The Ohio State University Columbus, OH Abstracts of Presentations · 2010. 2. 17. · Categorification of Krushkal’s Polynomial Sam E Calisch ([email protected]) Louisiana State

The sixth annual

Young Mathematicians Conference

August 28 th - 30 th, 2009

The Ohio State UniversityDepartment of Mathematics

Columbus, OH

Abstracts of PresentationsAbstracts of PresentationsAbstracts of Presentations

Supported by NSF Grant DMS-0841054

2

Plenary Lectures

Finding stability in chaos

Laura DeMarcoUniversity of Illinois at Chicago

Abstract of Lecture: I will explain the concept of a dynamical system, with examples from nature andsimple mathematical models. The mathematical challenge is to find and describe the stable structuresinside these systems. Stability of a system means its long-term behavior is unchanged by small-scaledisturbances. The examples produce beautiful fractal pictures which are often easy to generate with acomputer.

You can’t hear the shape of a drum

Carolyn GordonDartmouth College

Abstract of Lecture: In spectroscopy, one attempts to determine the chemical composition of an objectsuch as a star from the characteristic frequencies of emitted light. Analogously, one asks the extent towhich the shape of a vibrating membrane, e.g., a drumhead, is encoded in the characteristic frequenciesof vibration. In the words of Mark Kac, ”Can one hear the shape of a drum?” We will look both atpositive answers, e.g., a round drum has no imposters, and negative answers. In particular, we willconstruct exotic shaped sound-alike drums. We will also listen to Dennis DeTurck’s simulation of thesounds of these drums.

Diophantine Analysis of Apollonian Packings

Peter SarnakPrinceton University & I.A.S.

Abstract of Lecture: The curvatures of the circles in certain Apollonian packings are all integers.Afterreviewing the plane geometry and group theory that underlies this remarkable fact, we will examine thediophantine properties of these integral packings. As is the case with most diophantine problems, the”obvious” questions that present themselves are difficult to answer but one can make progress.

3

Student Presentations(Alphabetically by last name of primary presenter)

Harmonic Maps on Cayley Graphs and Compactification

Robert J. Abramovic ([email protected])

Canisius College [Mentor:Stratos Prassidis]

Abstract of Report Talk: We introduce the analogue of harmonic maps on graphs and give some resultsthat parallel ones from Complex Analysis. On Cayley graphs, we compare general harmonic mapsto those induced by group homomorphisms. In particular, we prove that the free group and abeliangroups on more than one generator admit harmonic functions that are not induced by homomorphisms.Furthermore, we introduce the Floyd compactification of a geodesic space, particularly that of a graph.Our goal is to compare the Floyd boundary to other graph boundaries up to homeomorphism. Inthis context, the Dirichlet Problem naturally arises. It asks for conditions for extending a continuousfunction on the boundary to a harmonic map on the entire space. The results of Anders Karlsson statethat the solvability of the Dirichlet problem depends on the size of the Floyd boundary. We examinethe connections of this result to properties of the differentials of harmonic maps and give applicationsto the hyperbolic groups of M. Gromov. [AR24143957]

Received: August 7, 2009

Visualizing Algebraic Surfaces

Jennifer E Bonsangue ([email protected])

CSU Channel Islands [Mentor:Ivona Grzegorczyk]

Abstract of Poster Presentation: Algebraic surfaces in Rn and Pn are the collection of points satisfyinga finite number of polynomial equations of the form p(a1, a2, . . . , am) = 0 . They may be smooth,singular, connected, or disjoined. They may have symmetries and group actions, or special curves lyingon them. The general problem of classifying algebraic surfaces for any degree is very difficult. However,new technology helps us to better visualize and describe their properties. We study in depth familiesof quadratic and cubic surfaces, and analyze their features. We find that there are only 16 possibledistinctive types of degree two surfaces. We provide a reference guide for these surfaces, analyzingfeatures such as symmetries, singularities, self-intersections, etc. Then, we describe some degree threesurfaces in terms of the aforementioned features, as well as exploring the straight lines that lie on degreethree surfaces. [BJ02160403]


4

2-Selmer groups of quadratic twists of elliptic curves

George A Boxer ([email protected])

Peter Z Diao ([email protected])

University of Wisconsin, Madison [Mentor:Ken Ono]

Abstract of Report Talk: In our work we investigate families of quadratic twists of elliptic curves. Givenan elliptic curve E/Q let E(d) denote the quadratic twist of E by d and let LBSD(E)(E/Q, 1) denote thealgebraic part of the central L value predicted by the Birch and Swinnerton-Dyer conjecture. Addressinga speculation of Ono, we identify a large class of elliptic curves, which we call good elliptic curves, forwhich we can show that the parity of LBSD(E(d), 1) is essentially that of the coefficient a|d| of L(E/Q, s).More precisely we prove that if E is a good elliptic curve and d is a square free integer, then we have{

LBSD(E(d), 1) ≡ a|d| (mod 2) if (d,∆) = 1LBSD(E(d), 1) ≡ 0 (mod 2) if (d,∆) > 1,

where ∆ is the discriminant of E.We achieve this by controlling the 2-Selmer rank of E(d) (a la Mazur and Rubin) when the Tamagawa

factors appearing in LBSD(E(d), 1) do not already dictate the parity. [BG03001509]


Multivariate Tutte polynomial of graphs and HOMFLYPT polynomial of links.

Robert M. Bradford ([email protected])

Ohio State [Mentor:Sergei Chmutov]

Abstract of Report Talk: Theorems of F. Jaeger and L. Traldi state that the HOMFLYPT polynomial ofa link constructed from a planar graph by putting a double crossing tangle on every edge of the graphcan be obtained by a suitable substitution to the multivariate (weighted) Tutte polynomial of the graph.The class of links covered by these theorems is very restricted. Using a double weighted multivariateTutte polynomial firstly we can reformulate the theorem in more elegant form. Secondly we generalizethe theorem to a larger class of links using either horizontal or vertical chain of arbitrary even numberof crossings. [BR01232132]

[Joint with Minh Nguyen, Michael Dworkin, Kaushik Bagchi] Received: August 7, 2009

Categorification of Krushkal’s Polynomial

Sam E Calisch ([email protected])

Louisiana State University [Mentor:Neal Stoltzfus]

Abstract of Report Talk: Khovanov homology not only provides a strictly stronger link invariant thanthe Jones polynomial, but also establishes an entirely new way of thinking about this well-known invari-ant. Since Khovanov’s paper appeared, there has been a flurry of categorifications of other invariantswith state sum formulae, including the chromatic polynomial, the Tutte polynomial, and the Bollobas-Riordan polynomial. This project continues the trend, constructing a categorification of the Krushkalpolynomial of ribbon graphs. This Krushkal homology relies on the duality inherent in the Krushkalpolynomial and exhibits a long exact sequence giving rise to the contraction-deletion property of thepolynomial. [CS02134455]


5

A Census of Two Dimensional Toric Codes over Galois Fields of Sizes 7, 8, and9

Alejandro U Carbonara ([email protected])

Berkeley [Mentor:John Little]

Abstract of Report Talk: J. Hansen introduced toric codes using the geometry of polygons in R2 and theconvex polytopes P in Rm more generally. But collections of points more general than the sets P ∩Zmcan also be used to define generalized toric codes. In our research using the Magma computer algebrasystem, we search to find the generalized toric codes with the best parameters for some dimensions overthe Galois fields of size 7, 8 and 9. Our work produced new toric codes at least one of which possessesbetter minimum distance than that of previously known codes. Improving the minimum distance of acode allows for the correction of more errors which occur during transmission or storage and hence theimprovement gives us a better code for use in communication. [CA02104014]

[Joint with Juan Murillo, Abner Ortiz] Received: August 8, 2009

The Topology of Spaces of Triads

Michael J Catanzaro ([email protected])

Wayne State University [Mentor:Robert Bruner]

Abstract of Report Talk: We may think of triads in different musical scales as triangles, with each note inthe triad representing a vertex. Using this type of visualization, we may attach triads (or triangles) toone another along edges if they share two notes. After attaching all the different triads in our musicalscale together, we may ask what types of topological spaces are formed. In this presentation, we classifyall such spaces of triads.

It is well known that major and minor triads in the diatonic scale form a torus. In fact, we showthat the the most likely space of triads is a torus (or several copies of a torus). A natural choice oftriads in the whole tone scale turns out to form a space made of three tetrahedra joined circularly alongopposite edges. Other connected spaces which occur are cylinders and Mobius bands.

It turns out that our spaces may be made up of components not attached to one another. When thishappens, all components of the space must be identical. Really, this means that there are disjoint setsof triads with no shared notes, running ‘in parallel’ to, but unrelated to, one another. So long as thesteps between the notes in the triads have no common divisor, our space will be connected, i.e. it willcontain only one component. This implies that any triad can be converted to any other by a sequenceof L, R, or P type transformations, which change only one note at a time. For example, the space ofmajor and minor triads in the diatonic scale is connected (there are 3, 4, and 5 steps between the notesin these types of triads).

The mathematical tools required include a bit of number theory and algebraic topology. [CM29143421]


6

Beautifully Ordered Balanced Incomplete Block Designs

Hau Chan ([email protected])

College of Charleston [Mentor:Dinesh Sarvate]

Abstract of Report Talk: A Balanced Incomplete Block design, BIBD(v, k, λ), is a collection of k-subsets(called blocks) of a v-set such that each pair of distinct points occurs in exactly λ blocks where k<v. Ifeach of the blocks of a BIBD(v, k, λ) is ordered such that for any k1 indices i1, i2, . . . , ik1 the sub-blocks{ai1 , ai2 , . . . , aik1

} of all ordered blocks {a1, a2, . . . , ak} of the BIBD(v, k, λ) form a BIBD(v, k1, λ1) thenwe say that the collection of ordered blocks gives a Beautifully Ordered Balanced Incomplete BlockDesign, BOBIBD(v, k, λ, k1, λ1) where 2 ≤k1≤k − 1. We prove that necessary conditions are sufficientfor the existence of BOBIBD with block size k=3 and k=4 for k1=2 except possibly for eleven exceptions.Existence of BOBIBDs with block size k=4 and k1=3 is demonstrated for all but one infinite familyand the non-existence of BOBIBD(7, 4, 2, 3, 1), the first member of the unknown series, is shown. Thetechniques used to construct families of designs are standard but interesting ad hoc methods are alsoused to construct specific designs, for example using counting and pattern recognition. There are severaldesigns and design families whose existence is still an open problem and hence there is scope for furtherinvestigation. [CH10173201]


The Finte Weyl Group of Type Bn as the Automorphisms of the n-Cube: Iso-morphism and Conjugacy

David D Chen ([email protected])

University of Georgia [Mentor:Leonard Chastkofsky]

Abstract of Report Talk: The Weyl group associated to a root system of type Bn and the group ofgraph automorphisms of the n-cube Aut(Qn) are known to be isomorphic to Zn2 o Sn. We provide adirect isomorphism between them via correspondence of generators. Geck and Pfeiffer have provided aparametrization of conjugacy classes and an algorithm to compute standard representatives. We believewe have a more transparent account of conjugacy in the Weyl group by looking at Aut(Qn). We givea complete description of conjugacy in the automorphism group. We also give an algorithm to recovera canonical minimal length (in the Weyl group sense) representative from each conjugacy class, and analgorithm to recover that same representative from any other in the same conjugacy class. Under thecorrespondence with the Weyl group, this representative coincides precisely with the minimal lengthrepresentative given by Geck and Pfeiffer, leading to an easier derivation of their result. [CD31201019]


7

Matrices and Tensors of Small Norm

Elizabeth M Collins-Wildman ([email protected])

Matthew J Hoffman ([email protected])

Kent State University [Mentor:Andrew Tonge]

Abstract of Report Talk: For matrices and tensors with ±1 entries, norm estimates as a function of theorder are important tools in functional analysis and operator theory. The operator norm of an order nmatrix or tensor can never be smaller than

√n; we determine sharp lower bounds.

Using properties of Hadamard matrices (±1 entry matrices H of order n such that HHT = nI) alongwith combinatorial bounding arguments, we prove that for matrices of order 4n+ 1, 4n+ 2, 4n+ 3, and4n+ 4 with ±1 entries the minimal operator norm is

√4n+ 4 provided that there exists an Hadamard

matrix of order 4n + 4. If the Hadamard Conjecture is correct, there exist Hadamard matrices of allorders divisible by 4. This would give sharp lower bounds on the norms of ±1 entry matrices of allorders.

We use an inductive argument to prove that for ±1 entry tensors of order 2, the minimal norm is√2, regardless of the dimension. Finally, we give examples of ±1 entry tensors of order n 6= 2 with

norm√n. [CE31140257]


Factorial and Noetherian Subrings of Power Series Rings

Damek Davis ([email protected])

University of California, Irvine [Mentor:Daqing Wan]

Abstract of Report Talk: Let F be a field. We show that certain subrings contained between F [X] =F [X1, . . . , Xn] and F [X][[Y ]] = F [X1, . . . , Xn][[Y ]] have Weierstrass factorization, which allows us todeduce both unique factorization and the Noetherian property. These subrings are obtained fromelements of F [X][[Y ]] by bounding their total X-degree above by a positive, real-valued monotonic upfunction, λ, on their Y -degree. These rings arise naturally in studying p-adic analytic variation of zetafunctions over finite fields. Future research into this area may study more complicated subrings in whichY = (Y1, . . . , Yp) for p ≥ 1, or for which there are multiple degree functions, λ1, . . . , λr, r ≥ 1. [DD03030218]


8

Effective nonvanishing of canonical Hecke L-functions

Peter Z Diao ([email protected])

George A Boxer ([email protected])

University of Wisconsin [Mentor:Riad Masri]

Abstract of Report Talk: Let 3 < p ≡ 3 (mod 4) be a prime and Kp = Q(√−p). For each such imaginary

quadratic field, there are families of weight 2k−1 “canonical” Hecke characters ψ ∈ Ψp,k. Gross defineda family of elliptic curves Ap that have the factorization formula

L(Ap, s) =∏

ψ∈Ψp,1

L(ψ, s).

The nonvanishing of the central value L(Ap, 1) is related to the Birch and Swinnerton-Dyer conjecture forelliptic curves. Kim, Masri, and Yang have found further arithmetic applications for the nonvanishingof the central values of “canonical” Hecke L-functions for higher weight k > 1. Using the work ofRodriguez-Villegas and Zagier, we show that there are nonvanishing central values provided that p ≥6.5(k − 1)2 and the root number is 1. Moreover, we deduce a corollary that implies that in the case2k − 1 is relatively to prime to the class number of Kp, all such central values vanish in our effectiverange of p. [DP03104204]


Laplacians on the Airplane, Basilica in Rabbit and Dendrite Julia Sets

Chu Yue (Stella) Dong ([email protected])

Cornell University [Mentor:Strichartz Robert]

Abstract of Poster Presentation: We study Laplacians on the Airplane, Basilica in Rabbit and DendriteJulia sets, extending earlier research on Laplacians on the Basilica and Rabbit Julia sets. It is difficultto study Laplacians on these Julia sets directly, so we apply an onto (Riemann) map to map the pointsof each Julia set to a unit circle. We investigate the equilibrium measure and the energy on the circle,and then transform them back to the corresponding Julia set. The way that we calculate the changeof energy from one level to the next is based on the subdivision rules we define for each Julia set. Weimplement an algorithm for the Laplacian, and explicitly compute the spectrum and the eigenfunctionsand study their properties. In order to get accurate results for the spectrum of the Laplacian, we applyboth finite different method and finite element method and compute their average. Since the structureof the Basilica and Rabbit Julia sets is similar to that of Airplane and Dendrite Julia Sets respectively,we compare the properties of eigenfunctions and the spectrum of the Laplacian on them and expect toobtain some same structures. We did observe certain similarities, but we also found a few interestingphenomenons which did not occur in the earlier research. [DC03115224]


9

Relative Tutte polynomial and virtual links

Yishun Dong ([email protected])

OSU [Mentor:Sergei Chmutov]

Abstract of Report Talk: A relative Tutte polynomial of a graph with respect to a subset of edges wasintroduced (in a language of matroids) by S. Chiken in 1989. Recently it was rediscovered in a preprintof Y. Diao and G. Hetyei where it was applied to a generalization of Thistlethwaite’s theorem to virtuallinks. We will present a corank-nullity formula for the relative Tutte polynomial and use it to simplifythe proof of the Diao–Hetyei theorem. Then we will discuss how other invariants of virtual links can beobtained by suitable specialization of the relative Tutte polynomial. [DY31150539]

[Joint with Jeffries Jack, Carnovale Marc] Received: August 12, 2009

Conjectures on Twisted Kloosterman Sums

Anastassia Etropolski ([email protected])

University of Wisconsin - Madison [Mentor:Ken Ono]

Abstract of Report Talk: N. Katz and R. Evans conjectured some evaluations of twisted sums of tracesof the nth symmetric power of twisted Kloosterman sheaves, Tn. We prove some of these explicitevaluations for n = 4 and a cubic character. R. Evans has also conjectured relationships betweenthese twisted Kloosterman sheaf sums and the coefficients of certain modular forms. For three of thesemodular forms, each of weight 3, Evans conjectured that the coefficients are related to the squares ofthe coefficients of weight 2 modular forms. We prove these relationships using the theory of complexmultiplication. [EA03134042]

[Joint with Jeremy Booher, Amanda Hittson] Received: August 8, 2009

Computing the number of complete Sperner families of linear chromatic number2

Maxwell S Forlini ([email protected])

Miami University [Mentor:Reza Akhtar]

Abstract of Poster Presentation: A Sperner family on a set S is a collection of subsets of S, no two ofwhich are contained in each other; it is called complete if the union of the subsets in the family equalsS. The problem of enumerating the Sperner families on a given (finite) set remains open and datesback to Dedekind. Recently, Yusuf Civan introduced the notion of the linear chromatic number of aSperner family as a generalization of an analogous concept for simplicial complexes. For a Spernerfamily S on a set S, write Sx for the collection of sets in S which contain a particular element x ∈ S.Then the linear chomatic number is the smallest positive integer k for which there exists a surjectivemap f : S → {1, . . . , k} with the property that if f(x) = f(y), then Sx ⊆ Sy or Sy ⊆ Sx. We give analgorithm leading to a formula for the number of complete Sperner families of linear chromatic number2 on any finite set. [FM03151524]


10

The Relative Class Number for Real Quadratic Feilds

Amanda Furness ([email protected])

Mount Holyoke University [Mentor:Giuliana Davidoff]

Abstract of Report Talk: We examine relative class numbers, associated to class numbers of orders inquadratic fields Q(

√d) with field discriminants d or 4d, for d > 0 and square -free. These are subrings of

the full ring of integers that can be shown to correspond to discriminants d0 = f2d of indefinite binaryquadtratic forms. The relative class number is the quotient:

Hd(f) = h(f2d)/h(d),

and it is not known if for every d there exists an f for which this ratio is one. Hd(f) depends ona function φ(f), the minimal power of the fundamental unit, εd, that lies in the order. φ(f) is notmultiplicative, but it does have some nice properties over the prime numbers. We give estimates onthe growth rate, the proportion of small values, and other properties of the function Hd(p) for certainfixed d and increasing values of prime p. Our results take into account examples of discriminants forwhich the sign of the norm of εd is positive and for which it is negative, so we are able to measure thesensitivity of Hd(p) to that property of the quadratic field. [FA03152900]

[Joint with Charlie Marshak, Elliot Lee, Katherine Poulson, and Hudson Harper] Received: August 9, 2009

(0,2)-graphs From Roots Systems Using Tableaux

Alexander C Garver ([email protected])

University of Georgia [Mentor:Leonard Chastkofsky]

Abstract of Report Talk: A (0,2)-graph is a connected graph Γ where any pair of vertices a, b ∈ V (Γ) haveeither 0 or 2 common neighbors. Known examples of (0,2)-graphs include hypercubes and incidencegraphs of projective planes. Recently a construction of (0,2)-graphs from roots systems associated withlie algebras of type An has been found. Given a target vector β, form a graph denoted by Γ(An, β)consisting of the subsets of the set of positive roots of An such that the sum of the roots in that subsetequals β. For example, Γ(An, α0) where α0 = α1 + α2 + . . . + αn is associated to the graph of thehypercube. It is natural to consider other cases with target kα0 for an integer k > 1. We have Weyl’sdimension formula for counting the vertices in these graphs, but this formula is difficult to use. We showhow tableaux can be used to represent the vertices of these graphs and show how this representation canbe used to count the vertices of Γ(An, kα0) using an inductive method. In particular, we have found aclosed recursive formula for the vertices of Γ(An, 2α0) and now we are trying to find a recursive formulathat will work for Γ(An, 3α0) with the intention of generalizing to kα0. I will discuss our progress onthis construction. [GA16162656]


11

Triangulations of n-gons and their corresponding monomial ideals

Jessica M. Glover ([email protected])

Mount Holyoke College [Mentor:Jessica Sidman]

Abstract of Report Talk: Consider a triangulation of a polygon on n vertices, and its correspondinggraph. Let R be a polynomial ring in n variables. We study the ideals generated by the products ofthree variables in R corresponding to the vertices of each triangle in our triangulation.

We know every ideal in a polynomial ring is finitely generated, and unlike a basis for a vector space,there are nontrivial relations on the generators of an ideal. Viewing these ideals through the lensof algebraic geometry, we study all of the relations on the generators, through which we can gleangeometric information such as the dimension of the ideal from the numerical invariants found in theideal’s minimal free resolution.

For some of these ideals corresponding to our family of graphs, when we construct a map whosekernel is the ideal, we find that its generators can be determined by combinatorial features of the graph.We were able to prove results about these generators after computing the resolutions of many ideals inour family using Macaulay 2 software.

In this talk, we give examples of these ideals, and show how the relations on their generators can bedescribed combinatorially. We present an algorithm we developed for finding all of the non-isomorphictriangulations of the graph of an n-gon based on Wagner’s diagonal flip operation, and make conjecturesregarding the geometric properties of map generators for other triangulations. [GJ31153532]

[Joint with Rachel Cranfill] Received: August 10, 2009

Towards an ”average” version of Birch and Swinnerton-Dyer

John W Goes ([email protected])

Williams College [Mentor:Steven Miller]

Abstract of Report Talk: The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associatedL-function L(s, E). Previous investigations have focused on bounding how far we must go above thecentral point to be assured of finding a zero; in particular, Mestre showed the first zero occurs byO(1/ log logCE), where CE is the conductor of E, though we expect the correct scale to study thezeros near the central point is the significantly smaller 1/ logCE . We significantly improve on Mestre’sresult by averaging over a one-parameter family of elliptic curves and using some results from randommatrix theory and Fourier analysis, obtaining better bounds on the order of 1/ logCE as expected. Ourresults may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyerconjecture. These methods are then applied to additional families of L-functions. [GJ31151510]


12

Positive Hankel Matrices are Sums of Squares

Ali Gokal ([email protected])

Kailee A. Gray ([email protected])

University of Iowa [Mentor:Raul Curto]

Abstract of Report Talk: We will show that any given n× n positive semi-definite Hankel matrix Bk ofrank k where the first entry is 1 can be written as a convex combination of k rank 1 Hankel matrices.These in turn can be written in the form

[1 a a2 · · · an

]t [1 a a2 · · · an]

where a ∈ R. Morespecifically, we will show Bk =

∑kj=1 cj

[1 vj v2

j · · · vnj]t [1 vj v2

j · · · vnj]

where vj ∈ R, 0 <c1, . . . , ck < 1 and c1 + · · · + ck = 1. The tools used to reach this result will include a representationtheorem for positive polynomials and Smul’jan’s theorem for positive semi-definite matrices. [GA03134808]

[Joint with Jonah Ellman (Grinnell College)] Received: August 11, 2009

Nested Traps in Parity Games

Andrey V Grinshpun ([email protected])

Andrei Tarfulea ([email protected])

Cornell University [Mentor:Sasha Rubin]

Abstract of Report Talk: Parity games are games of infinite duration played on finite graphs; they areof particular significance in Computational Complexity as they remain one of the few natural problemsknown to be in NP ∩ co-NP but not known to be in P. To every parity game we associate a naturalnumber that we call its trap depth. This parameter measures how players can trap each other andis defined purely in terms of the structure of the graph; in particular, it makes no mention of gametheoretic notions such as plays and strategies. We show that the trap-depth of every game is boundedby the size of the game. Moreover, we supply an algorithm that runs in polynomial time which solvesparity games of trap-depth 1. We hope that this work can be extended to find a polynomial timealgorithm for parity games of arbitrary trap-depth. [GA31114216]

[Joint with Pakawat Phalitnonkiat, Alex Kruckman, Ben Zax, John Sheridan, James Worthington] Received: August 8, 2009

Manifolds with Density as Quotients of Riemannian Manifolds

Nate R Harman ([email protected])

Williams College [Mentor:Frank Morgan]

Abstract of Report Talk: A manifold with density is a Riemannian manifold equipped with a positivedensity function that weights both volume and hypersurface area. One place these come up naturallyis as quotients of Riemannian manifolds by a subgroup of their isometry group. First, we use thisinterpretation to solve the isoperimetric problem in certain manifolds with density by looking at theproblem in a suitable Riemannian manifold modulo symmetry. Second, we discuss conditions underwhich solving the isoperimetric problem in a Riemannian manifold can be reduced to solving it in amanifold with density of lower dimension. [HN27070548]

[Joint with David Thompson, Alexander Diaz, Sean Howe] Received: August 10, 2009

13

Asymptotics for Class Numbers and Strict Class Numbers for FundamentalDiscriminants

Hudson Harper ([email protected])

Katherine R Poulsen ([email protected])

Mount Holyoke College [Mentor:Giuliana Davidoff]

Abstract of Poster Presentation: Take h(d), h+(d) (respectively) to be the class number, strict classnumber associated to an integer d. The class number is typically defined in the context of equivalenceor strict equivalence classes of ideals in the field Q(

√d), for d a fundamental discriminant. In its original

context h+(d), for d ≡ 0, 1 mod 4 and not a perfect square, is the number of strict equivalence classesof binary quadratic forms of discriminant d. A major open question, conjectured by Gauss, asks, ford > 0, if h(d) = 1 infinitely often as d→∞. Using genus theory, one can greatly restrict the number ofclassical discriminants for which this can possible occur. We investigate the question for such candidatesamong fundamental discriminants, and derive asymptotics for h(d), h+(d), and related functions in therange d ≤ 5.2× 107. [HH03150101]


Stochastic difference equation with dependent coefficients

Diana Hay ([email protected])

Vivek Hirpara ([email protected])

Iowa State University [Mentor:Alexander Roitershtein]

Abstract of Report Talk: We consider the stochastic difference equation Rn = Qn + MnRn−1, n ∈ Z,for Markov-dependent sequences (Qn,Mn) ∈ R2. Under natural conditions on the coefficients, andregardless of the initial value R0, the series (Rn)n≥0 converges in distribution to its long-term equilibriumR. Furthermore, R is the unique stationary solution of this equation. In particular, if R0 = R, then Rnis distributed the same as R for all n ≥ 0.

We study the asymptotic behavior of the distribution tails of R and show that both P (R > t) andP (R < −t) are regularly varying at infinity. In previous work, two different set-ups have been explored.Assumptions of Kesten [73] and Goldie [91] imply that Mn is dominant in determining the asymptoticbehavior of the tails, whereas in Grincivicius [75] and Grey [94] Qn is dominant; all assuming that thepairs (Qn,Mn) form an i.i.d sequence.

We generalize the results of Grincivicius [75] and Grey [94] to a Markovian set-up. Using a Markovianrepresentation of certain chains with complete connections (Lalley [86]), we obtain an extension of thework of Grincivicius [75] and Grey [94] also to coefficients (Qn,Mn) induced by such processes. Thisextends previous work done for independent identically distributed coefficients to a more general setup,which is more realistic in some real-world applications and is desirable in certain theoretical models.[HD31165919]

[Joint with Arka Ghosh, Jiyeon Suh, Reza Rastegar, Ashley Schulteis] Received: August 12, 2009

14

The Dynamics of an Ammann Aperiodic Protoset

Vivian Olsiewski Healey ([email protected])

Notre Dame [Mentor:Arlo Caine]

Abstract of Report Talk: The Penrose aperiodic protoset is a set of two triangular tiles with accompa-nying adjacency rules that tile the plane nonperiodically. These two tiles can form uncountably manyincongruent nonperiodic tilings of the plane which are all locally isomorphic. A composition and re-scaling process gives rise to a dynamical system on the set of all Penrose tilings. A modification of thePenrose tiles discovered by Robert Ammann, which consists of three prototiles with no adjacency rules,has many properties analogous to the Penrose tiles. My original research explores the combinatorialproperties of this interesting set of tiles and the dynamics of an analogous re-composition and re-scalingprocess. In particular, by taking limits, I construct an uncountable family of incongruent nonperiodictilings with three proto-tiles which are all locally isomorphic and whose diffraction patterns displaysharp Bragg peaks and 5-fold symmetry. (Research started at Canisius College REU summer 2008 andcurrently continued at Notre Dame under Professor Arlo Caine.) [HV02225627]


Isoperimetric Regions in Planar Sectors with Density rp

Sean P Howe ([email protected])

Williams College [Mentor:Frank Morgan]

Abstract of Report Talk: A manifold with density is a manifold with a positive function weighting bothvolume and perimeter. They have received increasing attention in recent years, most notably appearingin Perelman’s proof of the Poincare conjecture. Building on the work of Dahlberg et al. in the plane,we consider the isoperimetric problem in planar sectors with radial density rp, a problem of particularinterest because of its relation to the Lp norm. We show that for p ∈ (−∞,−2) isoperimetric curves arecircular arcs about the origin and for p ∈ [−2, 0) isoperimetric curves do not exist. For p > 0, we showthat for sectors of small angle circular arcs about the origin are isoperimetric, for sectors of large anglecircular arcs through the origin are isoperimetric, and for a transition period in between isoperimetriccurves are members of a family of unduloids. We provide bounds on the angles of the transition interms of p and conjecture on the exact values. The conjecture is supported by theoretical results andnumerical evidence. [HS27063009]

[Joint with Alexander Diaz, Nate Harman, David Thompson] Received: August 9, 2009

Extensions of the Heisenberg Group

Laura E Janssen ([email protected])

Canisius College [Mentor:Byung-Jay Kahng]

Abstract of Poster Presentation: The Heisenberg group is a Lie group with applications in quantumphysics and Fourier analysis. Therefore, it is important to understand its extension groups and howto construct them. This poster discusses central extensions of the Heisenberg group and ways of con-structing them using 2-cocycles. It then goes on to consider extension groups of the groups resultingfrom central extensions of the Heisenberg group, on which not much has been written. This would serveas a helpful result in the field of representation theory. [JL23120423]


15

The Steiner Problem on Surfaces of Revolution

Ryan Jensen ([email protected])

Brigham Young University [Mentor:Denise Halverson]

Abstract of Report Talk: While the Steiner Problem has been extensively studied in the Euclidean plane,it remains an open problem to solve the Steiner Problem on arbitrary non-planar (piecewise smooth)surfaces. We solve the 3-point Steiner Problem on generalized surfaces of revolution by constructingan isometric framework on a plane endowed with a weighted metric, thus propelling a new analyticalavenue for studying the Steiner Problem on surfaces with non-constant curvature. [JR02145053]

[Joint with Elena Caffarelli] Received: August 11, 2009

C∗-algebras of Graph Products

Ann K Johnston ([email protected])

Andrew S Reynolds ([email protected])

Canisius College [Mentor:Byung-Jay Kahng]

Abstract of Report Talk: For certain graphs, we can associate a universal C∗-algebra, which encodesthe information of the graph algebraically. In this talk we examine the relationships between productsof graphs and their associated C∗-algebras, focusing on box and tensor product graphs. We presentan isomorphism between the tensor product of graph algebras and the algebra of a higher rank graphconstruction, related to the box product. We then show that the C∗-algebra of the tensor product oftwo graphs is a subalgebra of the tensor product of their graph algebras. [JA23135750]


Non-Hermitian Hamiltonians

William A. Karr ([email protected])

Indiana University-Purdue University Indianapolis [Mentor:Yogesh Joglekar]

Abstract of Report Talk: Motivated by the example of a quantum particle in an attractive short-rangedpotential, we numerically investigate the eigenvalues and eigenvectors of N×N non-Hermitian matricesthat satisfy f(q)M∗qp = f(p)Mpq where p, q ≥ 0 stand for the momentum label and f(u) is a generalfunction of u. We characterize the overlap of eigenvectors by γ2(N) = Tr

[(V †V )2 − 1

]/N(N − 1)

where V is the N × N matrix of eigenvectors and V † is its adjoint. We find that for a polynomialf the eigenvalues are real and the eigenvectors become orthogonal as the matrix size N diverges,γ(N) → g(1/N) → 0 where the scaling function g(u) is a monotonic function of u with g(0) = 0. Forf(u) = 1 and f(u) = u, we reproduce the analytical results from elementary quantum mechanics. Wethen consider matrices with random entries that decay algebraically (Mpq ∼ 1/|p−q|α) and exponentially(Mpq ∼ exp

[−|p− q|β

]). We find, again, that the eigenvectors become orthogonal as 1/N → 0 and we

obtain the corresponding scaling functions g(u). Our results show that random matrices which satisfyf(q)M∗qp = f(p)Mpq may serve as suitable Hamiltonians that have real spectra and generate a unitarytime-evolution consistent with the requirements of quantum mechanics. [KW03140636]


16

Reconstruction of a Family of Seperable Graphs

Hannah R Kolb ([email protected])

University of Nebraska-Lincoln [Mentor:Stephen Hartke]

Abstract of Report Talk: In the 1950’s, Ulam and Kelly posed the reconstruction conjecture in graphtheory: every graph with n ≥ 3 vertices is uniquely determined by its collection of vertex-deletedsubgraphs. While the problem remains open, many families of graphs have been shown to be recon-structible, such as disconnected graphs, regular graphs, and trees. In 1976, Manvel proved that graphswith connectivity 1 and no leaves are reconstructable. Here, we prove the reconstructability of graphswith connectivity 1 where no maximal end tree is a leaf. [KH01124554]

[Joint with Jared Nishikawa, Derrick Stolee] Received: August 8, 2009

The Conjugacy Classes of SO(5, Fp)Benjamin L Kornacki ([email protected])

University of Michigan [Mentor:Tatiana Howard]

Abstract of Report Talk: In our research we addressed the problem of identifying all conjugacy classesfor the special orthogonal group of rank 5 over a finite field with p elements where p is any prime greaterthan 2. By utilizing prior results of G.E. Wall we formulated theorems that we used to generate theconjugacy classes of SO(5,Fp). These results can further be used to study the representation theoryof the group SO(5,Fp). In this talk we will present our approach to the problem, the results that weobtained, and discuss their significance. [KB03143426]


Rook Polynomials in Three Dimensions

Nicholas B Krzywonos ([email protected])

Grand Valley State University [Mentor:Feryal Alayont]

Abstract of Report Talk: A rook polynomial counts the placements of non-attacking rooks on a board.One of the applications of rook polynomials is in matching type problems. Consider for example havingthree sandwiches and three packets of condiments, each of a different kind. We create a board in sucha way that the available sandwiches would correspond to the rows of the board while condiments wouldcorrespond to the columns. If one does not want to put a certain condiment, such as ketchup, on apeanut-butter sandwich, we can place a restriction on the tile at the intersection of the correspondingrow and column to replicate this restricted pairing. We then count the number of ways to place krooks on our board to find the number of ways to pair sandwiches with condiments. In our research wegeneralized the definition and properties of the rook polynomials to three dimensions. We also definegeneralizations of special two dimensional boards to three dimensions, including the triangle board andthe board representing the probleme des rencontres. The number of rook placements on these threedimensional families of rook boards are shown to be related to famous number sequences, such as centralfactorial numbers and the number of Latin rectangles with three rows. [KN31134858]


17

New Bounds on Some Ramsey Numbers

Daniel E Leven ([email protected])

Kevin M Black ([email protected])

Rochester Institute of Technology [Mentor:Stanislaw Radziszowski]

Abstract of Report Talk: The Ramsey number R(G,H) is the smallest positive integer n such that anygraph on n vertices contains G as a subgraph or H in the complement. We derive a new upper bound of26 for the Ramsey number R(K5 − P3,K5), improving on the previous upper bound of 28. This leaves25 ≤ R(K5 − P3,K5) ≤ 26.

We also show, with the help of a computer, that R(B2, B6) = 17 and R(B2, B7) = 18 by fullenumeration of (B2, B6)-good graphs and (B2, B7)-good graphs, where Bn is the book graph with ntriangular pages. [LD22115007]


The first coefficient of the Conway polynomial

Jeffrey W Lindquist ([email protected])

The Ohio State University [Mentor:Sergei Chmutov]

Abstract of Report Talk: Theorems of Hosokawa, Hartley, and Hoste state that for an m-component linkL the coefficients ci(L) of the Conway polynomial of L vanish when i ≤ m−2 and the coefficient cm−1(L)depends only on the linking numbers lij(L) between the i-th and j-th components of L. This coefficientis equal to the determinant of a certain matrix composed of the linking numbers. This determinant canbe computed using the matrix-tree theorem from graph theory.

For virtual links there are two different types of the linking number and two Conway polynomials,ascending and descending. We generalize the theorem above to virtual links. In this case the determinantrepresenting cm−1(L) is related to the oriented version of the matrix-tree theorem. [LJ02172433]

[Joint with Jack Cheng, Theodore Dokos] Received: August 11, 2009

Homotopy Classes of Linear 3-Fields on the 3-Sphere

Lauren R. McGough ([email protected])

MIT [Mentor:Todd Kemp]

Abstract of Report Talk: A collection of k vector fields on the n-sphere is called a linear k-field if thefields are pointwise linearly independent and are restrictions of linear maps on Rn+1. We characterizethe homotopy classes of linear 3-fields on the 3-sphere, where we restrict our attention to homotopiesthrough linear 3-fields. In particular, we determine when a linear 3-field is homotopic to one whichcan be represented by a collection of three skew-symmetric, signed permutation matrices (a so-called“Radon-Hurwitz” 3-field). We first use linear algebra to show the known result that there are twohomotopy classes of linear 1-fields, and that any linear 1-field is homotopic to a Radon-Hurwitz 1-field. We then use this proof along with further linear algebraic and analytic arguments to provide aconstructive proof that every linear 3-field is homotopic to a Radon-Hurwitz 3-field, and that there areexactly two homotopy classes of linear 3-fields. It follows that any linear 3-field is homotopic to a linear3-field which specifies an orthonormal basis to the tangent space at each point of S3. [ML03155222]


18

Self-Contact in Three Branch Planar Fractal Trees

Heather A McLendon ([email protected])

Ithaca College [Mentor:David Brown]

Abstract of Poster Presentation: Previous research with three-branch fractal trees has focused on three-dimensional fractal trees. Transferring three-branch trees into the plane imposes additional restrictionson their form and on the connectivity of the fractal attractor. A fractal tree is created via an iteratedfunction system of contractive similarity transformations with scaling ratio r and branching angle θ. Weinvestigate such trees, the canopies that result, equations which describe the r-θ relationship throughvarious ranges of theta, as well as the dimensions of the canopies for all θ.

Of particular interest is the special case in which theta is pi radians, where we show its attractor isthe classical Cantor set. [MH22134206]


Sudoku, Shidoku, and...Grobner Bases?: An Algebraic Approach To CountingBoards

Matthew J Menickelly ([email protected])

Katharina M Carella ([email protected])

James Madison University [Mentor:Elizabeth Arnold]

Abstract of Poster Presentation: We investigate various counting proofs for Shidoku boards and relatedvariants, such as the number of possible solution boards from incomplete puzzles. We also look intothe algebraic group derived from symmetries of Shidoku boards. We use these group isomorphisms toclassify all possible numbers of solutions from incomplete puzzles. We use Grobner Basis representationsof Shidoku and Sudoku to obtain these results. We provide a complete classification of all the possiblenumber of solutions that can result from incomplete Shidoku puzzles. [MM23152124]


Description of the space of Cartier boundary divisors on Mn,1(A)

Khoa L Nguyen ([email protected])

Rutgers University [Mentor:Christopher Woodward]

Abstract of Report Talk: Denote by A the affine line over C. Let Mn,1(A) denote the moduli space ofthe union of combinatorial types of scaled n-marked affine lines. It turns out that Mn,1(A) is a varietywith toric singularities. Based on the local description of Mn,1(A), we construct explicitly the localgenerating cones and describe the relations of the local Cartier boundary divisors. We then finallyprove that the relations on the space of Cartier boundary divisors of Mn,1(A) are the kernel of thepush-pull map defined by the incidence relation on subsets and partitions. [NK12005007]

[Joint with Christopher Woodward] Received: August 8, 2009

19

The Exponential and Worpitzky Series Modulo p

James E Pascoe ([email protected])

CSU Chico [Mentor:Benjamin Levitt]

Abstract of Summary Talk: In this presentation, we develop a theory of polynomials corresponding to theexponential bx modulo p. We show that these polynomials are equivalent to a single formal power seriesover Q in x−1 modulo the ideal generated by p and x−p. Furthermore, we show a strong connection withthe recursively defined Worpitzky numbers, which in turn give a relation to the more familiar Stirlingnumbers from combinatorics. This motivates the definition of the Worpitzky polynomials, which give asimple fomula for the coefficients of the aforementioned power series, and the development and definitionof the Worpitzky series, a formal power series over Z in two variables. [PJ29034032]


Low-Lying Zeros of Number Field L-functions

Ryan N Peckner ([email protected])

Williams College [Mentor:Steven Miller]

Abstract of Report Talk: One of the most important statistics in studying the zeros of L-functionsis the 1-level density, which measures the concentration of zeros near the critical point. Fouvry andIwaniec proved that the 1-level density for L-functions attached to imaginary quadratic fields agreeswith results predicted by random matrix theory (RMT). In this paper, we show a similar agreementwith RMT occurring in suitable sequences of number fields that are not necessarily imaginary quadraticfields. In addition to proving the correspondence of the main term with RMT, we derive the lower-orderterms of the 1-level density and connect them to the number field arithmetic. Furthermore, we provethat our results apply to test functions whose Fourier transforms have unusually large support. [PR01014200]


Algorithmic Game Theory: Selfish Set Packing

Jamie Quadri ([email protected])

Sam Cole ([email protected])

Oberlin College [Mentor:Alexa Sharp]

Abstract of Poster Presentation: The General Set Packing problem states: given a collection of subsets,what is the largest selection of these subsets with the property that all chosen subsets are pairwisedisjoint? We apply a game theoretic model to this classic computational problem and define the selfishset packing game, in which the subsets themselves are greedy agents that decide how they will be“packed.” We use the price of anarchy, a standard metric from algorithmic game theory, to compare theworst stable solutions in this game to optimal ones. We show that, in general, the selfish, uncoordinatedbehavior of the agents can produce arbitrarily bad stable outcomes. However, for certain special casesof set packing, we show that the worst stable solutions are within a constant factor of optimal. We alsoexamine the outcomes of the game when players are allowed to collaborate and form coalitions. Ourresults show that allowing coalitions of arbitrary size will always produce an optimal stable outcome.[QJ02112818]

[Joint with Kevin Woods (mentor)] Received: August 10, 2009

20

On the Maximum Number of Isosceles Right Triangles in a Finite Point Set

David B Roberts ([email protected])

California State Univ., Northridge [Mentor:Bernardo Abrego]

Abstract of Summary Talk: Let Q be a finite set of points in the plane. For any set P of points in theplane, we let SQ(P ) be the number of similar copies of Q contained in P . For a fixed n, Erdos andPurdy asked to determine the maximum possible value of SQ(P ), denoted by SQ(n), over all sets Pof n points in the plane. Although good bounds have been found for the general problem, not muchmore is known for specific patterns Q. We consider this problem when Q = 4 is the set of verticesof an isosceles right triangle. We give exact solutions for small values of n ≤ 9, and show that as napproaches infinity, S4(n) has quadratic order. Lastly, we provide new upper and lower bounds for thelimit of S4(n)/n2 as n approaches infinity. [RD02072139]


Perfect state transfer, integral circulants, and join of graphs

Matthew C. Russell ([email protected])

Clarkson University [Mentor:Christino Tamon]

Abstract of Report Talk: Quantum walks on graphs provide a way to develop fast quantum algorithms.An important topic in this field is the idea of quantum perfect state transfer, which refers to when aquantum state can be perfectly transferred from one vertex to another. We propose new families ofgraphs which exhibit perfect state transfer. Our constructions are based on the join operator on graphsand its circulant generalizations. We build upon the results of Basic et al. and construct new integralcirculants and regular graphs with perfect state transfer. More specifically, we show that the integralcirculant ICGn({2, n/2b} ∪ Q) has perfect state transfer, where b ∈ {1, 2}, n is a multiple of 16 andQ is a subset of the odd divisors of n. Using the standard join of graphs, we also show a family ofdouble-cone graphs which are non-periodic but exhibit perfect state transfer. This class of graphs isconstructed by simply taking the join of the empty two-vertex graph with a specific class of regulargraphs. Their existence answers a question posed by Godsil. [RM31121538]

[Joint with Ricardo Javier Angeles Canul, Rachael Norton, Michael Opperman, Christopher Paribello] Received: August 10, 2009

Self-Contact in Ternary Planar Fractal Trees

Bolanle O Salaam ([email protected])

Ithaca College [Mentor:David Brown]

Abstract of Poster Presentation: We establish the properties of ternary fractal trees in planar form.Previous research explores ternary trees in three dimensions, as well as binary trees in both planar andthree dimensions. We extend popular results concerning binary planar trees into the three-branch case,namely, the method for determining the location of any tip of the fractal tree in the r-θ plane. We alsoexplore the structure of the resulting canopies, prove the relationship between angle measure and scalingratio used in the Iterated Function System, discuss dimension for any given ternary planar fractal tree,and suggest a generalization for determining the location of any branch tip in an n-ary tree. [SB16082523]


21

Discrete Complex Analysis

Joseph W Seaborn ([email protected])

Taylor University [Mentor:Philip Mummert]

Abstract of Poster Presentation: This poster presents a discrete analogue of complex analysis we de-veloped on the integer lattice of the complex plane. With a new finite difference definition of thederivative, ∆f(z) = f(z+1)−f(z)−i(f(z+i)−f(z))

2 and the use of the Cauchy Riemann equations, we statewhat it means to be discrete complex differentiable. From this notion arises tools which we developedsuch as the discrete path integral, the exponential function, falling power series and the Cauchy inte-gral. Some theorems from classical complex analysis such as Cauchy’s theorem and Taylor’s theoremcarry over nicely while others such as the Schwarz reflection principle do not. Along with the theoremswhich do carry over, I’ll present some unproven conjectures such as an analogue of Liouville’s theorem.[SJ31141502]


Duality Properties of Indicatrices of Knots and their Relationship to PhysicalKnot Invariants

Charmaine J Sia ([email protected])

Williams College [Mentor:Colin Adams]

Abstract of Report Talk: The bridge index and superbridge index of a knot are important invariantsin physical knot theory. We define the bridge map of a knot conformation, which is closely relatedto these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation.Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we showthat the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and somenumber of great circles. Similarly, we define the intersection map of a knot conformation, interpret it interms of the binormal indicatrix and express its graph in terms of the tangent indicatrix. This dualityrelationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of aknot conformation. Analogous concepts are defined and results derived for stick knots. [SC26215642]

[Joint with Daniel Collins, Katherine Hawkins, Robert Silversmith, Bena Tshishiku] Received: August 9, 2009

Zeta Functions of Graph Sequences

Mollie E Stein ([email protected])

Canisius College [Mentor:Stratos Prassidis]

Abstract of Report Talk: The Ihara zeta function, originally defined for finite graphs, is an analogue ofthe Riemann zeta function, that combines arithmetic and combinatorial properties of graphs. Usinganalytic methods, it can be extended to infinite graphs which are approximated by finite graphs. Thissituation arises when the Cayley graph of a residually finite group is approximated by the Cayleygraphs of its finite quotients. The advantage of this setting is that maps between consecutive graphsin the graph sequence are regular graph coverings. Using the formula for the Ihara zeta function forgraph covers and the filtration of the projective modular group PSL2(Z) by its principal congruencesubgroups, we look for a formula for the Ihara zeta function of the projective modular group. [SM24113301]


22

Spiraling to My Doom: The Revenge (In 3D!)

Kiri E Sunde ([email protected])

Jessalyn G Bolkema ([email protected])

Hope College [Mentor:Aaron Cinzori]

Abstract of Report Talk: Given n ≥ 3 points P0, P1, . . . , Pn−1 in R2, we examine piece-wise linear spiralsgenerated by taking convex combinations of m of those points where m ∈ {2, . . . , n}. In particular, foreach k ≥ 0, let Pn+k = t1Pk+t2Pk+1+· · ·+tmPk+m−1 where 0 ≤ t1, t2, . . . , tm ≤ 1 and t1+t2+· · ·+tm =1. This generates a sequence of points {Pk}∞k=0. For each k ∈ N, let Qk = Pk − Pk−1 be the vectorconnecting consecutive points. For each choice of n, m, and t1, . . . , tm, we establish necessary andsufficient conditions on the starting points P0, P1, . . . , Pn−1 that result in the lengths ‖Qk‖ forming ageometric sequence. We call the resulting piece-wise linear spirals geometric spirals. We prove severalproperties of geometric spirals including the fact that the geometric ratio is the modulus of an eigenvalueof a certain naturally arising matrix and that the angle between consecutive Qk’s is constant andcalculable from the starting points. This work generalizes work done by two earlier REU groups on them = 2 case. Our research culminates in the following theorem: Given a spiral in R3 defined by theaforementioned recursive formula, if the spiral is geometric then the points of the spiral are coplanar.We employed the tools of linear algebra, infinite series, and complex analysis in a combination ofexperimental computer mathematics and rigorous composition of proofs. [SK28143608]


Characterizing Inscribable Polyhedral Graphs

Giang T.H. Tran ([email protected])

William Vickery ([email protected])

Bard College [Mentor:Lauren Rose]

Abstract of Poster Presentation: The planar graph of a polytope is said to be of inscribable type if it iscombinatorially equivalent to the set of vertices and edges of the convex hull of a set of noncoplanarpoints on the surface of the sphere. Graphs of inscribable type have been characterized by usingnumerical (edge weighing, as in Rivins condition), graph theoretical (1-hamiltonian) or combinatorialinformation (1-supertough, as in Dillencourts condition). We made use of geometric properties ofinscribable polytopes by developing a new and simpler set of necessary and sufficient conditions forinscribability that involved assigning angles to stereographic projections of inscribable polytopes. Wehave established their relationship with the previously mentioned conditions and developed methods togenerate certain families of graphs that are not of inscribable type. [TG03111837]

[Joint with Jim Belk, Maria Belk, Sining Leng, Lionel Barrow] Received: August 7, 2009

23

A systematic census of generalized toric codes over F4, F5, F16

Brian M Vega ([email protected])

James E Amaya ([email protected])

UC Berkeley [Mentor:John Little]

Abstract of Poster Presentation: Toric codes are a specific class of linear codes. In our work we studiedgeneralized toric codes generated by subsets of Zmq−1, where q is a power of a prime. The orbits of thesesets of points determine equivalent codes with the same minimum distance and weight distribution. Wewished to find and distinguish the codes for a given blocklength n and dimension k, with n = (q − 1)m

and k being equal to the number of points in Zmq−1 used to generate the code. To accomplish this weused various Magma processes to compute minimum distances and weight distributions of many codes.In our analysis, we sought to find codes over the finite fields F4 and F5 that have minimum distancesexceeding the currently best known for codes with given parameters n and k. In the process, we noticedan unusual property about the average weight of words in toric codes and found codes over finite fieldsF5 and F16 with good minimum distances. [VB02230710]

[Joint with April Harry] Received: August 9, 2009

Taking Curvature to the Extreme!

Lindsay M. Willett ([email protected])

Hope College [Mentor:Stephanie Edwards]

Abstract of Poster Presentation: Let F be a real polynomial of degree N . Then the curvature of F isdefined to be

κ =F ′′

(1 + (F ′)2)32

.

An interesting problem is to determine the maximum number of relative extreme values for the functionκ. In 2004, Edwards and Gordon showed that if all the zeros of F ′′ are real, then F has at most N − 1points of extreme curvature. Using a computer, my research group examined the level curves of certainauxiliary functions in the complex plane to try to remove the hypothesis that the zeros of F ′′ have tobe real. We proved a partial solution to the problem, showing that aF has at most N − 1 points ofextreme curvature, given that a ∈ R is smaller than a given bound. The conjecture that F has at mostN − 1 points of extreme curvature remains open. [WL23162558]

[Joint with Edward Niedermeyer, Tarah Jensen] Received: August 11, 2009

24

Selfish Set Covering

Elara Willett ([email protected])

David Leibovic ([email protected])

Oberlin College [Mentor:Kevin Woods]

Abstract of Poster Presentation: In a variation on the NP -hard combinatorial problem, Set Cover, wepresent a game with k selfish agents who each must choose one subset out of a fixed set of subsetsof underlying elements. The elements split their worth evenly among agents that cover them, whilethe agents choose subsets that (given the choices of the other agents) maximize their payoff. We givea constant bound on the Price of Anarchy (PoA) and Price of Stability (PoS), ratios that measurethe quality of stable solutions to optimal solutions. We also show that Best Response Dynamics isguaranteed to converge only if agents cannot collaborate, and we bound the corresponding PoA andPoS when agents do collaborate. Lastly, we consider our model on Vertex Cover and Edge Cover.[WE02113139]

[Joint with Alexa Sharp] Received: August 11, 2009

Constant Curvature Curves on Polyhedra

Leah Wolberg ([email protected])

Indiana University [Mentor:Matthias Weber]

Abstract of Poster Presentation: Given a few hundred meters of fencing, how would you go aboutenclosing the largest amount of land possible? This is better known as the isoperimetric problem, andit has interested mathematicians since antiquity. It has been shown that the boundary of the optimaldomain is a curve of constant curvature.

The isoperimetric profile of a polyhedron gives the length of the shortest boundary needed to enclosea given area. What information about the geometry of the polyhedron is contained in the profile?

While there have been some studies of geodesics—curves with constant zero curvature—on polyhedra,little is known about more general constant curves. We will describe techniques for finding curves ofconstant curvature on arbitrary polyhedra. We will also present several nice results regarding theisoperimetric profile of the doubled triangle. [WL03151419]


25

Joint Dilation Wavelet Set

Zhiwei Wu ([email protected])

Texas A&M University [Mentor:David Larson]

Abstract of Report Talk: A wavelet set is defined to be a measurable subset E of R for which 1√2πχE is the

Fourier transform of a wavelet. In particular, a wavelet set is a 2π-translation and 2-dilation generatorof a partition of the real line. (A set W is 2-dilation if ∪n2nW is a disjoint union and ∪n2nW = Rmodulo a null set.) We also define a joint dilation wavelet set as a wavelet set under dilation by bothd1 and d2 for d1 6= d2. We have proven that a set W is a joint (2) and (−2) dilation wavelet set ifand only if

⋃j∈Z

4jW is symmetric. By applying this result to some families of 2-dilation wavelet sets

given by David Larson and Xingde Dai in their AMS memoir, we have proven that all of the Journesets are joint (2) and (−2) dilation wavelet sets. Using the same tool, we can also find all 2-interval and3-interval joint dilation wavelet sets. A characterization of all 3-interval wavelet sets different from theliterature is included for our purpose.This work was done by the author while participating in Matrix Analysis and Wavelet Theory summerREU 2009 in Texas A&M University mentored by David Larson. Part of this is work joint with DanielPoore, another REU participant. [WZ30010314]

[Joint with Daniel Poore] Received: August 10, 2009

Spectral Theory of Mele Graphs

Michelle T Zagardo ([email protected])

Casey L McKnight ([email protected])

University of Wyoming [Mentor:Jeff Selden]

Abstract of Poster Presentation: The aim of this paper is to develop mele graphs which are graphsthat incorporate both discrete and continuous graphs in a unified framework. In order to discover theappropriate Laplacian operator to use with these graphs, we start with totally discrete graphs andpopulate one edge with equidistributed vertices. Increasing the number of vertices on this one edge toinfinity might reveal the behavior of the graph in the continuum limit. The eigenvalues of the Laplacianoperator (that we currently work with) fall into two categories: those coming from the discrete part ofthe graph and those coming from the populated (or string-like) edges. The eigenvectors corresponding tothe continuous eigenvalues involve the entire graph while the eigenvectors corresponding to the discretevertices isolate behavior to the discrete subgraph. What we end up with is a non-standard eigenvalueproblem and a first order approximation of the spectrum of a mele graph. [ZM02124137]

[Joint with Peter Muller] Received: August 10, 2009

26

Isoperimetric Inequalities for Wave Fronts and Menzin’s Conjecture on Sur-faces of Constant Curvature

Valentin Zakharevich ([email protected])

Sean Howe ([email protected])

Penn State University REU [Mentor:Sergei Tabachnikov]

Abstract of Report Talk: The classical isoperimetric inequality relates the lengths of curves to the areasthat they bound. More specifically, we have that for a smooth, simple closed curve of length L boundingarea A on a surface of constant curvature c, L2 ≥ 4πA − cA2 with equality holding only if the curveis a geodesic circle. We prove generalizations of the isoperimetric inequality for both spherical andhyperbolic wave fronts (i.e. piecewise smooth curves which may have cusps). We then discuss “bicyclecurves” using the generalized isoperimetric inequalities. The euclidean model of a bicycle is a unitsegment AB that can move so that it remains tangent to the trajectory of point A (the rear wheel isfixed on the bicycle frame). We extend this definition to a general Riemannian manifold, and concernourselves in particular with bicycle curves in the hyperbolic plane H2 and on the sphere S2. Given acurve Γ, let it represent the motion of the front wheel. Up to choice of relative initial position of theback wheel, the path of the back wheel is determined by Γ. The monodromy map associated to Γ isthe function that sends each choice of starting position for the back wheel to its terminal position. Weresolve both spherical and hyperbolic versions of Menzin’s conjecture, which relates the area boundedby a smooth convex curve to its associated monodromy map. [ZV28190044]

[Joint with Matthew Pancia] Received: August 9, 2009

Greedy Algorithms for Generalized k-Rankings of Paths

Andrew N Zemke ([email protected])

RIT [Mentor:Darren Narayan]

Abstract of Report Talk: A k-ranking of a graph is a labeling of the vertices with positive integers1, 2, . . . , k so that every path connecting two vertices with the same label contains a vertex of largerlabel. An optimal ranking is one in which k is minimized. Let G be a graph containing a Hamiltonianpath on vertices v1, v2, . . . , vn but no Hamiltonian cycle. We use a greedy algorithm to successivelylabel the vertices assigning each vertex with the smallest possible label that preserves the rankingproperty. We show that when G is a path the greedy algorithm generates an optimal k-ranking. Wethen investigate two generalizations of rankings. We first consider bounded (k, s)-rankings in whichthe number of times a label can be used is bounded by a predetermined integer s. We then considerkt-rankings where any path connecting two vertices with the same label contains t vertices with largerlabels. We show for both generalizations that when G is a path, the analogous greedy algorithmsgenerate optimal k-rankings.

We then proceed to quantify the minimum number of labels that can be used in these rankings. Wedefine the bounded rank number χr,s(G) to be the smallest number of labels that can be used in a

(k, s)-ranking and show for n ≥ 2, χr,s(Pn) =⌈n−(2i−1)

s

⌉+ i where i = blog2(s)c + 1. We define the

kt-rank number, χtr(G) to be the smallest number of labels that can be used in a kt-ranking. We presenta recursive formula that gives the kt-rank numbers for paths, showing χtr(Pj) = n for all an−1 < j ≤ anwhere {an} is defined as follows: a1 = 1 and an =

⌊t+1t an−1

⌋+ 1. [ZA21134908]

[Joint with Sandra James (Concordia University, St. Paul)] Received: August 10, 2009

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Student Index with Home Institutions

Abramovic, Robert J. Johns Hopkins University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Amaya, James E The College of New Jersey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Black, Kevin M Harvey Mudd College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Bolkema, Jessalyn G Hope College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Bonsangue, Jennifer E California State University of Channel Islands . . . . . . . . . . . . . .3Boxer, George A Princeton University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Boxer, George A Princeton University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Bradford, Robert M. Ohio State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Calisch, Sam E Grinnell College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Carbonara, Alejandro U California Institute of Technology . . . . . . . . . . . . . . . . . . . . . . . 5Carella, Katharina M Ithaca College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Catanzaro, Michael J Wayne State University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Chan, Hau College of Charleston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Chen, David D University of California, Los Angeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Cole, Sam Oberlin College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Collins-Wildman, Elizabeth M Carleton College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Davis, Damek University of California, Irvine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Diao, Peter Z Princeton University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Diao, Peter Z Princeton University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Dong, Chu Yue (Stella) Polytechnic Institute of New York University . . . . . . . . . . . . . . 9Dong, Yishun Upper Arlington High School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Etropolski, Anastassia Bard College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Forlini, Maxwell S Miami University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Furness, Amanda Wittenberg ’10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Garver, Alexander C Augsburg College 09-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Glover, Jessica M. UC Santa Cruz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Goes, John W University of Illinois at Chicago . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12Gokal, Ali Bradley University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Gray, Kailee A. University of South Dakota . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Grinshpun, Andrey V Carnegie Mellon University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Harman, Nate R University of Massachusetts Amherst . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Harper, Hudson University of South Carolina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Hay, Diana California State University, Monterey Bay . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Healey, Vivian Olsiewski Notre Dame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Hirpara, Vivek Iowa State University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14Hoffman, Matthew J University of Akron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Howe, Sean University of Arizona . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Howe, Sean P University of Arizona . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Janssen, Laura E University of Nebraska - Lincoln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Jensen, Ryan Brigham Young University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Johnston, Ann K Harvey Mudd College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Karr, William A. Indiana University-Purdue University Indianapolis . . . . . . . . . . . . . . 16Kolb, Hannah R Illinois Institute of Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Kornacki, Benjamin L University of Michigan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Krzywonos, Nicholas B Grand Valley State University . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Leibovic, David Oberlin College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25Leven, Daniel E Rutgers University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Lindquist, Jeffrey W The Ohio State University. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18McGough, Lauren R. MIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18McKnight, Casey L Austin Peay State University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26McLendon, Heather A SUNY Brockport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18Menickelly, Matthew J Miami University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

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Nguyen, Khoa L MIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Pascoe, James E University of North Texas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Peckner, Ryan N UC Berkeley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Poulsen, Katherine R Columbia University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Quadri, Jamie Oberlin College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Reynolds, Andrew S UCLA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Roberts, David B California State Univ., Northridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Russell, Matthew C. Taylor University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Salaam, Bolanle O Howard University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21Seaborn, Joseph W III Taylor University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Sia, Charmaine J MIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22Stein, Mollie E Barnard College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Sunde, Kiri E University of North Carolina - Chapel Hill . . . . . . . . . . . . . . . . . . . . . . . . . 23Tarfulea, Andrei University of Chicago . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Tran, Giang T.H. Bard College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Vega, Brian M California State Polytechnic University, Pomona . . . . . . . . . . . . . . . . . . 24Vickery, William Bard College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Willett, Elara Oberlin College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Willett, Lindsay M. Grove City College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Wolberg, Leah Bowdoin College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Wu, Zhiwei Bard College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Zagardo, Michelle T Mount Holyoke College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Zakharevich, Valentin Polytechnic Institute of NYU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Zemke, Andrew N Rochester Institute of Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Documents

The Ohio State University Columbus, OH Abstracts of Presentations · 2010. 2. 17. · Categorification of Krushkal’s Polynomial Sam E Calisch ([email protected]) Louisiana State