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Advanced Topics Courses

Fall 2016
Course Section CRN Title and Description
Math 595 CLS 49185 Correlations and Local Spacings in Number Theory (Zaharescu)
See full course description.
Math 595 CP 64474 Classic Papers in Algebraic Geometry and Geometric Representation Theory (Nevins)
See full course description.
Math 595 DLG 64464 Dynamics of Large Groups (Solecki)
See full course description.
Math 595 HCT 66669 Higher category theory and quasicategories (Rezk)
See full course description.
Math 595 HM 64469 Homological methods in group theory and topology (Mineyev)
See full course description.
Math 595 IA 64467 Introduction to Affine Algebraic Groups (Haboush)
See full course description.
Math 595 LF 64475 Local Fields (Allen)
See full course description.
Math 595 STP 58570 Special Topics in Probability (Dey)
See full course description.
Math 598 OA 60303 Operator Algebras (Boca)
Math 598 REN 47921 Research Experience in Number Theory (Berndt)

Spring 2016
Course Section CRN Title and Description
Math 595 AC 52596 Algebraic Combinatorics (Yong)
See full course description.
Math 595 AT 45994 Applied Topology (Baryshnikov)
See full course description
Math 595 CRM 63212 Computational Risk Management of Equity-Linked Life Insurance (Feng)
See full course description.
Math 595 FT 39571 Two Theorems in Contact and Symplectic Topology (Kerman)
See full course description.
Math 595 GTG 63213 Geometry, Topology and Group Theory (Leininger)
See full course description.
Math 595 HM 63214 Holomorphic Mappings (D’Angelo)
See full course description.
Math 595 IT 63215 Intersection Theory (Dutta)
See full course description.
Math 595 MG2 63216 Modern Algebraic Geometry, II (Nevins)
See full course description.
Math 595 PW 63217 Polyhedra in the Wild (Davidson)
See full course description.
Math 598 OA 61174 Operator Algebras (Junge)
Math 598 REN 45371 Research Experience in Number Theory (Berndt)
See full course description.

Fall 2015
Course Section CRN Title and Description
Math 595 CA 64464 Computer Animation (Francis) 1st 8-week course
See full course description.
Math 595 CLA 64465 Cluster Algebras (Di Francesco)
See full course description.
Math 595 ECM 64475 Elliptic Curves & Modular Forms (Luu)
See full course description.
Math 595 EG 64466 Expander Graphs in Number Theory (Fuchs)
See full course description.
Math 595 ES 64467 Exponential Sums (Zaharescu)
See full course description.
Math 595 GMT 64468 Geometric Measure Theory (Tyson)
See full course description.
Math 595 IAG 64470 Introduction to the Theory of Algebraic Groups (Haboush)
See full course description.
Math 595 MC 39224 Moduli of Curves (Katz)
See full course description.
Math 595 PDE 64472 Topics in PDEs (Agbanusi)
See full course description.
Math 595 SEE 49185 Smooth and Etale Extensions (Dutta)
See full course description.
Math 595 SPG 64474 Symplectic & Poisson Geometry (Fernandes)
See full course description.
Math 595 VDF 64476 Valued Differential Fields (van den Dries)
See full course description.
Math 598 OA 60303 Operator Algebras (Boca)
Math 598 REN 47921 Research Experience in Number Theory (Berndt)
See full course description.

Spring 2015
Course Section CRN Title and Description
Math 595 AGI 61537 Algebraic Geometry II (Lo)
See full course description.
Math 595 SRS 61538 Super Riemann Surfaces (Katz)
See full course description.
Math 595 CET 61539 Classification in Ergodic Theory (Tserunyan)
See full course description.
Math 595 CT 61536 Coding Theory (Duursma)
See full course description.
Math 595 CGC 61540 Geodesic Currents on Free Groups (Kapovich)
See full course description.
Math 595 LC 39675 Local Cohomology (Dutta)
See full course description.
Math 595 LD 38183 Large Deviations (Sowers)
See full course description.
Math 598 REN 47921 Literature Seminar in Number Theory (Berndt)
See full course description.
Math 595 LF 62534 L-Functions (Zaharescu)
See full course description.
Math 598 OA  

Literature Seminar in Operator Algebras (Junge)

Math 598 REN 47921 Literature Seminar in Number Theory (Berndt)
See full course description.

Fall 2014
Course Section CRN Title and Description
Math 595 AGI 42955 Algebraic Geometry I (Katz)
See full course description.
Math 595 BS 46711 Banach Spaces (Ruan)
See full course description.
Math 595 CCT 62765 Curve Complexes and Surface Topology (Dowdall)
See full course description.
Math 595 CFT 39222 Class Field Theory (Duursma)
See full course description.
Math 595 GT 62756 Graph Theory (Balogh)
See full course description.
Math 595 IMS 62764 Integrable Models, Statistical Physics and Combinatorics (Di Francesco)
See full course description.
Math 595 MT 51377 Model Theory (van den Dries)
See full course description.
Math 595 MTF 63378 Mock Theta Functions (Berndt)
See full course description.
Math 595 QIT 62754 Quantum Information Theory (Junge)
See full course description.
Math 595 RGT 62758 Reflections of Group Theory (Mineyev)
See full course description.
Math 595 RIS 62753 Representation Theory & Integrable Systems (Bergvelt)
See full course description.
Math 595 TTA 62762 Top Ten Algorithms (Baryshnikov)
See full course description.
Math 598 OA 60303 Operator Algebras (Junge)
Math 598 PFM 60173 Preparing Future Mathematicians (Laugesen)
See full course description.
Math 598 REN 47921 Literature Seminar in Number Theory (Berndt)
See full course description.

Spring 2014
Course Section CRN Title and Description
Math 595 BVA 60052 Bifurcation Theory (Kirr)
See full course description.
Math 595 DST 49974 Descriptive Set Theory (Solecki)
See full course description.
Math 595 ET 60050 Ergodic Theory (Rosenblatt)
See full course description.
Math 595 MTC 54387 Mathematical Tools (Kutzarova)
See full course description.
Math 595 OS 54383 Operator Spaces (Junge)
See full course description.
Math 595 PG 60051 Poisson Geometry (Fernandes)
See full course description.
Math 595 SFY 44536 Symmetric Functions and Young Tableaux (Yong)
See full course description.
Math 595 TP 43499 Theory of Partitions (Berndt)
See full course description.

Fall 2013
Course Section CRN Title and Description
Math 595 AG2 42955 Algebraic Geometry II (Katz)
See full course description.
Math 595 ANT 51380 Additive Number Theory (Ford)
See full course description.
Math 595 FAO 61080 Faces of amenability:  groups, actions, measures, space (Mineyev)
See full course description.
Math 595 IEG 61079 (In)finite ergodic theory (Cellarosi)
See full course description.
Math 595 SE2 42964 Smooth and Etale Extensions II (Dutta)
See full course description. (to be posted)
Math 595 SGT 61081 Symplectic geometry and topology (Kerman)
See full course description.
Math 595 STG 61082 Selected Topics in Graph Theory, III (Kostockha)
See full course description.
Math 595 OMS 55580 O-Minimal Structures (Hieronymi)
See full course description.
 
Math 598 ARB 61421 Around Rolling Balls (Baryshnikov and Arnold) See full course description.
Math 598 PFM 60173 Preparing Future Mathematicians (Laugesen) See full course description.
Math 598 OA 60303 Operator Algebras (Junge)
See full course description.
Math 598 REN 47921 Literature Seminar in Number Theory (Berndt) See full course description.

Spring 2013
Course Section CRN Title and Description
Math 595 GIT 39565 Geometric Invariant Theory (Haboush)
See full course description.
Math 595 BS 38186 Banach Spaces (Junge)
See full course description.
Math 595 IC 51218 Integrable Combinatorics (Di Francesco)
Meets 14-Jan-13 - 08-Mar-13.
See full course description.
Math 595 IQS 58449 Introduction to q-series and Entries from Ramanajun’s Lost Notebook (Berndt)
See full course description.
Math 595 MGO 58451 Metric Geometry and orderability of Groups (Mineyev)
See full course description. (to be posted)
Math 595 MNT 58452 Multiplicative Number Theory (Zaharescu)
See full course description.
Math 595 OA 58453 Topics in Operator Algebras (Brannan)
See full course description.
Math 595 SEE 56132 Smooth and Etale Extensions (Dutta)
See full course description.
Math 595 TGT 58450 Selected Topics in Graph Theory (Kostochka)
See full course description.

Fall 2012
Course Section CRN Title and Description
Math 595 AC 40114 Algebraic Combinatorics (Yong)
See full course description.
Math 595 EC 42963 Elliptic Curves (Duursma)
See full course description.
Math 595 L21 59513 L2 Invariants in Topology and Group Theory (Mineyev)
See full course description.
Math 595 MF 46714 Modular Forms (Ahlgren)
See full course description.
Math 595 STG 60384

Selected Topics in Graph Theory I (Balogh)

Spring 2012
Course Section CRN Title and Description
Math 595 ACA 57626 Analytic Combinatorics and Algorithms (Baryshnikov)
See full course description.
Math 595 ACS 56590 Analytic Continuation in One and Several Variables (Tumanov)
See full course description.
Minicourse: 17-Jan-12 - 09-Mar-12
Math 595 CGV 48262 Computer Graphics and Geometric Visualizations (Francis)
See full course description.
Minicourse: 17-Jan-12 - 09-Mar-12
Math 595 LC 39575 Local Cohomology (Dutta)
See full course description.
Math 595 MTV 56594 Model Theory of Valued Fields (van den Dries)
See full course description.
Math 595 RF 57509 Geometric Topology (La Nave)
See full course description.
Math 595 RLA 56591 The Regularity Lemma and its Applications (Balog)
See full course description.
Math 595 SIC 56592 A Stern Introduction to Combinatorial Number Theory (Reznick)
See full course description.

Fall 2011
Course Section CRN Title and Description
Math 595 AG2 42955 Algebraic Geometry II (Schenck)
See full course description.
Math 595 AI 57741 Anatomy of Integers (Ford)
See full course description.
Math 595 BS 46711 Banach Spaces (Ruan)
See full course description.
Math 595 DS 52329 Distribution of Sequences in Number Theory (Zaharescu)
See full course description.
Math 595 GT 57742 Geometric Topology (Leininger)
See full course description.
Minicourse: 22-Aug-11 - 14-Oct-11.
Math 595 HT1 57743 Topics in Homotopy Theory (Ando)
See full course description.
Minicourse: 22-Aug-11 - 14-Oct-11.
Math 595 HT2 57744 Topics in Homotopy Theory (Rezk)
See full course description.
Minicourse: 17-Oct-11 - 07-Dec-11.
Math 595 MT 51377 Morse Theory with applications to Riemannian geometry and
symplectic topology (Kerman)
See full course description.
Math 595 MW 49184 Mathematical Writing (Hildebrand)
See full course description.
Minicourse: 22-Aug-11 - 14-Oct-11.
Math 595 RWT 57749 Random walks and queueing theory in biology (Bauer)
See full course description.
Minicourse: 22-Aug-11 - 14-Oct-11.
Math 595 SLE 58616 An Introduction to SLE (Bauer)
See full course description.
Minicourse: 17-Oct-11 - 07-Dec-11.
Math 595 STP 58570 Spectral Theory of Partial Differential Equations (Laugesen)
See full course description.
Minicourse: 17-Oct-11 - 07-Dec-11.

Spring 2011
Course Section CRN Title and Description
Math 595 CS 55673 Closure Systems (Jamison)
See full course description.
Math 595 DSL 54379 Dynamics of SL(2)-actions (Athreya)
See full course description.
Math 595 HA 55938 Homological Algebra (Schenck)
See full course description.
Mini-Course: 18-Jan-11 - 11-Mar-11
Math 595 MAM 53316 Methods of Applied Mathematics (DeVille)
Math 595 MTM 39573 Mapping Theory in Metric Spaces (Tyson)
See full course description.
Math 595 OS 54383 Operator Spaces (Ruan)
See full course description.
Math 595 RTM 54384 Random Techniques in Metric Spaces with Applications to Compressed Sensing (Junge)
Math 595 RZF 54386 The Riemann Zeta Function and L-functions (Zaharescu)
See full course description.
Math 595 SEE 56132 Smooth and Etale Extensions (Dutta)
See full course description.
Math 595 SS 54388 Stochastic Simulation (Sowers)
See full course description.
Mini-Course: 18-Jan-11 - 11-Mar-11
Math 595 TG 54387 The Grassmannian (Yong)
See full course description.
Mini-Course: 14-Mar-11 - 4-May-11

Fall 2010
Course Section CRN Title and Description
Math 595 AG2 42955 Algebraic Geometry II (T. Nevins)
See full course description.
Math 595 BC 55577 Bounded Cohomology (I. Mineyev)
See full course description.
Math 595 CAS 41485 Curves on an Algebraic Surface (W. Haboush)
See full course description.
Math 595 CFT 39222 Class Field Theory (I. Duursma)
See full course description.
Math 595 DM 55578 Discharging Methods (A. Kostochka)
See full course description.
Math 595 SG 42972 Symplectic Geometry (S. Tolman)
See full course description.
Math 595 TG 55580 The Grassmannian (A. Yong)
See full course description.
Mini-Course: 18-Oct-10 - 08-Dec-10
Math 595 TP 42970 Theory of Partitions (B. Berndt)
See full course description.
Math 595 UCA 55581 Uniformization and Rigidity in Complex Analysis (S. Merenkov)
See full course description.
Mini-Course: 18-Oct-10 - 08-Dec-10

Spring 2010
Course Section CRN Title and Description
Math 595 BS 38186 Banach Spaces (M. Junge)
See full course description.
Math 595 CEV 53655 The Cauchy-Riemann Equations in Several Variables (J. D'Angelo)
See full course description.
Mini-Course: 19 Jan 2010 - 12 Mar 2010
Math 595 CGV 48262 Computer Graphics & Geometrical Visualization (G. Francis)
See full course description.
Mini-Course: 19 Jan 2010 - 12 Mar 2010
Math 595 DN 53838 Dynamics on Networks (E. Lerman)
See full course description.
Math 595 ENT 52597 Elliptic Functions with Applications to Number Theory (B. Berndt)
See full course description.
Math 595 HA2 52598 Harmonic Analysis II (X. Li)
See full course description.
Math 595 HFG 53440 Hermitian forms and CR geometry (J. Lebl)
See full course description.
Mini-Course: 15 March 2010 - 5 May 5 2010
Math 595 LC 39575 Local Cohomology (S. Dutta)
See full course description.
Math 595 LP 52599 Levy Processes (P. Kim)
See full course description.
Math 595 MAM 53316 Methods of Applied Mathematics (L. DeVille)
See full course description.
Math 595 PM 46003 Probabilistic Combinatorics (R. Song)
See full course description.
Math 595 PNT 52600 Polynomials in Analysis and Number Theory (K. Stolarsky)
See full course description.

Fall 2009 Math 595 -- Regular Courses
Course Section CRN Title and Description
Math 595 AG2 42955 Algebraic Geometry II (L. Li)
See full course description.
Math 595 AMA 53550 Asymptotic Methods in Analysis (A.J. Hildebrand)
See full course description.
Math 595 ANT 51380 Algorithmic Number Theory (I. Duursma)
See full course description.
Math 595 MCG 53538 Mapping Class Groups (C. Leininger)
See full course description.
Math 595 OA 54652 Operator Algebras (Z. Ruan)
See full course description.
Math 595 TV 53566 Toric Varieties II (H. Schenck)
See full course description.
Math 595 TIL 53560 Finite Model Theory, Measure Theory, and Structure of Polish Groups (S. Solecki)
See full course description.

Fall 2009 Math 595 Mini Courses
Course Section CRN Title and Description
Math 595 GHS 53542 Gromov Hyperbolic Spaces and their Boundaries (J. Mackay)
See full course description.
Meets Aug 24 - Oct 16, 2009.
Math 595 KT 55232 K-Theory (M. Junge)
See full course description.
Meets Oct 19 - Dec 10, 2009.
Math 595 LD 53563 Large Deviations (R. Sowers)
See full course description.
Meets Aug 24 - Oct 16, 2009.
Math 595 SFM 53543 Symmetric functions and Macdonald polynomials (R. Kedem)
See full course description.
Meets Oct 19 - Dec 10, 2009.

 

Spring 2009 Math 595 Regular Courses
Course Section Course Reference Number Title and Description
Math 595 AEN 46008 Advanced Topics in Elementary Number Theory (P. Pollack)
In a typical first course in number theory, there are many theorems mentioned but not proved; these often include Gauss's characterization of the integers which are sums of three squares, Dirichlet's theorem on primes in arithmetic progressions, Waring's assertion on sums of kth powers of integers, the prime number theorem, etc. Many of these problems have solutions which, while intricate, do not require advanced ideas. See full course description.
Math 595 APA 49979 Analytic and Probabilistic Aspects of Continued Fractions (F. Boca)
Continued fractions, both one and multi-dimensional, arise in a large number of instances in mathematics. We plan to discuss certain connections between continued fractions and functional analysis, probability, ergodic theory, and dynamical systems. Topics will include: Gauss-Kuzmin-Levy theory, Perron-Frobenius type operators, elements of dynamics on homogeneous spaces, symbolic dynamics, multi-dimensional continued fraction algorithms, AF algebras. Prerequisite: Math 540 or approval of instructor.
Math 595 BC 49982 Bounded Cohomology (I. Mineyev)
The course will be an introduction to bounded cohomology and its various faces and appli cations. If time allows, at the end of the course we might go deeper into homological algebra and discuss Hochschild-Serre spectral sequence for bounded cohomology. No textbook is required, we will be using papers by various authors. See full course description.
Math 595 BVM 51394 Bifurcation and Variational Methods in Nonlinear Partial Differential Equations (E. Kirr)
The course focuses on two powerful methods in studying properties of solutions of nonlinear partial differential equations (pde's). Both methods are based on studying nonlinear maps between Banach spaces via calculus in Banach spaces.

The first method views solutions of pde's as zeroes of a nonlinear map between Banach spaces. The implicit function theorem in Banach spaces combined with Lyapunov-Schmidt reduction gives information on the number of solutions of the pde and pinpoints the bifurcation points, namely values of coefficients in the equation where the number of solutions jumps. Recent applications of this technique to symmetry breaking phenomena in optics, statistical physics and molecular chemistry will be presented. If time permits extensions to Nash-Moser type implicit function theorems and their applications to Kolmogorov-Arnold-Moser (KAM) theory will be discussed.

The variational method exploits the fact that certain equilibrium or time periodic solutions of pde's are given by critical points of nonlinear functionals. Existence of (constrained) minima or maxima of such functionals implies the existence of equilibrium/periodic solutions for the pde. However, as opposed to the classical theory the functionals related to pde's are in general not convex nor are their constraints compact. We will discuss how to compensate for their abscence with methods like the generalized Rellich compactness or concentration compactness. Moreover, if the functional is of Lyapunov type, in other words it is nonincreasing in time along solutions of the time dependent pde then its minima gives stable solutions of the time dependent pde. This can be refined to saddle points in case the dynamics is prevented to move in the decreasing directions of the saddle by, for example, conserved quantities in the dynamics. All these situations will be exemplified with recent results for physical models.

The course will attempt to be self-contained. While familiarity with Banach space in particular with Sobolev spaces will be useful the necessary material will be reviewed.

Math 595 CSF 45995 Symmetric Functions and Young Tableaux (A. Yong)
The goal of this course is to provide an introduction to the combinatorics of symmetric functions, and in particular the Schur functions. These objects appear throughout algebra, geometry and combinatorial enumeration. I'll take a purely combinatorial approach. Specifically, I will discuss the subject of tableau algorithms, such as the Robinson-Schensted-Knuth correspondence, jeu de taquin, the Littlewood-Richardson rule, and dual equivalence. Time permitting I will discuss related special topics. There are no prerequisites for graduate students. The grading will be based on problem sets and/or a presentation by the student on a paper to be chosen in consultation with the instructor." See full course description.
Math 595 DST 49974 Descriptive Set Theory (S. Solecki)
The course will concentrate on continuous actions of Polish groups. Most of the material will come from descriptive set theory, but several important connections will be made with topological dynamics, ergodic theory, Ramsey theory, and model theory. We will start with studying the complexity of the orbit equivalence relation induced by the partition of a Polish space into orbits of a continuous action of a Polish group. This topic is closely related to important classification problems in various areas of mathematics, and we will spend some time explaining these connections. This part of the material has a descriptive set theoretic flavor, and we will cover the needed background from this field. Next we will study the internal structure of Polish groups that are important in other areas of mathematics. We will concentrate on the group of all measure preserving automorphism of the Lebesgue measure space, the group of all isometries of the Urysohn metric space, and the group of all homeomorphism of the pseudo-arc. This part of the material has connections with ergodic theory and Ramsey theory and the needed background in these areas will be reviewed. Background: basic metric topology, basic analysis, some descriptive set theory will also be helpful but, strictly speaking, not necessary. Relevant books/papers will be announced/distributed in class.
Math 595 HCT 49977 Higher Category Theory (C. Rezk)
Higher category theory is the study of structures which are like categories, but are "higher-dimensional": while a category has objects (0 dimensions), and morphisms between objects (1 dimensions), higher dimensional analogues are allowed to have morphisms between morphisms (2 dimensions), and so on. The goal of this course is to describe some of the approaches to this topic. See full course description.
Math 595 IC 51218

Integrable Combinatorics (P. DiFrancesco)
Classical combinatorics is the art of counting, guessing and proving. It is the simplest way into many sophisticated physics and mathematics problems. In this course, we address various counting problems arising in theoretical physics, mostly statistical physics and field theory, and unravel their very rich mathematical structures, inherited from either classical or quantum integrability. We target an audience of both mathematicians and physicists. We will develop various technical tools, such as: matrix integrals, orthogonal polynomials, tree bijections, lattice paths and associated de- terminants, transfer matrices, (quantum) R-matrices, divided difference equations such as the quantum Knizhnik-Zamolodchikov (qKZ) equation and their multiple contour integral solutions. While we always put a special emphasis on the combinatorial aspects, each technique will be applied within its original physical context. However, each of the problems addressed will be put into a simple combinatorial form that does not require any prior knowledge. Conversely, all techniques will be self-contained and only basic mathematical knowledge is required. Applications range from quantum gravity to algebraic geometry, always in relation to simple two-dimensional lattice models. The scope of this course is to expose the extent and depth of various connections between mathematics and physics, as both inspirational tools and fields of application.

Math 595 IMF 49972 Introduction to Modular Forms (J. Rouse)
This course will cover the basic theory of modular forms for congruence subgroups, Hecke operators, and further topics chosen based on the available time and student interest. See full course description.
Math 595 LC2 43505 Local Cohomology II (S. Dutta)
This course can be viewed as a continuation of "Smooth and Etale extensions" offered in Spring 2007. The main goal of this course is to cover Popeseu's proof of Artin's conjecture on solution of polynomial equations on excellent rings-Swan calls the whole process "Neron-Popescu desingularization". The main topics will include the following: quasi-unramified, quasi-smooth and quasi-etale extensions, geometric regularity; Excellent rings; Hochster's construction of big Cohen-Macaulay Modules, Popescu's proof of Artin's Conjecture in characteristic o, Neron desingularization (special case), Popescu's proof of Artin's conjecture in characteristic p>o and in the mixed Characteristic and applications. Prerequisite: Math 502 Text: Neron-Popescu Desingularization - Expository paper by R. Swan, University of Chicago. 10:30-11:50 Tu-Th
Math 595 MBB 49971

D-Modules and Beilinson-Bern Localization (T. Nevins)
In this course, we’ll develop some basics of representation theory of finite-dimensional, simple complex Lie algebras, the geometry of flag varieties, and D-modules, and then put them all together to understand Beilinson-Bernstein localization and how it can be used to prove the Kazhdan-Lusztig conjecture.
The course will assume some cultural familiarity with complex Lie algebras and with algebraic varieties, but not with D-modules. A student who has taken a first course in representation theory and knows what an algebraic variety is, or who has taken a course in algebraic geometry and knows a few basic definitions about Lie algebras, should be able to follow the course. See full course description.

Math 595 NA 39574

Nonstandard Analysis (C. Ward Henson)
Nonstandard Analysis (NSA) is a framework for systematically applying some of the basic ideas of model theory to all areas of mathematics. It is especially effective in analysis, geometry, topology, and related areas of mathematics where the concept of limit is central. Forty years ago, the logician Abraham Robinson observed that the construction of nonstandard extensions could provide a rigorous foundation for the use of infinitesimals in basic analysis.1 Since then, applications of this set of ideas have spread through all of mathematics, greatly extending Robinson's original use of infinitely small and infinitely large numbers, and NSA has become an active branch of research in its own right.

In order to reach advanced applications of NSA in this course, we will assume a knowledge of first-order logic extending at least through the compactness theorem. Students should be able to formulate mathematical statements within first-order logic and should have some experience with nonstandard models. We will also use some tools (such as the construction of saturated models) from the beginning parts of model theory. After developing the basic framework of NSA we will give a substantial indication of how NSA is developed within two areas of advanced mathematics:

  • probability and stochastic analysis (based on the Loeb measure construction);
  • geometry and functional analysis (based on the nonstandard hull construction).
Prerequisites: A knowledge of first-order logic through the compactness theorem; what is covered in the first half of Math 570 at UIUC or in a good undergraduate course in logic will be sufficient.
See full course description.

Math 595 RB 41543 Cryptography - Theory and Practice (R. Blahut)
Course description to be posted.
Spring 2009 Math 595 Mini Courses
Course Section CRN Title and Description
Math 595 ANT 51267 Additive Number Theory (J. Balogh)
The course will focus on Roth's Theorem and on Szemeredi's Theorem on the existence of
arithmetic progressions in dense subsets of integers. We look at this theorem from several aspects: analytical, graph theoretical, and hypergraph theoretical points of view. In particular, I will spend lots of time on the hypergraph regularity lemma, based on Gower's paper. See full course description.
Meets Jan 20 - Mar 13, 2009.
Math 595 AVA 49992 Abelian Varieties and their Arithmetic (M. Sabitova)
The course will consist of three parts:
(1) basic geometric properties of abelian varieties,
(2) abelian varieties over the field of complex numbers,
(3) arithmetic of abelian varieties.
In part (1) we will talk about basic definitions and properties of abelian varieties over an arbitrary field K. In part (2) we will focus on the case K = C and discuss the related topics such as Riemann surfaces, the Riemann-Roch, Abel-Jacobi, and Lefschetz's theorems. Part (3) will be devoted to the discussion of the Tate's and Mordell's conjectures, L-functions and E-constants attached to abelian varieties and the Birch and Swinnerton-Dyer conjecture. See full course description.
Meets Mar 16 - May 6, 2009.
Math 595 HDS 51887 Hamiltonian Dynamics and Symplectic Topology (E. Kerman)
Hamiltonian dynamical systems are the general mathematical framework which describe classical mechanical systems such as a charged particle moving under the influence of an electromagnetic field, or the motion of celestial bodies under their mutual gravitational attraction. The first part of this course will be a survey of Hamiltonian dynamics with an emphasis on the presentation of many examples; from billiards to geodesic flows. Since energy is conserved in classical mechanical systems, Hamiltonian flows are highly recurrent. A tremendous amount of research has been devoted to the study of the orbits of these flows which are genuinely periodic. This includes a large part of the work of Poincare, and one can trace the roots of a great deal of modern mathematics to the study of these periodic motions. In the second part of the course we will discuss the variational approach to detecting periodic orbits and will survey several of the landmark results concerning their existence.
Meets Mar 16 - May 6, 2009.
Math 595 SGA 49969 Sub-Riemannian Geometry and Analysis (J. Tyson)
This minicourse will cover the foundations of (first-order) analysis, geometric measure theory and differential geometry in the Heisenberg group and more general sub-Riemannian manifolds.See full course description.
Meets Mar 16 - May 6, 2009.
Fall 2008 Math 595 Courses
Course Section CRN Title and Description
Math 595 ADS 51374 Advanced Descriptive Set Theory (C. Rosendal)
description to be posted.
Math 595 AG2 42955 Algebraic Geometry II (W. Haboush) This course will be based on the third chapter of Hartshorne, Algebraic Geometry. After a brief recollection of the geometry of projective schemes, I will give a brief overview of homological algebra. Then I will discuss injective and flabby (flasque) sheaves and cohomology as the right derived functor of global sections. Then I will discuss Cech cohomology and I will explicitly compute the cohomology of projective space. Then I will discuss Serre duality, smooth and etale morphisms and flatness. Time permitting, I will discuss birational morphisms, Zariski's main theorem and the semicontinuity theorem.
Math 595 ANT 51380 Additive Number Theory (K. Ford)
Prerequisites: Math 53l/equivalent or consent of the instructor Recommended Text: There is no official text for the course, but the following books contain much of the material. 1. Sequences, 2nd ed., by H. Halberstam and K. F. Roth, Springer-Verlag, New York- Berlin, 1983. 2. Additive Number Theory: The classical bases, by M. Nathanson, Springer GTM 164, 1996; Additive Number Theory: Inverse problems and the geometry of sumsets, by M. Nathanson, Springer GTM 165, 1996. 3. The Hardy-Littlewood method, 2nd ed., by R. C. Vaughan, Cambridge Tracts in Mathematics, vol. 125, 1997. Full course description at http://www.math.uiuc.edu/timetable/595KF_fall08.pdf
Math 595

CD

 

51375 Complex Dynamics (A. Hinkkanen)
Prerequisite: Math 542. Full course description is at http://www.math.uiuc.edu/timetable/595CD_fall08.pdf
Math 595 CFT 39222 Class Field Theory (S. Ullom)
Let K be an extension field of the rationals of finite degree. Class field theory is the study of all abelian extensions of K, that is, Galois extensions L of K such that the Galois group is an abelian group. By the Kronecker-Weber theorem every abelian extension of the rationals is a subfield of a field of roots of unity. Via the Artin symbol we will prove a general reciprocity law and derive the quadratic reciprocity and cubic reciprocity laws as special cases. We will develop the basic properties of abelian L-series and use these to outline the proofs of the main results of class field theory. Rather than prove every result in detail we will give several applications such as the local-global principle for quadratic forms over number fields. We will prove that the ideal class group of K is isomorphic via the Artin map to the Galois group over K of the maximal abelian extension of K that is unramified at all primes of K. Considerable emphasis will be on working out specific examples of class fields which illustrate general theory. Prerequisite: Math 530 or equivalent background in algebraic number theory. Recommended text: S. Lang, Algebraic Number Theory (not required). Jim Milne's notes available on the web are a good source.
Math 595 GFV 52315  Generalized Flag Varieties (W. Haboush) The geometry and representation theory of generalized flag varieties. The Bruhat decomposition, the Borel Weil theorem, the Kempf vanishing theorem, structure theory and intersection theory of generalized Schubert cells, Bott Samelson desingularizations, the Weyl and Demazure character formulae, the Chow ring and the Grothendieck ring of the generalized flags. The course will be self contained and will not require extensive knowledge of algebraic groups.Other topics in the structure theory of flag varieties.
Math 595 HA 49180 Homological Algebra II (I. Mineyev)
Among the intended topics: some applications of the Leray-Serre and Lyndon- Hochschild-Serre spectral sequences, cup product, Gysin sequence, Kunneth formula for complexes, universal coecients theorem, Eilenberg-Moore spectral sequence, the generalized Mayer-Vietoris spectral sequence, homology of groups with coecients in a chain complex, equivariant homology, homology of amalgamations and HNN extensions of groups, Adams spectral sequence for stable homotopy groups of spheres.
Math 595

MT

 

51377 Morse Theory (E. Kerman)
Morse theory is the study of the relation between the functions on a space and its topology. It is an extremely powerful tool which plays an important role in many areas of geometry and topology. Some applications of Morse theory include; Smale’s proof of the Poincare conjecture in dimensions greater than four, the Bott periodicity theorem, and several theorems on the existence of closed geodesics. In this course we will first discuss the basic machinery of Morse theory starting with the material described in Milnor’s classic text. We will also study Morse-Bott theory, and the Morse theory of manifolds with boundary. We will then discuss the modern formulation of these ideas due to Thom, Smale, Witten and Floer. This goes under the name of Morse homology, and is a finite-dimensional model of Floer homology. The remainder of the class will be devoted to applications of these tools. These will be chosen according to the tastes of the participants and are subject to the limitations of the instructor. They may include; the existence of closed geodesics, the Bott periodicity theorem, the Morse theory of moment maps in symplectic geometry, and the Morse-Novikov theory of closed one–forms. Prerequisites: The prerequisites for this class are basic differential topology and algebraic topology at the level of Guillemin and Pollack’s book Differential Topology and Vassiliev’s book Introduction to Topology. If you have taken Math 520 you should be well equipped for this class. Reference (Suggested): Morse Theory by J.W. Milnor, Annals of Math. Studies, vol. 51, Princeton University Press, 1963.
Math 595

PM

 

46715 Advanced Methods in Probabilistic Combinatorics (J. Balog)
The Probabilistic Method is a powerful tool in tackling many problems in discrete mathematics. It belongs to those areas of mathematics which have experienced a most impressive growth in the past few decades. This course provides an extensive treatment of the Probabilistic Method, with emphasis on methodology. We will try to illustrate the main ideas by showing the application of probabilistic reasoning to various combinatorial problems. The topics covered in the class will include (but are not limited to): Linearity of expectation, the second moment method, the local lemma, correlation inequalities, martingales, large deviation inequalities, Janson and Talagrand inequalities, pseudo-randomness, random graphs, random regular graphs, Szemeredi Regularity Lemma, percolation, bootstrap percolation. TEXTBOOKS: Most of the topics covered in the course appear in the books listed below (especially the first one). Other topics appear in recent papers. The Probabilistic Method, by N. Alon and J. H. Spencer, 2nd Edition, Wiley, 2000. (Note that 3rd edition might be coming out soon); Random Graphs, by B. Bollobas, 2nd Edition, Cambridge University Press, 2001; Random Graphs, by S. Janson, T. Luczak and A. Rucinski, Wiley, 2000; Graph Coloring and the Probabilistic Method, by M. Molloy and B. Reed, Springer, 2002. PREREQUISITES: There are no official prerequisites, but students need the mathematical maturity and background for graduate-level mathematics. For example, basics of linear algebra, probability and graph theory are assumed to be known.
Math 595 SFM 51373 Symmetric Functions and Macdonald Polynomials (R. Kedem)
Symmetric polynomials are basic objects in representation theory, commutative algebra, algebraic geometry, combinatorics and mathematical physics. Despite their simple definition, there are many interesting applications, connections and open problems. Macdonald polynomials are currently the most active subject of research in this field. The course will start with the basics of symmetric functions (for example, as covered in the Appendix to Fulton and Harris, Representation Theory, or Macdonald’s classic and highly recommended book on Symmetric Functions and Hall Polynomials). We will approach the subject of Macdonald polynomials from the combinatorial and the algebraic points of view. We will also introduce the geometric point of view when appropriate. Students are assumed to have some basic knowledge of the definitions of Lie algebras and representation theory. The course will introduce some problems suitable for graduate research. Text: Handouts will be distributed. Recommended: I. Macdonald, Symmetric functions and Hall polynomials. Prerequisites: Some knowledge of Lie algebras.
Math 595 SG 42972 Symplectic Geometry (A. Malkin)
Symplectic geometry studies manifolds equipped with a closed non-degenerate differential 2-from (symplectic form). Such manifolds arise in many contexts: classical mechanics, quantization, microlocal analysis of PDEs, Kahler geometry, mirror symmetry, etc.. This course provides a gentle introduction to symplectic geometry. It covers basic concepts: symplectic linear spaces and manifolds, Lagrangian submanifolds, local structure, symplectomorphisms group, Hamiltonian dynamics, symmetries and reduction. More advanced topics such as quantization, nonsqueezing theorem, toric manifolds, will also be mentioned but without complete proofs. The only prerequisite is basic differential geometry: manifolds, vector fields, differential forms.

Math 595

(mini-course)

CVO 51368 Culler-Vogtmann Outer Space (I. Kapovich)
Full course description is at http://www.math.uiuc.edu/timetable/595CVO_fall08.pdf

Math 595

(mini-course)

VAM 51371 Vertex Algebras and the Monster Group (M. Bergvelt)
This is a short course (meets 20-Oct-08 - 10-Dec-08). The Monster group is the largest of the 26 sporadic finite simple groups. The Monster group is related to modular functions, a basic topic in number theory. The group is the symmetry group of the Moonshine Module, a vertex algebra. Vertex algebras are the building blocks of conformal field theories, a fundamental topic in string theory. The aim of the course is to give an accessible introduction to these topics. Recomended background reading: • Gannon, Moonshine Beyond the Monster, Cambridge University Press.

Math 595

(mini-course)

DS 52329
Dynamical Systems and Ergodic Theory (J. Rosenblatt)
This mini-course is meant to introduce the ideas and
basic theorems of dynamical systems and ergodic theory. The firsst goal will be to understand the Ergodic Theorem in its norm and pointwise
versions, and to see how these results are related to similar results for martingales and Lebesgue derivatives. The second goal will be to describe the structure of a variety of dynamical systems (ergodic,
weakly mixing, strongly mixing, etc.) and to understand which of these types are typical, which are common, and which are rare. The third goal, as time allows, will be to survey how various recurrence theorems for dynamical systems have played a role in both combinatorics and number theory.
Reference materials: The text "Ergodic Theory" by Karl Petersen provides a basic introduction to ergodic theory. Another text that would be good for the last part of the course is "Recurrence in ergodic
theory and combinatorial number theory" by Hillel Furstenberg. See Joe Rosenblatt's homepage in the online departmental faculty pages, and go to the link to publications, to view a range of articles by the
lecturer on these subjects. Required Background: Some experience in analysis and a knowledge
of Lebesgue measure (even just on Euclidean spaces) is enough background for this course. The rest of the background needed will be provided through lectures or recommended reading as the course progresses.

See past timetables for past Math 595 offerings:
Fall 2007
Spring 2007
Fall 2006
Spring 2006
Fall 2005
Spring 2005
Fall 2004