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Advanced Graduate Courses 2011/12

See below for the proposed syllabi for the advanced seminar courses. All the other courses are described in the catalog.

FALL 2011 WINTER 2012 SPRING 2012
510 Statistics Models and Methods 510 Statistics Models and Methods
H. Sadofsky To be announced
511 Intro to complex analysis I 512 Intro to complex analysis II
P. Gilkey P. Gilkey
513 Intro to analysis I 514 Intro to analysis II 515 Intro to analysis III
M. Bownik M. Bownik P. Gilkey
520 Differential equations I 521 Differential equations II 522 Differential equations III
M. Yattselev P. Lu P. Lu
531 Intro to topology I 532 Intro to topology II 533 Intro to diff’l geometry
N. Proudfoot N. Proudfoot B. Botvinnik
544 Intro to algebra I 545 Intro to algebra II 546 Intro to algebra III
V. Ostrik V. Ostrik V. Ostrik
556 Networks and combinatorics
A. Vaintrob
557 Discrete dynamical systems
V. Ostrik
567 Stochastic processes
D. Levin
607 Sheaf theory 607 Triangulated categories 607 Algebraic groups
N. Proudfoot V. Vologodsky A. Kleshchev
607 Quantum Topology
A. Vaintrob
607 Comparison Geometry
P. Lu
607 Elliptic PDE II
W. He
616 Real analysis I 617 Real analysis II 618 Real analysis III
C. Sinclair Y. Xu Y. Xu
619 Complex analysis
M. Yattselev
634 Algebraic topology I 635 Algebraic topology II 636 Algebraic topology III
D. Sinha D. Sinha D. Sinha
647 Abstract algebra I 648 Abstract algebra II 649 Abstract algebra III
J. Brundan J. Brundan J. Brundan
672 Probability theory I/II 673 Probability theory II/III
C. Sinclair C. Sinclair
681 Commutative algebra 682 Algebraic geometry 683 Algebraic geometry
M. Vitulli A. Polishchuk A. Polishchuk
684 Elliptic PDE I 685 Analysis on spheres 686 Analysis on spheres
W. He Y. Xu Y. Xu
690 Differential forms & de Rham 691 K-theory and geometry 692 WETSK
P. Gilkey D. Dugger H. Sadofsky



Nick Proudfoot, 607: Homological algebra via sheaf theory

Homological algebra is the study of abelian categories and the functors between them. A very important example of an abelian category is the category of sheaves on a topological case. (Exactly what sheaves you want to consider depends on whether you want to do topology, differential geometry, or algebraic geometry.) We will develop the foundations of homological algebra with categories of sheaves as our motivating examples, culminating in a sheaf-theoretic interpretation of the singular cohomology groups of a manifold.

The prerequisites for this course will be the 600 algebra and 600 topology sequences. In turn, this class will provide useful background for two classes that are proposed for the winter term. The sheaf theory that we develop will be useful for algebraic geometry, and the abstract homological algebra will be a prerequisite for the course on triangulated categories.

Vadim Vologodsky, 607: Triangulated Categories

A basic example of a triangulated category is the derived category of an abelian category, which is roughly the category of complexes over the abelian category “up to quasi-isomorphisms”. The theory of derived and triangulated categories was developed by Grothendieck and Verdier as a tool to formulate (and prove) general duality theorems in Topology (so called Verdier’s duality which is far reaching generalization of the Poincare duality for smooth compact manifolds) and Algebraic Geometry (Serre’s duality). We will develop the foundations of the theory of triangulated categories and then focus on its geometric applications (including those mentioned above).

The prerequisites for this course will be the 600 algebra sequence and “Sheaf theory” (Math 607, F). It is recommended to take this course together with the Algebraic Geometry sequence.

Alexander Kleshchev, 607: Algebraic groups

No course description is available.

Arkady Vaintrob, 607: Quantum topology

The focus of this course will be topological quantum field theories (TQFTs) in two and three dimensions and their applications in low-dimensional topology.

We will start with a discussion of the mathematical formalism of TQFTs and related algebraic structures (such as tensor categories, Frobenius and Hopf algebras, quantum groups, e.a.). After constructing first non-trivial examples of TQFTs and a crash course in knot theory, we will study related invariants of knots, links and three-manifolds which include the famous Jones polynomial and the Reshetikhin-Turaev invariants.

In the third part of the course, depending on time and other constraints, we may study additional topics such as the perturbative invariants (a.k.a. the finite type or Vassiliev invariants), the Kontsevich integral, or the relationship between between TQFTs and conformal field theories, moduli spaces of Riemann surfaces and operads.

The prerequisites for this course will be the 600-level algebra and topology sequences. No prior knowledge of physics or knot theory will be assumed.

Peng Lu, 607: Comparison geometry

The main theme is the comparison theorems in Riemannian geometry from the school of Cheeger and Gromov. The results include Hessian comparison theorem under sectional curvature bound, the Lapacian comparison and volume under Ricci curvature bound, Bochner techniques. The minor theme will be more analytical but not relying on the partial differential equation theory: maximum principles and its application in deriving gradient estimate, Harnack inequalities, and etc.

The purpose is to bridge the gap between the basic differential geometry classes and research in differential geometry.

Weiyong He, 607: Elliptic PDE II

This course is a continuation of Math 684 from Fall 2011. See the course description there.

Maxim Yattselev, 619: Complex analysis

Potential theory has its root in electrostatics. Distribution of equally charged particles on a conductor provides intuition to as well as illustrates many results in potential theory. Main motivational example during the course will be the Dirichlet problem of finding a harmonic function on a given domain in complex plane with prescribed boundary values. This application of potential theory is by far not unique as it is successfully used in such areas as orthogonal polynomials, random matrices, number theory and etc.

Textbook: T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, 1995

Contents: Harmonic function: harmonic conjugates, mean-value property, maximum principle, Poisson integral, reflection principle, Harnack’s inequality, Harnack’s theorem. Subharmonic functions: upper semicontinuity, maximum principle, Phragmen-Lindelof principle, criteria of sumharmonicity, integrability. Potentials: continuity, minimum principle, polar sets, equilibrium measures, weak-star convergence, Frostman’s theorem. Dirichlet Problem: Perron’s solution, regular boundary points, harmonic measure, two-constant theorem. Capacity: properties, transfinite diameter, Chebyshev constant, Evans’ theorem, Bernstein-Walsh inequality.
Marie Vitulli, 681: Commutative algebra

In addition to being a beautiful and powerful subject on its own right, commutative algebra serves as the foundation for modern day algebraic geometry and algebraic number theory. This seminar will introduce notions from both areas but the bias will be towards algebraic geometry. We will probably use the text by H. Matsumura [M] to gain a basic understanding of the subject and the 2006 text by I. Swanson and C. Huneke [SH] to delve into some topics that are important in much current research by the commutative algebra community. I will presume that the students who enroll have already seen the basic results on commutative rings and modules, including localization. I will start with associated primes and primary decompositions (the end of Chapter 2 in Matsumura) and aim to discuss regular sequences and depth. I will talk about graded rings and modules, which are critical in the study of projective varieties and their sheaves. The other topics to be covered might change after consultation with Alexander Polishchuk.

Alexander Polishchuk, 682–683: Algebraic geometry

No course description is available.

Weiyong He, 684: Elliptic PDE I

This description covers both Math 684 and a continuation as Math 607 in Winter 2012.

We would like to follow Evans’ text book: Partial differential equations, including the basic material in PDE theory, like Holder space and Sobolev space. For elliptic theory, we plan to cover basically linear theory, including classical Schauder theory and weak solutions. If time permits, we would introduce some techniques to deal with non-linear theory.

All the material requires calculus and maybe basic knowledge of real analysis, and a little bit about functional analysis. In particular, for elliptic theory, we would like to emphasize the bare-handed techniques, such as technique using integration by parts and maximum principle. So any graduate student who finished a 500 level analysis course should be able to follow that.

Yuan Xu, 685–686: Analysis on spheres

This course is not a continuation of Math 684 from Fall 2011, and Math 684 is not a prerequisite.

The course is about Approximation Theory and Harmonic Analysis on the unit sphere in the Euclidean space. The analysis on the sphere requires a whole set of tools that are useful in many other areas, from mathematical physics to group representations. The course will start from basics, cover central results on approximation theory and harmonic analysis on the sphere, develop necessary tools along the way, and end up at recent advances in this research area. The topics will include spherical harmonics, orthogonal expansions, best approximation, moduli of smoothness, K-functionals, maximal functions, multiplier theorems, analysis in weighted spaces with reflection invariant weight functions, and analysis on related domains such as balls and simplexes.

Peter Gilkey, 690: Differential forms and de Rham cohomology

Global Analysis intersects with both Differential Geometry and Algebraic Topology. The Hodge- DeRham theorem is one such example. Let M be a compact Riemannian manifold. The DeRham cohomology groups are the space of closed p-forms modulo the space of exact p-forms. The Hodge theorem identifies the kernel of the p-form valued Laplacian with the DeRham cohomology groups. Sheaf theory can then be used to identify the DeRham cohomology groups with the usual real valued cohomology groups of algebraic topology. So an analytic invariant (kernel of the Laplacian) agrees with a differential geometric notion (DeRham cohomology) agrees with an invariant of algebraic topology (ordinary cohomology).

The course will develop the necessary machinery to prove these results. Some knowledge of differential geometry is required — the 600 level course in differential geometry or Math 515. And a smattering of algebraic topology would be helpful (ordinary cohomology). But the necessary analytic machinery will be developed as needed. The Chern-Gauss-Bonnet formula for the Euler characteristic of the manifold will be presented as an example.

Dan Dugger, 691: K-theory and geometry

K-theory is a generalized cohomology theory for topological spaces, one of the easiest to understand after singular cohomology. It is a basic tool used in several areas of topology and geometry. This course will be an introduction to vector bundles and K-theory, focusing on the geometric aspects of these subjects. The course will accentuate the connections to algebraic geometry (and commutative algebra), based on the correspondence between vector bundles and projective modules. Topics will include Serre’s formula for intersection multiplicities, the formal group law for the K-theory Euler class, the Riemann-Roch formula, Thom isomorphism, spin bundles, and the beginnings of index theory; in all these topics we will focus on the interplay between algebra, geometry, and topology. Students will be assumed to have completed the 600 algebra and topology sequences.

Hal Sadofsky, 692: WETSK

WETSK – or “What every topologist should know” – is a course in which students read and present important papers in topology. Depending on the interests of the students and the instructor, these paper may or may not be coordinated around a central topic.