Advanced Topics Courses
Spring 2017  
Course  Section  CRN  Title and Description 
Math 595  ACE  64841  Algebraic Curves & Surfaces (Cliff) See full course description. 
Math 595  AIG  64840  Asymptotic Invariants of Groups (Sapir) See full course description. 
Math 595  ELI  63212  Equity Linked Insurance (Feng) See full course description. 
Math 595  GTC  64839  Geometry, Topology, and Groups (Leininger) See full course description. 
Math 595  IPS  64837  Interacting ManyParticle Systems (Kirkpatrick) See full course description. 
Math 595  MAG  64836  Modern Algebraic Geometry II (Katz) See full course description. 
Math 595  MFF  64833  Modular Forms (Ahlgren) See full course description. 
Math 595  PFO  64835  PerronFrobenius Operators (Boca) See full course description. 
Math 595  SPD  64838  Spectral Theory of PDEs (Laugesen) See full course description. 
Math 595  TPD  64834  Theory of Partitions (Berndt) See full course description. 
Math 598  ACA  65929  Analytic Combinatorics, with Applications (Baryshnikov) See full course description. 
Math 598  OA  61174  Operator Algebras (Junge) 
Math 598  REN  45371  Research Experience in Number Theory (Berndt) 
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 (Junge) 
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 EquityLinked 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 8week 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  LFunctions (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  OMinimal 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 14Jan13  08Mar13. See full course description. 
Math 595  IQS  58449  Introduction to qseries 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: 17Jan12  09Mar12 
Math 595  CGV  48262  Computer Graphics and Geometric Visualizations (Francis) See full course description. Minicourse: 17Jan12  09Mar12 
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: 22Aug11  14Oct11. 
Math 595  HT1  57743  Topics in Homotopy Theory (Ando) See full course description. Minicourse: 22Aug11  14Oct11. 
Math 595  HT2  57744  Topics in Homotopy Theory (Rezk) See full course description. Minicourse: 17Oct11  07Dec11. 
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: 22Aug11  14Oct11. 
Math 595  RWT  57749  Random walks and queueing theory in biology (Bauer) See full course description. Minicourse: 22Aug11  14Oct11. 
Math 595  SLE  58616  An Introduction to SLE (Bauer) See full course description. Minicourse: 17Oct11  07Dec11. 
Math 595  STP  58570  Spectral Theory of Partial Differential Equations (Laugesen) See full course description. Minicourse: 17Oct11  07Dec11. 
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. MiniCourse: 18Jan11  11Mar11 
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 Lfunctions (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. MiniCourse: 18Jan11  11Mar11 
Math 595  TG  54387  The Grassmannian (Yong) See full course description. MiniCourse: 14Mar11  4May11 
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. MiniCourse: 18Oct10  08Dec10 
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. MiniCourse: 18Oct10  08Dec10 
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 CauchyRiemann Equations in Several Variables (J. D'Angelo) See full course description. MiniCourse: 19 Jan 2010  12 Mar 2010 
Math 595  CGV  48262  Computer Graphics & Geometrical Visualization (G. Francis) See full course description. MiniCourse: 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. MiniCourse: 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  KTheory (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 multidimensional, 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: GaussKuzminLevy theory, PerronFrobenius type operators, elements of dynamics on homogeneous spaces, symbolic dynamics, multidimensional 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 HochschildSerre 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 LyapunovSchmidt 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 NashMoser type implicit function theorems and their applications to KolmogorovArnoldMoser (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 selfcontained. 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 RobinsonSchenstedKnuth correspondence, jeu de taquin, the LittlewoodRichardson 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 pseudoarc. 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 "higherdimensional": 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) 
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 ringsSwan calls the whole process "NeronPopescu desingularization". The main topics will include the following: quasiunramified, quasismooth and quasietale extensions, geometric regularity; Excellent rings; Hochster's construction of big CohenMacaulay 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: NeronPopescu Desingularization  Expository paper by R. Swan, University of Chicago. 10:3011:50 TuTh 
Math 595  MBB  49971  DModules and BeilinsonBern Localization (T. Nevins) 
Math 595  NA  39574  Nonstandard Analysis (C. Ward Henson) In order to reach advanced applications of NSA in this course, we will assume a knowledge of firstorder logic extending at least through the compactness theorem. Students should be able to formulate mathematical statements within firstorder 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:
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 RiemannRoch, AbelJacobi, and Lefschetz's theorems. Part (3) will be devoted to the discussion of the Tate's and Mordell's conjectures, Lfunctions and Econstants attached to abelian varieties and the Birch and SwinnertonDyer 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  SubRiemannian Geometry and Analysis (J. Tyson) This minicourse will cover the foundations of (firstorder) analysis, geometric measure theory and differential geometry in the Heisenberg group and more general subRiemannian 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, SpringerVerlag, 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 HardyLittlewood 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 KroneckerWeber 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 Lseries 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 localglobal 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 LeraySerre and Lyndon HochschildSerre spectral sequences, cup product, Gysin sequence, Kunneth formula for complexes, universal coecients theorem, EilenbergMoore spectral sequence, the generalized MayerVietoris 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 MorseBott 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 finitedimensional 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 MorseNovikov 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, pseudorandomness, 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 graduatelevel 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 nondegenerate differential 2from (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 (minicourse) 
CVO  51368  CullerVogtmann Outer Space (I. Kapovich) Full course description is at http://www.math.uiuc.edu/timetable/595CVO_fall08.pdf 
Math 595 (minicourse) 
VAM  51371  Vertex Algebras and the Monster Group (M. Bergvelt) This is a short course (meets 20Oct08  10Dec08). 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 (minicourse) 
DS  52329  Dynamical Systems and Ergodic Theory (J. Rosenblatt) This minicourse 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