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Mod p Points on Shimura Varieties

Mark Kisin
Harvard University
March 8, 2012

A conjecture of Langlands-Rapoport predicts the structure of the mod p points on a Shimura variety. The conjecture forms part of Langlands' program to understand the zeta function of a Shimura variety in terms of automorphic L-functions.

I will report on progress towards the conjecture in the case of Shimura varieties attached to non-exceptional groups.

Arnold Diffusion via Normally Hyperbolic Invariant Cylinders and Mather Variational Method, Part I

Vadim Kaloshin
Pennsylvania State University; Member, School of Mathematics
March 7, 2012

In 1964 Arnold constructed an example of instabilities for nearly integrable systems and conjectured that generically this phenomenon takes place.
There has been big progress attacking this conjecture in the past decade. Jointly with Ke Zhang we present a new approach to this problem. It is based on a construction of crumpled and flower Normally Hyperbolic Invariant Cylinders. Once existence of these cylinders is shown to construct diffusion we apply Mather variational mechanism. A part of the project is also joint with P. Bernard.

Various Approaches to Semiclassical Quantum Dynamics

George A. Hagedorn
Virginia Tech
March 6, 2012

I shall describe several techniques for finding approximate solutions to the time-dependent Schr\"odinger equation in the semiclassical limit. The first of these involves expansions in "semiclassical wave packets" that are also sometimes called "generalized squeezed states." The second is a time-dependent WKB approach that is related to the work of Maslov. I also plan briefly to describe the use of Bargmann transforms and Wigner.