the benefit of using a sparse U in order to computeS faster. I will also present improvements on minimal ma-trix generator calculation that allow to use fast polynomialmatrix multiplication while preserving the original Copper-smith approach or using an online (relaxed) computationmodel.

Pascal GiorgiLIRMM

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Univ. Montpellier 2, [email protected]

MS85

On the Complexity of Multivariate Interpolationwith Multiplicities and of Simultaneous PolynomialApproximations

The first step in Guruswami and Sudan’s list-decoding al-gorithm amounts to bivariate interpolation with prescribedmultiplicities and degree constraints. Essentially two ap-proaches have been proposed so far: one involves poly-nomial lattices, the other solves so-called (extended) keyequations. Focusing on the latter approach, we extend it tothe multivariate case and show how to reduce this problemto a problem of simultaneous polynomial approximations,which can be solved thanks to existing fast structured lin-ear solvers.

Vincent NeigerLIP, ENS-Lyon, [email protected]

Muhammad F.I. ChowdhuryORCCA, Western University, London ON, [email protected]

Claude-Pierre JeannerodINRIA Rhone-AlpesENS Lyon, [email protected]

Eric SchostComputer Science DepartmentUniv. of Western [email protected]

Gilles VillardCNRS, Ecole Normale Superieure de Lyon, U. [email protected]

MS85

Simultaneous Computation of Row and ColumnRank Profiles

We present a new full pivoting strategy in Gaussian elim-ination, generating a PLUQ decomposition, that makes itpossible to recover at the same time both row and columnrank profiles of the input matrix and of any of its leadingsub-matrices. We propose a rank-sensitive quad-recursivealgorithm that computes the latter PLUQ decompositionof an m × n matrix of rank r in O(mnrω−2) field opera-tions, with ω the exponent of matrix multiplication. Overa word size finite field, this algorithm also improves thepractical efficiency of previously known implementations.

Clement PernetUniiversite de [email protected]

MS85

Lattice Reduction of Polynomial Matrices

Compared the analogous problem on integer matrices, lat-tice reduction of polynomial matrices is considered to beeasy: in this talk I’ll give a survey of existing algorithmsfor lattice reduction of polynomial matrices that demon-strate this. I’ll also discuss some related results, including

derandomization of previously Las Vegas algorithms, tech-niques for handling matrices of arbitrary shape and rank,and algorithms for normalizing a reduced basis to arrive atthe canonical Popov form.

Arne StorjohannD. Cheriton School of C.S.Univ of Waterloo, ON, [email protected]

MS85

Parallel Exact Gaussian Elimination of Rank Defi-cient Matrices

This work deals with the parallelization of Gaussian elim-ination triangular matrix decomposition algorithms over afinite field Z/pZ, even in case of rank deficient matrices.Indeed these kinds of computation are known and exten-sively investigated in numerical computation and the cut-ting of the matrix is done in a static regular way. But inexact computation, particularly in rank deficient case thecutting proves to be more difficult and the size of blocs isdetermined dynamically during the computation. Further-more, since the dimension of blocs is varying, these adap-tive applications have workloads that are unpredictableand change during the computation. Such applicationsrequire dynamic load balancers that adjust the decompo-sition as the computation proceeds. We studied differentapproaches : OpenMP tasks parallelization, pragma ompfor loops and tasks parallelization with KAAPI that havedynamic data flow synchronizations. In this work we showalso the advantages of the latter by comparing our imple-mentations with KAAPI and with OpenMP.

Ziad SultanINRIA/LIG-MOAISGrenoble [email protected]

MS86

An Algorithm forComputing Degrees of Parametrizations of Ellip-tic Curves by Shimura Curves

An elliptic curve, E, over Q of conductor N has a modu-lar parameterization from the modular curve X0(N) to E.The degree of this map is called the modular degree. Gen-eralizing to number fields, we no longer always have modu-lar curves. Takahashi and Ribet use the Jaquet-Langlandscorrespondence to parameterize elliptic curves over Q byShimura curves. I will examine how this generalizes tomodular abelian varieties over number fields.

Alyson DeinesUniversity of [email protected]

MS86

Genus 2 Curves with Good Reduction Away fromp = 3

In 1995, Smart classified all curves over Q of genus 2 whichhave good reduction at primes except p = 2. Chris Ras-mussen and I are working to extend Smart’s work to classifygenus 2 curves with good reduction for all primes except for3 (or 5). This project connects several interesting problemsand techniques, including solving S-unit equations and us-ing the LLL-lattice basis reduction algorithm to reduce the

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size of a very large search space.

Beth MalmskogColorado [email protected]

Chris RasmussenWesleyan [email protected]

MS86

Enumerating Abelian Varieties using MatrixGroups

The Frobenius endomorphism of an abelian variety A/Fq

acts as a symplectic similitude on the torsion subgroupsA[n](F q). In 2003, Gekeler used an equidistribution as-sumption on the elements of GL2(Z/r) to show that thenumber of elliptic curves with certain characteristics is re-lated, via results of Sato-Tate and the class number, tothe Euler factors of the L-function of a quadratic imagi-nary field. By computing the sizes of conjugacy classes ofFrobenius elements in the groups GSp2g and applying atheorem of Everett Howe, we extend Gekeler’s heuristic toabelian varieties with complex multiplication.

Cassie L. WilliamsJames Madison [email protected]

MS86

2-Adic Images of Galois Representations Associ-ated to Elliptic Curves over Q

We give a classification of all possible 2-adic images of Ga-lois representations associated to elliptic curves over Q. Tothis end, we compute the ‘relevant’ tower of 2-power levelmodular curves, develop new techniques to compute theirequations, and classify rational points on these curves.

David [email protected]

Jeremy RouseWake [email protected]

MS87

The Neural Ring: An Algebraic Tool for AnalyzingNeural Codes

Neurons in the brain represent stimuli via neural codes.An important problem confronted by the brain is to inferproperties of represented stimulus spaces using only theintrinsic structure of neural codes. How does the brain dothis? To address this question we define the neural ring, analgebraic object that encodes the full combinatorial dataof a neural code, and show how it can be used to extractrelevant features from the code.

Carina CurtoDepartment of MathematicsUniversity of [email protected]

Vladimir ItskovDepartment of Mathematics

University of Nebraska, [email protected]

Alan Veliz-CubaUniversity of Nebraska [email protected]

Nora YoungsUniversity of [email protected]

MS87

Data Characterization and Identification for Net-work Inference

Recently polynomial dynamical systems (PDSs) have beenused as models of gene regulatory networks (GRNs). Anadvantage of using PDSs as models is that the set of allmodels that “fit” a given data set for a GRN admits an al-gebraic structure, similar to a vector space. However, fewresults other than a criterion by L. Robbiano have beenpublished on the amount and type of data necessary foridentifying PDS models and there are no methods for gen-erating the specific data sets which would unambiguouslyidentify the network. In this talk, necessary and sufficientcriteria will be presented for a data set to uniquely identifyPDSs of GRNs using tools from computational algebra andalgebraic geometry.

Elena S. DimitrovaClemson [email protected]

Brandilyn StiglerSouthern Methodist [email protected]

MS87

Exact Hypothesis Tests for Biological NetworkData

We develop methodology for exact statistical tests of hy-potheses for models of biological network dynamics. Themethodology formulates Markovian exponential familieswith a dispersion parameter phi, then uses sequential im-portance sampling to compute expectations within basinsof attraction and within level sets of a sufficient statis-tic. Comparisons of hypotheses can be done conditionalon basins of attraction. Examples use recent data on can-cer and stomatal regulation.

Ian Dinwoodie, Kruti PandyaPortland State [email protected], [email protected]

MS87

Geometric Approach to Learning Bayesian Net-works with Applications to Biology

This talk will cover descriptions of probabilistic conditionalindependence (CI) models and learning graphical modelswhich has applications in biology (epistasis, gene regula-tory networks, protein signaling, systems biology), Markovrandom processes, probabilistic reasoning, artificial intelli-gence and more. Given observed data, the goal is to findthe CI structure which best explains the data. I will moti-vate the problem with some interesting biological applica-tions. Next, I will quickly overview graphical approaches

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to the description of CI structures. Then, I will describe asuperior algebraic description of CI structures introducedby Studeny et al. which has many elegant properties, suit-able for applications of linear programming methods. Theremainder of the talk will be devoted to linear optimiza-tion approaches to learning Bayesian networks, which arespecial graphical models.

David HawsIBM T.J. Watson Research [email protected]

MS87

Encoding Simplicial Complexes by Neural Net-works

Networks of neurons in the brain encode memories via theirsynaptic connections, yet the relationship between networkconnectivity and encoded memory patterns is still poorlyunderstood, especially in cases where the set of patterns ishighly structured. Motivated by well-studied neural codes,we tackle the problem of encoding simplicial complexes infeedforward and recurrent networks. We find surprisinglystrong results using various geometric and combinatorialtools, including the inverse nerve construction and Cayley-Menger determinants.

Vladimir ItskovDepartment of MathematicsUniversity of Nebraska, [email protected]

Carina CurtoDepartment of MathematicsUniversity of [email protected]

Chad GiustiUniversity of [email protected]

PP1

Mallows Mixture Model and Its Vanishing Ideal

The Mallows model is simple statistical model for rankeddata. We study a mixture model based on the originalMallows model with two mixture components. We examinethis model from and algebraic geometry standpoint. Wethen fully characterize the degree 1 and degree 2 generatorsof the vanishing ideal for this mixture model when thereare two mixture components.

Brandon W. Bock, Seth SullivantNorth Carolina State [email protected], [email protected]

PP1

Geometrical Algorithms for Civil Helicopter (CH)Rotor-Blades Instantaneous Rotation Center De-termination in Deformable/Turbulence ConditionsUsing Numerical Reuleaux Method (MRM)

Civil Helicopter (CH) rotor blades are usually manufac-tured/designed with deformable materials.Their movementis governed mainly by four angles,rotor, flapping,laggingand torsional ones.In severe turbulence/windy conditionsstructural dynamics of blades could become deformedand/or damaged (blade flapping/lagging bending moment

and torsional deflections angles could appear and be signif-icant,we do not analyze torsion). Then,there are changesin the IRC of rotor.Dynamically,this IRC biased posi-tion creates a variation chain in the Aerodynamics con-ditions of CH.We carry out a primary approximation toanalyze geometrically/numerically the IRC variation withthe NRM.Initial algorithms are developed with numeri-cal simulations.Aerodynamics consequences derived fromthis data are physically analyzed. F Casesnoves MSc MD.ASME (American Society of Mechanical Engineering. Indi-vidual Researcher Membership) References (Selected) The-ory and Primary Computational Simulations of the Numer-ical Reuleaux Method (NRM),Casesnoves,F.InternationalJournal of Mathematics and Computation(http://www.ceser.in/ceserp/index.php/ijmc/issue/view/119).Volume13,Issue Number D11.Year 2011.

Francisco CasesnovesAmerican Mathematical Society (Individual ResearcherMember)[email protected]

PP1

Cad Numerical Vertebral Surface Refinements andGeometrical Data Development for Surgical De-vices Design

Lumbar Vertebra CAD constitute an useful method bothfor anatomical study of bone shape/structure and surgi-cal instrumentation design.We present a polynomial dis-cretization and refinement derived from previous rougherpolynomials fits of vertebral surface.Non-Linear Opti-mization Methods/specific-software were used for the ob-jective.So we can get a more accurate and real sur-face representation.It is shown a group of geometrical-mathematical formuli that can be obtained from re-finement data.Industrial applications of these techniquesare related to manufacturing data/design of lumbarspine instrumentation and artificial implants. F Cas-esnoves MSC MD ASME (Individual Researcher Affilia-tion) Refs:’Computational Simulations of Vertebral Bodyfor Optimal Instrumentation Design’.Casesnoves,F. ASMEJOURNAL OF MEDICAL DEVICES.May 23,2012. J.Med. Devices/June 2012/Volume 6/Issue 2/021014 (11pages)http://dx.doi.org/10.1115/1.4006670

Francisco CasesnovesAmerican Mathematical Society (Individual ResearcherMember)[email protected]

PP1

Distance-Based Phylogenetic Algorithms Around aPolytomy

Distance-based phylogenetic algorithms map an arbitrarydissimilarity map representing biological data to a treemetric. The set of all dissimilarity maps is a Euclideanspace containing as a subset the space of all tree met-rics, which is a polyhedral fan and a tropical variety. Westudy the behavior of distance-based tree reconstructionalgorithms such as UPGMA and Neighbor-Joining near apolytomy using the polyhedral geometry of the subdivisionof Euclidean space induced by the algorithm.

Ruth E. Davidson, Seth SullivantNorth Carolina State [email protected], [email protected]

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PP1

Perturbed Regeneration for Finding Singular Solu-tions

Homotopy continuation is used to find solutions to systemsof polynomial equations. Numerical algebraic geometry, arelatively young and quickly growing field, utilizes contin-uation methods to find and classify isolated and positive-dimensional solutions. We will present some developmentson approximating singular solutions efficiently. Our tech-nique is especially useful for certain large problems. Inparticular, perturbed regeneration will find such solutionswithout the cost of deflation, which is necessary in stan-dard regeneration to regularize them.

Brent R. DavisColorado State [email protected]

Daniel J. BatesColorado State UniversityDepartment of [email protected]

Chris PetersonColorado State [email protected]

Eric HansonDepartment of MathematicsColorado State [email protected]

David EklundColorado State [email protected]

PP1

A Fractal Model of Time

In traditional quantum mechanics, time is considered tobe an observable; no time operator has been established.One of the most familiar results of quantum mechanics isthe quantization of energy. Inherent in Planck’s constant,with its units of time multiplied by energy, lies the conceptof the quantization of time. For a certain class of statefunctions, time can be quantized in time-energy quanta,based upon the existence of a quantum-mechanical timeoperator. This time operator would be a function of theHamiltonian, the energy operator, and yield four results foreach energy level. The applications of this model includehigh energy fusion and cosmology; however, our results areentirely theoretical, and have to be confirmed with a realsystem, such as a hydrogen atom. The question of whethertime can be quantized or no is more than academic. If suchquantization does indeed exist, it might serve as a basis fora unified field theory.

Jorge Diaz-CastroUniversity of P.R. School of LawUniversity of P.R. at Rio [email protected]

PP1

A Set of Polynomial Systems from Population Bi-ology

An important problem in population biology is determin-

ing equilibria of haplotype frequencies. It concerns a squareparametrized polynomial system of dimension N equal to apopulation’s number of haplotypes. Bertini, with a linear-product formulation, captures the generic solutions with-out paths diverging. The disparity between the linear-product count and the total degree, 2N − 1 and 3N , re-spectively, grows rapidly with N . Often most haplotypesare lethal; sparse linear algebra provides an efficient ap-proach to solving such systems.

Jesse W. DrendelColorado State [email protected]

PP1

Parameterized Polynomial Systems and NumericalAlgebraic Geometry

Given a parameterized system of polynomial equations, itis natural to consider how different choices of parametersaffect the zero set. This poster will highlight some resultsrelating to this question and methods involving parame-terized polynomial systems. Specifically, we will considerusing Numerical Algebraic Geometry (NAG) to study pa-rameterized systems. We will give a high-level overview ofperturbed regeneration, fiber products for finding excep-tional parameter values, and solving systems over (semi)-algebraic sets of parameters.

Eric HansonDepartment of MathematicsColorado State [email protected]

Dan BatesColorado State [email protected]

PP1

A Hybrid Numerical-Symbolic Algorithm for Com-puting the Solutions of Fewnomial Systems

Gale duality is an isomorphism between solutions of a poly-nomial system and those of its Gale dual system of masterfunctions. This transformation, known as the Gale trans-form, is completely symbolic. The solutions of the masterfunctions can be computed (by previously known methods)and an unwrapping algorithm can be applied to obtain thesolutions of the original fewnomial system. This posterfocuses on the construction and optimization of the dualsystem.

Matthew Niemerg, Dan BatesColorado State [email protected], [email protected],

Jon HauensteinNorth Carolina State [email protected]

Frank SottileTexas A&M [email protected]

PP1

Machine Learning for Phylogenetic Invariants

We build on Eriksson and Yaos work, which used machine

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learning to optimize the power of phylogenetic invariantsto reconstruct evolutionary trees. While previous work fo-cused on selecting a good set of invariants for the con-struction of quartet trees, we extend this work to treeswith more taxa. In addition to the invariants, we includeinequalities arising from the study of the real points on thephylogenetic variety into the metric learning algorithm.

Hannah M. Swan, Joseph P. Rusinko, Emili PriceWinthrop [email protected], [email protected],[email protected]