Graduate Program: Syllabus of Graduate Courses, Winter 2007

412-3 (Complex) Analysis

Wilkinson, Lunt 101, 1pm MWF

The material in the course will not rely on extensive knowledge of the first two quarters, and I encourage all grad students who have not had a course in complex analysis to take (or sit in on) the course. The course will be taught out of Ahlfors, at a reasonably elementary level. The topics I hope to cover are: conformal mapping, analytic continuation, topics in entire and meromorphic functions, elliptic and modular functions, and a proof of the prime number theorem.

425-2 Hyperbolic Partial Differential Equations

G.-Q. Chen, Lunt 102, 2pm MWF

This is a second-year graduate course on the theory of hyperbolic partial differential equations and its applications to singularity theory, wave propagation, stochastic analysis, differential geometry, dynamical systems, mechanics, relativity, among others. We plan to start with rigorous mathematical derivation of general hyperbolic partial differential equations from mathematical setups and physical principles in various areas, with emphasis on important ideas and motivations for the whole subject. This is followed by a systematic introduction to the linear and nonlinear theory of hyperbolic partial differential equations and important applications in mathematics and physics.

There will be no specific prerequisites, but some basic PDE, analysis, and algebra will be assumed.

Basic References:

Lectures on Nonlinear Hyperbolic Differential Equations, by Lars Hormander, Springer-Verlag: New York, 1997.
Hyperbolic Conservation Laws in Continuum Physics, by Constantine M. Dafermos, Springer: Berlin-Heidelberg, 2005.
Lectures on Nonlinear Wave Equations, by C. D. Sogge, International Press: Boston, 1995.
Methods of Mathematical Physics, by Richard Courant and David Hilbert, John Wiley & Sons: New York, 1989.
The Evolution Problems in General Relativity, by Klainerman-Nicolo, Birkhauser: Boston, 2003.
Supersonic Flow and Shock Waves, by Courant-Friedrichs, Springer: New York, 1976.
Lecture notes by G.-Q. Chen

465-3 Algebraic Topology II - A-infinity algebras and categories

Yu. Manin, Lunt 101, TTh, 1-2.20pm (first class: April 5, 2007)

One of the classical insights which formed the background of modern algebra was the discovery of matrices and matrix multiplication: matrices are similar to numbers, but their product is non-commutative.

It took another two hundred years to realize, that associativity of multiplication also can get broken in a highly systematic and non-trivial way. This realization came from homotopy theory. Its impact is slowly but interestingly widens: from physics (Mirror Symmetry à la Kontsevich) to Non-Commutative Geometry where A-infinity (or A_∞) algebras might replace associative rings as basic models of rings of functions.

In this course I will present a systematic introduction to this relatively recent formalism.

Previous exposure to Homological Algebra is helpful, but not necessary. For a brief overview, see B. Keller, Introduction to A-infinity algebras and modules

482-1 Algebraic Number Theory - Local Fields

Calegari, Lunt 102, 1pm, MWF

Topics to be covered include: p-adic numbers, Hensel's lemma, Completion, Ostrovskii's theorem. [Review: inverse limits, discrete valuation rings]. Extensions, the Norm map, Galois Extensions, Frobenius. The discriminant. Unramified extensions. Krasner's Lemma. The inertia group. Higher ramification groups (lower numbering). The discriminant revisited. Tamely ramified extensions. Higher ramification groups (upper numbering). The local Kronecker-Weber theorem for Q_p. Possible further topics may include: Witt Vectors. The local Artin map: statements and consequences. Formal groups: Examples of Lubin-Tate.

483-2 Algebraic Geometry

Tamarkin, Lunt 103, 2pm MWF

486-1 Algebraic K-Theory

Suslin, Lunt 102, 12pm, MWF

513-1 Topics in Dynamical Sustems - Measurable Rigidity

Kra and Wilkinson, Lunt 103, 2pm MWF

In this course, we will study various rigidity phenomena that appear in connection with problems in number theory. The topics we will address include: invariant measures for abelian actions (e.g. Furstenberg's ×2×3 conjecture), recent progress on the Littlewood conjecture by Einsiedler, Katok and Lindesnstrauss, and (time permitting) the Oppenheim Conjecture and Ratner's Theorem.