Graduate Program: Syllabus of Graduate Courses, Winter 2007

415-1 Introduction to Functional Analysis

Wunsch, Lunt 101, 11am MWF

The course will be a one-quarter introduction to functional analysis, with a stress on "applications" in other areas of mathematics. The stress will especially be upon the spectral theorem and its ramifications. We will discuss tempered distributions and Sobolev spaces; weak convergence and weak topologies; compact operators; resolvents and the spectrum of a bounded operator; compact operators; the spectral theorem for compact operators, bounded operators, and, time permitting, unbounded operators.

We will try to keep at hand applications in ODEs, particularly to Sturm-Liouville problems and one-dimensional Schrödinger operators (as a substitute for more technically demanding PDE problems); applications in ergodic theory will also be touched on.

Textbook: Reed and Simon, Methods of Modern Mathematical Physics, Vol. 1.

428-0 Geometric Measure Theory and Applications

G.-Q. Chen, Lunt 103, 3pm, MWF

Geometric measure theory has applications in partial differential equations, calculus of variations, differential geometry, dynamical systems, differential topology, mathematical physics, among others. This is a one-quarter introductory course on geometric measure theory and its applications. It is planned to start with a quick review of general measure theory in basic real analysis.

This is followed by an introduction of Hausdorff measures, fine properties of functions and sets, Lipschitz functions and rectifiable sets, the area and co-area formula, BV functions and sets of finite perimeter, Divergence-measure fields, normal traces and the Gauss-Green theorem, theory of varifolds, theory of currents, and mass minimizing currents. Some important applications of this theory will also be discussed if time permits. Basic real analysis (Math 410-1 & 410-2 or their equivalent) is the only essential prerequisite.

Basic references include:

Geometric Measure Theory: An Introduction by F. Lin and X.-P. Yang.
Geometric Measure Theory by Herbert Federer.
Measure Theory and Fine Properties of Functions by Lawrence C. Evans and Ronald F. Gariepy.
Weakly Differentiable Functions by William P. Ziemer.
Geometric Measure Theory: A Beginners Guide by Frank Morgan (out of print).
Lecture notes by G.-Q. Chen.

The first of the these is assigned as the course's textbook.

435-0 Ergodic Theory

Kra, Lunt 102, 1pm, MWF

This course is an introduction to abstract ergodic theory, focusing on the asymptotic behavior of measure preserving transformations. Topics to be covered include:

Measure preserving transformations and flows. We consider examples from probability, smooth dynamical systems, number theory, stationary stochastic processes and physics.
Convergence theorems. Some convergence results we'll study are the mean ergodic theorem, Birkhoff ergodic theorem, subadditive ergodic theorem, multiplicative ergodic theorem.
Recurrence properties. We consider the qualitative behavior of orbits and study ergodicity, strong mixing, weak mixing, recurrence, spectral problems, K-property, Bernoulli.
Entropy. Entropy is one of the most important invariants of a dynamical system and is used to study the classification of dynamical systems.

The recommended book is Walters, Introduction to Ergodic Theory.

455-1 Stochastic Analysis

Zabell, Lunt 102, 11am, MWF

The following is a list of topics to be covered in each week of the course.

Brownian motion
Stochastic integral
Extensions of the stochastic integral
Ito's formula
Applications of the Ito formula 1
Applications of the Ito formula 2
Stochastic differential equations 1
Stochastic differential equations 2
Applications to finance
Applications to filtering, Feynman-Kac formula

Textbook: H.H. Kuo, Introduction to Stochastic Integration.

465-2 Algebraic Topology II - Stacks

Getzler, Lunt 101, 2pm MWF

This course will be an introduction, through examples, to some of the ideas of the modern theory of stacks, and to some of the abstract ideas (simplicial sheaves, closed model categories, cosimplicial spaces, homotopy diagrams and their rectification) which are at the heart of a large part of contemporary research in geometry and topology.

The course will stay in the topological and differentiable worlds, although many of the ideas we discuss will be relevant to algebraic geometry too.

Here is a selection of the topics to be covered:

Lie groupoids and Lie algebroids, with applications to classical mechanics (Poisson manifolds) and the theory of integrable systems.
Simplicial sets and manifolds, and the nerves of categories and groupoids.
Monoidal categories, bicategories, and the nerve of a bicategory.
n-groupoids, according to Duskin, and Lie n-groupoids, according to Henriques. Examples of Lie 2-groupoids.
Lie n-algebroids, and their associated Lie n-groupoid: the Campbell-Hausdorff formula for Lie 2-algebras.
Descent, as a nonabelian analogue of cohomology. Applications to quantum mechanics.

Prerequisites: Differentiable manifolds and vector bundles, vector fields and differential forms, and singular cohomology of spaces.

483-2 Algebraic Geometry

Suslin, Lunt 101, 1pm MWF

484-0 Lie Theory

Vilonen, Lunt 102, 12pm, MWF

The topic of the Lie theory course for the Winter quarter will be Hodge theory. I will give a general introduction to the subject intended for an audience of users of the theory. It should be useful to people who are interested in, for example, representation theory, algebraic geometry, or number theory. In some sense this a second course in the subject, without assuming the first one. The first course in the subject has been taught here before and usually covers the basics of Kähler manifolds and the theory of harmonic forms. I will simply explain and quote these results. The emphasis of the course will be on the Hodge structures themselves.

The course will begin with a historical perspective to the subject, i.e., I will explain a little bit of the work of Riemann, Poincaré, Lefschetz, and Hodge. Most of the course will concentrate on explaining the second phase of the development of the subject: the work of Griffiths, Deligne, and Schmid. Of course, in a quarter I will manage to cover a part of this story only. In some sense this second phase was the golden era of Hodge theory and since the mid 1970's the subject has played a less central role in mathematics. The reason for my teaching this course at this time is that I believe that we are close to the third phase where the full force of more advanced forms of the theory, like Saito's theory of (mixed) Hodge modules, will play a crucial role in other parts of mathematics.

The book by Griffiths and Harris Principles of Algebraic Geometry is a good source for some basic things. To some extent I will be following the survey by Griffiths and Schmid, Recent Developments in Hodge Theory: A discussion of Techniques and Results. This paper and others are available on the (password-protected) course website.

There will be no classes during the weeks starting January 8 and 15. These classes will be made up by shifting them to certain Tuesdays and Thursdays, 2:30pm-4pm, during the quarter. One of these will be January 4; the others are to be determined.

517-1 Recent Advances in Hochschild Homology

Kaledin, Lunt 101, 1pm-2.20pm, TTh

Hochschild homology and cohomology of associative algebras was introduced by G. Hochschild 60 years ago, but it is still the topic of active research. Hochschild and cyclic homology theories form the homological foundation of non-commutative geometry, a subject of much interest recently, partially because of its connections to contemporary theoretical physics. If anything, the field of Hochschild homology and cohomology has seen more new results in the last ten years than in the preceding fifty. We plan to overview some of these recent advances. Specifically, we will discuss the following topics.

Hochschild homology and cohomology of abelian categories; invariant definitions.
Invariant definition of cyclic homology through a derived version of Drinfeld's double.
Braid group action and Deligne's Hochschild Cohomology Conjecture.
Deformations of abelian categories.
Non-additive functors, Topological Hochschild Homology
Hochschild homology in positive characteristic, generalized Cartier map
Non-commutative Hodge-to-de Rham degeneration through reduction to positive characteristic

The course will be largely self-contained. Previous knowledge of Hochschild homology/cohomology and cyclic homology will be helpful, but we will in any case recall the definitions. We will make heavy use the usual machinery of homological algebra, e.g. the contents of the first four chapters of Gelfand and Manin's book Homological Algebra, so that it will be essential to be comfortable with these techniques. We will probably need some facts from algebraic topology and algebraic geometry, but we will not assume any previous knowledge of either.