During this academic year, Henryk Iwaniec and Peter Sarnak will be in residence at the Institute for Advanced Study and there will be a program with the purpose to bring together specialists in analytic number theory and specialists in the analytic theory of automorphic forms. John Friedlander and Dinakar Ramakrishnan will also be in residence for the academic year and Dennis Hejhal will be here for a term. Some topics to be covered are:
The main emphasis is recent developments in non-linear PDE's and the related analysis. This includes themes such as dispersive Hamiltonian equations with critical nonlinearity, the structure of singularity formations for NLS and generalized KDV type equations, Strichartz theory with nonsmooth variable coefficients, aspects of Ginzburg-Landau theory. Over the recent years, there have been a number of significant advances on these various topics, often involving a considerable analytical technology.
James Arthur from the University of Toronto will be in residence at the Institute for the academic year 2000-01 and will be giving an advanced course on the trace formula and applications. In addition, Robert Kottwitz, Diana Shelstad, M. -F. Vigneras, G. Henniart and J. -P. Labesse will be in residence for term II.
A1-homotopy theory is the homotopy theory for algebraic varieties and more generally for schemes which is based on the analogy between the affine line and the unit interval. During this year we will concentrate on two topics. One is the extension of the existing theory of triangulated motives from varieties over fields to general schemes. The main remaining problem there can be reformulated in terms of the A1-homotopy theory as the problem of finding a good recognition principle for T-loop spaces.
During the academic year 2000-2001, the School of Mathematics will host a special program on computational complexity theory at the Institute for Advanced Study. Several senior researchers will be in residence at the Institute for the year, and we expect a large number of junior visitors and post-doctorate fellows. Some topics on which special focus is planned are:
Mini Conference December 10-12th
During term I of the year, School faculty member Jean Bourgain and Van Vu of Rutgers University will lead a program on arithmetic combinatorics. The following is preliminary information about the program.
During the 2007-08 academic year, Roman Bezrukavnikov of MIT will lead a special program on algebraic geometry and physics in representation theory.
The focus of the year will be on related recent developments in representation theory, algebraic geometry and physics.
The first conference, "Gauge Theory and Representation Theory", will be held during the week of November 26-30, 2007.
During the academic year 2006-07, the School of Mathematics will have a special program on algebraic geometry. We don't want to focus on any single aspect, but rather aim to have many flavors of algebraic geometry and its applications represented, including (not exhaustive list) cohomology theories, motives, moduli spaces, Shimura varieties, complex or p-adic analytic methods and singularities.
In recent years new and important connections have emerged between discrete subgroups of Lie groups, automorphic forms and arithmetic on the one hand, and questions in discrete mathematics, combinatorics, and graph theory on the other. One of the first examples of this interaction was the explicit construction of expanders (regular graphs with a high degree of connectedness) via Kazhdan's property T or via Selberg's theorem (lambda1 is greater than 3/16).
During the academic year 2004-2005 the School of Mathematics will host a program on the Bloch-Kato conjecture relating Milnor's