Timetable Information for Spring 2009

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.

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.

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.

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).