Course Descriptions Spring 2018

Department of Mathematics University of Illinois

Spring 2018

Math 503: Introduction to Geometric Group Theory

MWF, 10am, Altgeld 445

Instructor: Ilya Kapovich

Geometric Group Theory is an actively developing area of mathematics drawing on theideas and techniques from Riemannian geometry, low-dimensional topology, combinatorics,analysis, probability, logic as well as the traditional group theory. A key underlying ideaof Geometric Group Theory is to study the interaction between algebraic properties of afinitely generated group and geometric properties of a space admitting a “nice” isometricaction of this group.

Prerequisites: Math 500 “Abstract Algebra I”.

Text:• Gilbert Baumslag “Topics in Combinatorial Group Theory”, Birkhuser Verlag, Basel, 1993;(recommended text)• Pierre de la Harpe “Topics in geometric group theory. Chicago Lectures in Mathematics”.University of Chicago Press, Chicago, IL, 2000; (recommended text)• Cornelia Drutu and Michael Kapovich, “Geometric Group Theory”, American Mathemat-ical Society, (recommended text, when and if it becomes available; Expected publicationdate March 24, 2018)

Approximate Syllabus.

(1) Free groups and their subgroups via Stallings subgroup graphs.

(2) Groups given by generators and relators. Cayley graphs and the word metric. VanKampen diagrams and van Kampen Theorem. Time permitting, a discussion of smallcancellation theory.

(3) Bass-Serre theory: Amalgamated free products and HNN-extensions, graphs of groupsand group actions on simplicial trees.

(4) Derived series, upper and lower central series, nilpotent and solvable groups, commu-tator calculus. Finitely generated and finite nilpotent groups. Semi-direct productsand wreath products.

(5) Quasi-isometries, Milnor-Swarc theorem, geometric properties and invariants. Exam-ples of quasi-isometric invariants: ends, growth, isoperimetric functions, amenability,solvability of the word problem and asymptotic cones.

(6) A sample advanced topic, e.g. one or more of the following: word-hyperbolic groups;the Novikov-Boone Theorem; the Higman Embedding Theorem; Burnside groups;Grigorchuk groups of intermediate growth; automatic groups; relative hyperbolicity;actions on R-trees, etc.

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Math 511

Professor Dodd

This comps course provides an introduction to classical algebraic geometry, using Shavarevich's textbook as a guide. The material will prepare interested students to go on to work towards more advanced topics in algebraic geometry, namely the further courses in scheme theory. We'll discuss the basic notions concerning quasi-projective varieties over an algebraically closed field, such a irreducibility and smoothness, and prove basic results such as Bezout's theorem, reviewing the relevant commutative algebra along the way.

Math 519

Differentiable Manifolds II

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Instructor: Ely Kerman

Lectures: Tuesday–Thursday, 12:30 to 1:50

Course Description. In this second course on the theory of smooth mani-folds we will study a variety of rich structures on manifolds including vectorbundles, principle bundles, Riemannian metrics, covariant derivatives, con-nections and curvature. These structures will lead us to several theoremsin which analysis is used to reveal deep connections between the geometryand topology of manifolds. These will include de Rham theory, the theoryof characteristic classes, Chern-Weil theory, and the Gauss-Bonnet-Cherntheorem.

Grading. Grades will be based on Problem Sets.

Prerequisites. Basic differential topology at the level of Math 518 isrecommended.

Math 525, Topology (Spring 2018)

Instructor: Pierre Albin

Office: Illini Hall 237

Email: palbin [at] illinois .edu

Lectures: TBA

Office Hours: TBA

 

Home   Problem Sets   Links

Web page: https://faculty.math.illinois.edu/~palbin/Math525.Spring2018/home.html 

Text: Hatcher, Algebraic Topology available on the author's webpage

Supplementary Texts: Bredon, Topology and Geometry

 May, A concise course in Algebraic Topology , available on the author's webpage 

Assignments: There will be homework each week.You are allowed (and encouraged) to work with other students whiletrying to understand the homework problems. However, the homework that you hand in should be your work alone. Latehomework will not be accepted, but the lowest score will be dropped.

Exams: There will be two midterms (in lecture) and a final. 

Holidays: This semester we will not have classes on: Spring break March 19­23

 Grading percentages: Problem sets (30%)

 Two Midterms (20% each) Final Exam (30%)

Description: This is a first course in Algebraic Topology. 

 Topology is the study of those properties of a space that are unchanged by a continuous transformation. At first this wasstudied by assigning integers to spaces that did not change under a reversible continuous map or homeomorphism. ThusRiemann classified surfaces by the minimum number of simple closed curves along which to cut in order to obtain a simplepresentation of the surface (namely, twice the `genus' of the surface), and much later Rado showed that two connectedclosed oriented surfaces are homeomorphic if and only if they have the same genus. For more complicated spaces we havemore sophisticated invariants. Instead of assigning a number to a space, we might assign a group, a ring, or a complex ofrings to a space in a homeomorphism invariant way. These richer invariants extend the older numerical ones, e.g., the groupmight have its size determined by the numerical invariant as happens with the fundamental group of a surface and its genus.In this course we will study some of the earliest algebraic invariants of topological spaces: the fundamental group, coveringspaces, and homology. Introduced by Poincaré in 1895, these are now fundamental tools for anyone interested in geometry.

Math 530: Algebraic Number Theory

Spring 2018

Instructor: Patrick Allen (pballen@illinois.edu)

Description: This will be a first course in algebraic number theory, in which we will studythe arithmetic of the rings of integers in finite field extensions of Q. This is a subject thatis both classical and modern, and in this course we will develop the foundations while alsogiving vistas of modern questions and developments. Concrete topics we will discuss include:

• integral closure and Dedekind domains;

• unique factorization of ideals and the class group;

• Minkowski theory, the finiteness of the class group, and Dirichlet’s Unit Theorem;

• valuations and p-adic fields;

• ramification and Galois theory.

Time permitting, we will also dicsuss a selection of topics among the following:

• local–global principles for Diophantine equations;

• class field theory;

• Hecke and Artin L-functions;

• special value formulas for Dedekind ζ-functions.

Prerequisites: Math 500 (Graduate Algebra) or equivalent.

Grading: Grades will be determined based on assigned homework, midterms, and a finalexam.

References: There is no required text. The following references may be useful:

• Milne, Algebraic Number Theory. Available at http://www.jmilne.org/math/CourseNotes/ant.html.

• Neukirch, Algebraic Number Theory. E-book available through the UIUC library.(One of the most comprehensive references, but also more challenging.)

• Marcus, Number Fields. E-book available through the UIUC library. (Very down-to-earth with lots of great exercises.)

SPRING 2018, MATH 532, ANALYTIC THEORY OFNUMBERS II : MULTIPLICATIVE NUMBER THEORY

INSTRUCTOR: ALEXANDRU ZAHARESCU

Math 532, TR 12:30 -1:50 PM

In this course we will discuss ideas from multiplicative number the-

ory centered on L−functions and their applications. The largest part

of the course will cover classical material focussing on the Riemann

zeta function and Dirichlet L−functions. For this part we will follow

Davenport’s book. In the last part of the course we will study some

recent papers on the distribution of zeros of the Riemann zeta function

and more general L−functions.

Prerequisite: MATH 531.

Recommended Textbook:

Harold Davenport, Multiplicative number theory. Third edition.

Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000.

xiv+177 pp. ISBN: 0-387-95097-4

There will be no exams. Students registered for this course will be

expected to give a couple of lectures on some topics related to the

content of the course. In addition some homework problems will be

assigned.

Office hours by appointment.

Office: 449 Altgeld Hall.

E-mail: zaharesc@illinois.edu

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Math 541: Functional AnalysisSpring 2018Instructor: Florin P. Boca (fboca@math.uiuc.edu)

This course will provide an introduction to Functional Analysis. The main topics willinclude:

• Review of abstract measure theory.• Basic topics on Banach spaces, linear and bounded maps on Banach spaces, open

mapping theorem, closed graph theorem.• Hahn-Banach theorem, Banach-Alaoglu theorem, extreme points, Krein-Milman the-

orem. Applications.• Compact operators, spectrum and the spectral theorem for compact operators on

Hilbert spaces.

Further topics (time permitting): Functional calculus, Fredholm alternative, unbounded op-erators, Riesz representation theorem, Haar measure for locally compact groups, non-linearfunctional analysis, distributions and Sobolev spaces.

Prerequisite: Math 540.

Textbook: There is no required textbook. The instructor will use his own notes.Recommended textbooks:

• J. B. Conway, A Course in Functional Analysis.• W. Rudin, Functional Analysis.• G. B. Folland, Real Analysis. Modern Techniques and their Applications.• Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis.

Grading: The final grade will be based on four homework assignments (90%) and classparticipation (10%).

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Math 553

Prof. Rapti

Basic introduction to the study of partial differential equations; topics include: the Cauchy problem, canonical forms, the method of characteristics, the wave equation, the heat equation, Laplace's equation, Sturm-Liouville problems and separation of variables, harmonic functions, potential theory, Fourier series, the Dirichlet and Neumann problems, and Green's functions. Prerequisite: Consent of instructor.

Math 565

Klara Buysse

Continuation of MATH 471. Introduction to tabular or parametric survival models

with single or multiple-life states; life insurance and annuity premium calculations;

reserving and profit measures; introductions to universal life insurances,

participating insurances, pension plans and retirement benefits.

Spring 2018 MATH 568

Actuarial Loss ModelsDr. Shu Li

This course provides an introduction to the theory of actuarial loss models that is useful for short-

term actuarial applications (such as auto insurance and homeowners insurance). Students will be

able to learn steps in the modeling process, various modeling methods as well their parameters

estimation, and the credibility techniques for prospective experience rating. Specific topics include:

• Actuarial modeling process: construction, selection and validation of empirical and paramet-

ric models;

• Survival, severity, frequency and aggregate loss models and coverage modifications;

• Statistical methods to estimate model parameters;

• Credibility theory: limited fluctuation; Bayesian; Buhlmann; Buhlmann-Straub; empirical

Bayes parameter estimation.

Prerequisite: MATH 408, MATH 461 or MATH 463. (Students without prerequisite can register

with instructor’s consent.)

References:

Loss Models: From Data to Decisions, (Fourth Edition), 2012, by Klugman, S.A., Panjer, H.H.

and Willmot, G.E., Wiley.

SET THEORY AND FORCING

Spring 2018 math 574 Anush Tserunyan

This course is an introduction to the Zermelo–Fraenkel set theory with Choice (ZFC): anaxiomatic system/theory generally accepted as the foundations of mathematics. As one ex-pects, the theory that captures most of mathematics is very complicated and, unsurprisingly,the study of its models and the boundaries of what it can and cannot prove has grown intoa complex and multifaceted mathematical discipline, which often has combinatorial flavor.

The basics and absoluteness. The course starts with the basics of set theory, namely: theaxioms of ZFC, ordinals, transfinite induction, cardinals, and some cardinal arithmetic—what arguably every mathematician should know. We will also discuss Shoenfield’s Abso-luteness theorem: if you manage to prove a statement (of a certain general form) in analy-sis/dynamical systems/probability theory using an extra ZFC-independent assumption (e.g.the Continuum Hypothesis), then Shoenfield’s Absoluteness implies that there is a proof ofyour statement without this extra hypothesis, thus yielding an unconditional theorem.

Godel’s universe L. This is the smallest submodel of any model of ZFC. What L is to amodel of ZFC is what the prime subfield is to a given field. In L, the Continuum Hypothesisholds, but many desirable statements fail: for example, you can start with a Borel sub-set of R3, apply innocuous operations (projection-complement-projection), and get a non-measurable subset of R.

Forcing and the Independence of the ContinuumHypothesis. Introduced by Paul Cohen,forcing is the most powerful technique of proving independence results. In a nutshell, it isa method of adjoining a new object to a given model of ZFC, analogous to adjoining, say,a root of x2 − 2 to Q: one gives this root a name,

√2, and builds the field Q(

√2) as the set

of all rational functions over Q and the name√

2. However, building a new structure thatsatisfies the field axioms is much easier than building a new structure that is a model ofZFC, i.e. most of mathematics! To make the method transparent, we will not only presentits traditional combinatorial treatment, but also its rephrasing in terms of Baire category.The course will culminate in the proof of the Independence of the Continuum Hypothesis.

Prerequisites: Familiarity with the basic concepts of mathematical logic (Math 570) suchas structures, definability, and elementarity. A motivated student can quickly pick theseup with the instructor’s guidance, using, for example, the instructor’s Mathematical Logiclecture notes available on her webpage.

Required work: Weekly homework with board presentations in weekly problem sessions.

Exams: None.

UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNDEPARTMENT OF MATHEMATICSCourse Description — SPRING 2018

MATH 581

EXTREMAL GRAPH THEORY

Instructor: A. Kostochka, 234 Illini Hall, 265-8037, kostochk@math.uiuc.edu.

TEXT: D. B. West, ”The Art of Combinatorics”, Volume I: Extremal GraphTheory. There also could be some handouts. Specialized texts covering the topics of thecourse will be listed on the course web site and will be on reserve in the library.

TOPICS: This is a companion course to Math 582 (Structure of Graphs). The two courses areindependent and discuss advanced material in graph theory. Extremal Graph Theory includestopics drawn from the following.

Trees and Distance: Optimization with trees; diameter and distance; encodings and embed-dings.

Matching and Independence: Bipartite matching, min-max relations, van der Waerden con-jecture; algorithms and applications (weighted matching, Menger’s Theorem, Hopcroft-Karpalgorithm, randomized on-line matching, stable matching); factors (Tutte’s f -Factor Theorem,Edmonds’ Blossom Algorithm); independent sets and covers, dominating sets and hypergraphtransversals.

Coloring: Vertex colorings (bounds, generalized colorings); structure of k-chromatic graphs(color-critical graphs, forced subgraphs); edge-colorings (Vizing’s Theorem and extensions);variations and generalizations (interval, list, circular colorings). Colorings of hypergraphs.

Perfect Graphs and Intersection Graphs: Perfect and imperfect graphs (Perfect Graph The-orem, partitionable graphs); classes of perfect graphs (chordal, interval, threshold, perfectlyorderable, etc.).

Other Extremal Problems: Forbidden subgraphs (Turan’s Theorem, Erdos-Stone Theorem);graph Ramsey theory; graph decomposition (arboricity, fractional arboricity, paths and cycles);representation parameters (intersection number, boxicity, interval number), etc.

COURSE REQUIREMENTS: There are no exams. There will be 5 problem sets, each requiring5 out of 6 problems for 50 points total. The problems require proofs related to or applyingresults from class. Roughly speaking, 85% of these points suffices for an A, 66% for a B.Discussions between students about problems can help understanding. Collaborations shouldbe acknowledged, and submitted homework should be written individually. Electronic mail isa good way to ask questions about homework problems or other matters.

PREREQUISITE: Math 580, or Math 412, or CS 473, or consent of instructor.

Math 584

Professor Balogh

PREREQUISITES: Math 580 or consent of instructor, obtainable by familiarity with elementary combinatorics. COURSE REQUIREMENTS: There will be 3 homework assignments. Homework allows you to choose five out of six problems to write up. Problems are worth 5 points each, so the maximum score/homework is 25 points. For Math 584 students A is from 80%, grade drops by 5% (so 75% is A-), for CS 575 A is from 75%, grade drops by 5% (so 70% is A-). Make up possibilities include giving lecture. TEXT: Jiri Matousek: 33 Miniatures, Mathematical and Algorithmic Applications of Linear Algebra + selected research papers. TOPICS: About half of the course is about the linear algebraic methods in combinatorics. Recent new breakthrough results should be included. Quarter of the course is focusing the Szemeredi Regularity Lemma and its applications; including basic Fourior methods. The remaining quarter will cover the recent "Container Method", and its applications

Math 595

Klara Buysse

In this course, Insurance companies will teach about different subjects of the Actuarial World. This Practicum course will give students the opportunity to have some more insight in real-world actuarial problems.

Math 595: Algebraic curves and surfaces

Spring 2018

Time: ???

Location: ???

Instructor: Emily Cliff.

Office hours: ???

Prerequisites: It will be assumed that most students will be (at least moder-ately) familiar with the material in Chapters II and III of Hartshorne’s AlgebraicGeometry, that is, foundations of schemes and their cohomology. However, ifyou are interested and motivated, and willing to do some independent reading,I will work with you to make it possible for you to follow and benefit from thecourse. In that case please contact me before registering.

Course content: We will cover selected topics from Chapters IV and V ofHartshorne’s Algebraic Geometry. The idea is to work with applications of theconcepts from Chapters II and III to concrete examples of curves and surfaces.

More specifically, we will begin our study of curves with a discussion of theRiemann-Roch theorem. We will then discuss two ways of explicitly representingand studying curves: as a branched covering of P1, or as a curve embedded in ahigher dimensional projective space. We will also discuss topics in the geometryof algebraic surfaces. In particular we will study ruled surfaces, and see howwe can use our knowledge of curves to gain a better understanding of thesesurfaces. We will also look at the example of non-singular cubic surfaces in Pn.

These topics are not set in stone; students with particular interests buildingon (or confusions relating to) the theory of schemes should feel free to suggesttopics that are exciting or helpful to them, before or during the term.

Grading: Grades will be assigned on the basis of problem sets, attendance,and participation. Problem sets will be assigned; only some questions will bemandatory but students should not expect to internalize much of the materialwithout engaging with examples such as those on the problem sets.

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University of Illinois Department of Mathematics 273 Altgeld Hall 244-0539

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C O U R S E D E S C R I P T I O N

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Spring 2018

MATH 595

L O C A L C O H O M O L O G Y

Prof. S. P. Dutta 11:00-12:20 Tu-Th

This course will be a study of Local Cohomology, introduced by Grothendieck, with various applications. The main topics will include: Cohen-Macaulay Rings and Modules, Injective Modules over noethierian rings, Gorenstein rings, local cohomology -- connection with dimension and depth, local duality theorem of Grothendieck, Cohomology of quasi-coherent and coherent sheaves, Serre’s Theorem on coherent sheaves on projective spaces, classification of Line-bundles on Pn, Hartshorne - Lichtenbaum Theorem and Faltings Connectedness Theorem.

Prerequisite: Math 502 Recommended Text: 1. Local Cohomology by R. Hartshorne; 2. Local Cohomology by Brodmann and Sharp, Cambridge University Press.

Math 595 Calculus on Meshes

Spring 2018 (Half-semester: Second 8 Weeks)

Instructor: Anil Hirani, hirani@illinois.edu

Course Description: Introduction to finite element discretization of exterior calculus on piecewise linearmanifold simplicial complexes (meshes). Approximate syllabus:

1. Finite element method (3 lectures)1.1 Introduction to Galerkin methods and finite elements (1 lecture)1.2 Consistency, convergence and stability of numerical methods (1 lecture)1.3 Computational examples (1 lecture)

2. Hilbert complexes (6 lectures)2.1 Basic definitions and Poincaré inequality (1 lecture)2.2 Hodge Laplacian and Hodge decomposition on Riemannian manifolds (1 lecture)2.3 Hodge decomposition in vector spaces and in Hilbert complexes (1 lecture)2.4 Abstract Hodge Laplacian and mixed formulation (1 lectures)2.5 Well-posedness of the mixed formulation (2 lectures)

3. Abstract approximation of Hilbert complexes (3 lectures)3.1 Bounded cochain projections and harmonic gap (1 lecture)3.2 Stability and convergence of the mixed method (2 lectures)

4. de Rham complex as a Hilbert complex (2 lectures)4.1 Square integrable forms and HΛk spaces (1 lectures)4.2 Hodge Laplacian problems in 1, 2 and 3 dimensions (1 lecture)

5. Approximations of the de Rham complex (7 lectures)5.1 Polynomial differential forms (P −

r Λk and PrΛ

k ) and the Koszul complex (2 lectures)5.2 Lowest order Whitney forms (P −

1 Λk spaces) and their properties (2 lectures)

5.3 Degrees of freedom and finite element differential forms (2 lectures)5.4 Constructing P −

r Λk from barycentric coordinates (1 lecture)

“Any young (or not so young) mathematician who spends the time to master this paper will have toolsthat will be useful for his or her entire career.” – AMS Mathematical Reviews, on the 2006 paper below.

Readings: Notes provided by instructor and two papers by Arnold, Falk and Winther: (i) Finite elementexterior calculus: from Hodge theory to numerical stability, Bulletin of the AMS, 2010; and (ii) Finite ele-ment exterior calculus, homological techniques, and applications, Acta Numerica, 2006.

Grading: Homeworks and a term paper.

Prerequisites: This course is designed for mathematics graduate students with knowledge of calculus onmanifolds (Math 481 or Math 518 or equivalent) and who have some basic knowledge of functional anal-ysis at the level of definitions (operator norms, bounded and unbounded operators). Some basic defini-tions and examples needed from elementary algebraic topology (chains, cochains, boundary, cobound-ary, homology and cohomology) will be either covered or notes provided, depending on the backgroundof the students. No prior knowledge of partial differential equations will be assumed or needed.

MATH 595 Advanced Topics in Mathematics: Water Waves Spring 2018

Water waves encompasses a wide range of phenomena, from ripples driven by surface tension

to rogue waves and tsunamis. The phrase describes the situation, where water lies below a

body of air and is acted upon by gravity.

While water waves have stimulated a considerable part of historical developments in the

theory of wave motion, they present profound and subtle difficulties for rigorous analysis,

modeling, and numerical simulation. Notably, the interface between the water and the air

is a priori unknown and to be determined as part of the solution. Free boundaries are

mathematically challenging in their own right. In addition, boundary conditions at the free

surface are severely nonlinear, presenting further challenges.

We shall focus on some latest developments in the mathematical aspects of water waves.

They include:

(1) Global regularity versus finite-time singularities for the Cauchy problem.

(2) Existence of traveling waves and their classification.

(3) Stability and instability of traveling waves.

Emphasis is on large scale dynamics and genuinely nonlinear behaviors, such as breaking

and peaking, which ultimately rely on analytical proofs for an acute understanding.

Instructor Vera Mikyoung Hur, verahur@math.uiuc.edu.

Text No official textbook. I will provide lecture notes and reading material.

Prerequisite MATH 540, MATH 553 or MATH 554 are useful, but not strictly required,

provided that you have a strong undergraduate background on real analysis and PDEs and

you are willing to work hard.

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Math 595 MNA Spring 2018Methods in Nonlinear Analysis and Applications to Differential Equations

Class time: TR

Lecturer: Eduard Kirr, e-mail: ekirr@illinois.edu

Description: The first part of the course will focus on fixed point theorems and theirapplications to differential equations. Metric and Banach spaces will be reviewed andCalculus in Banach spaces will be introduced before proving the Banach fixed pointtheorem and its consequences e.g., the implicit function theorem (IFT) in (infinite)dimensional Banach spaces. Its applications to existence, uniqueness and continuousdependence of data for systems of ordinary differential equations will be quickly re-viewed while the applications of the IFT to Lyapunov-Schmidt decomposition andlocal bifurcation theory will be discussed in detail with examples from nonlinear op-tics and statistical physics. Tha application to existence, uniqueness and continuousdependence of data for evolution partial differential equations is left for the sec-ond part of the course. The first part will continue with more sophisticated fixedpoint theorems based on the (Brouwer, Leray-Schauder) degree. The finite dimen-sional (Brouwer) degree will be introduced analytically albeit its deep relations tohomotopy and homology theory will be discussed as will its applications to Brouwer,Borsuk-Ulam, Ham and Sandwich Theorems. The degree will then be extended tocompletely continuous (compact but usually nonlinear) maps between Banach spacesand used to prove Leray-Schauder type fixed point theorems. Their applications tosolutions of partial differential equations will be discussed. Moreover an axiomaticview of the degree theory will be presented allowing for generalizations with appli-cations in local and global bifurcation theory. The first part of the course will finishwith a discussion of the contribution made by the degree theory in understandingthe collapse of Tacoma-Narrows bridge, a phenomenon that was not predicted bylaboratory simulations or the structural and dynamical stability theory preceding itsconstruction.

The second part of the course will focus on variational methods and, if type permitsthe extension of some of the results in the first part to non-completely continuous(non-compact) maps. The latter will be rather independent on the first part asdegree theory will be replaced for example by theory of real analytic functions toobtain an analytic global bifurcation theory. The former will be based on the rigorouscalculus in Banach spaces and replaces the more common but ad-hoc “calculus ofvariation” in which the definition of “variation” seems to change from problem toproblem. Moreover, when it comes to applications in finding certain equilibria orperiodic solutions in partial different equations, the variational methods have to copewith non-convex functional and non-compact constraints. We will discuss how tocompensate for these shortcomings via Rellich or concentration compactness and thenapply the classical theory which of course will be introduced.

References: I will follow my own notes (posted online) based on the following references:

1. Topics in Nonlinear Functional Analysis by L. Nirenberg.

2. Leray-Schauder degree: a half century of extensions and applications by J. Mawhinin Topol. Methods Nonlinear Anal. Volume 14, Number 2 (1999), 195-228.

3. Partial Differential Equation by Lawrence C. Evans.

4. Long time dynamics and coherent states in nonlinear wave equations by E. Kirrin Recent Progress and Modern Challenges in Applied Mathematics, Modelingand Computational Science, R. Melnik, R. Makarov, J. Belair, Eds. in FieldsInstitute Communications, 2017, pp. 59–88.

5. Large Torsional Oscillations in a Suspension Bridge: Multiple Periodic Solutionsto a Nonlinear Wave Equation by K.S. Moore in SIAM J. Math. Anal. Vol. 33,No. 6, pp. 1411-1429

Grading Policy: There are no homework assignments or exams for this course. The par-ticipants will be asked to make a presentation on the applications of these techniquesto a nonlinear PDE preferably from their own research area. Grades will be based onclass activity, and on the quality of the presentation.

Predictive Analytics Dr. Daniël Linders

Introduction Data are an important source for insurance companies and financial institutions to learn about their

business model and to develop future improvements. In this course we will give an overview of the

different methodologies one can employ to search for relationships in a dataset and to build a

predictive model. The theory will be applied in different fields of finance and insurance.

Outline 1. What is statistical learning

2. Programming in R

3. Linear regression

4. Classification

5. Linear model selection and regularization

6. Tree based methods

7. Application 1: auto insurance data

8. Application 2: predicting stock market returns

Prerequisites Basic knowledge about probability, statistics and programming is required.

Knowledge about programming is a plus.

BOUNDED COHOMOLOGY,

AMENABILITY AND HYPERBOLICITY.

MATH 595, the first half of Spring 2018.

Igor Mineyev, 3:00 pm MWF, 347 Altgeld Hall.

First we will define bounded cohomology of groups and topological spaces, then exhibit itsrelations to analytic, geometric, and group theoretic concepts, notably amenable groups andhyperbolic groups. The beauty of the subject comes from the fact that it connects severalareas of mathematics. Introduced by B.E. Johnson in the 70’s under the name “cohomologyof Banach algebras” (a purely analytic notion), bounded cohomology later attracted topolo-gists/geometers, especially after a seminal paper “Volume and bounded cohomology” by Gro-mov. This relates it to the notion of simplicial volume of a manifold. In the case of a hyperbolicmanifold, Gromov and Thurston showed that the simplicial volume (think of the surface of apretzel) is proportional to the usual volume. This gives an alternative definition of volume thatcan be defined for any topological space. I will also present a characterization of hyperbolicgroups in terms of bounded cohomology. Just as ℓ∞ is dual to ℓ1, bounded cohomology is dualto summable homology, which is defined in a dual fashion.

An orange can be cut into finitely many rigid pieces and the parts can be rearrangedso that the result is TWO oranges, each exactly the same as the original one. This is amathematically correct statement if we assume the axiom of choice, which we all do. Thisstatement is known as the Banach-Tarski paradox. A surprising ingredient in its proof is theuse of group actions, specifically, actions of free groups. Why can such a strange thing happento oranges? Because oranges, and most free groups, lack amenability. Amenable groups admitmany equivalent definitions: combinatorial, measure-theoretic, in terms of C∗-algebras, etc.They are also characterized by not having the above Banach-Tarski paradox. Already in hisoriginal paper, Johnson gave a characterization of amenable groups via bounded cohomology.We will discuss this analytic notion, talk about bounded cohomology of free groups and othergroups. One notoriously difficult open problem asks: Is the Thompson group amenable? If timeallows, we will also mention results of Linnell and Morris that show that for amenable groups,left-orderability is equivalent to local indicability, and also to certain geometric conditions onthe group.

There is no required prerequisite and no required textbook for the course.

MATH 595—HOMOLOGICAL MIRROR SYMMETRY—SPRING 2018

This course is actually two half-semester courses. Students mustregister for the two halves separately. An enrollment threshold mustbe met for the second half to run.

Course meets: MWF 1:00–1:50 p.m.

Instructor: James Pascaleff (jpascale@illinois.edu)

Course web page: faculty.math.illinois.edu/∼jpascale/courses/2018/595

Description: Homological Mirror Symmetry (HMS) is the study of the relations between three types ofmathematical objects:

symplectic manifolds←→ triangulated categories←→ algebraic varieties

For a symplectic manifold X, there is a triangulated category F(X) called the Fukaya category, and for analgebraic variety Y there is a triangulated category D(Y ) called the derived category. We then pose theproblem of finding pairs X and Y such that

F(X) ∼= D(Y )

The origin of this relation is in theoretical particle physics, where the two categories are interpreted ascollections of D-branes, and the relation expresses the duality between A-twisted topological string theoryon X and B-twisted topological string theory on Y .

The investigation of this relation raises many questions. How are the two sides actually defined? How dowe compute the two sides, and what should the “answer” of such a computation look like? What generalstructure is present that constrains the problem? The goal of this course is to set up the machinery andunderstand the solution in a specific case: when X is a hypersurface in projective space, including the quinticthreefold, following Seidel and Sheridan. Topics to include:

• Categories: triangulated, differential graded, A∞.

• Algebraic varieties, categories of coherent sheaves.

• Symplectic manifolds, Lagrangian Floer cohomology, Fukaya categories.

• Case of surfaces, HMS for the two-torus, other relatively simple models.

• Hypersurfaces in projective space.

Prerequisites: This is an advanced topic that connects to many areas of mathematics, so I could list manythings here. However, that would defeat the purpose of this course, which is to give students access to thisarea of research. That said, some prior knowledge of abstract algebra and differential topology is necessaryto get anything out of this course. The course on Symplectic Geometry taught concurrently by Prof. Tolmanwould be helpful, but is not required.

Texts:

• P. Aspinwall, et al., Dirichlet branes and mirror symmetry.

• P. Seidel. Fukaya categories and Picard-Lefschetz theory.

• P. Seidel. Homological mirror symmetry for the quartic surface.

• N. Sheridan. Homological mirror symmetry for Calabi-Yau hypersurfaces in projective space.

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