Joint IAS/PU Number Theory

Multiple Zeta Values

Francis Brown
CNRS/Institut de Math. de Jussieu, Paris
April 19, 2012

I will report on some recent work on multiple zeta values. I will sketch the definition of motivic multiple zeta values, which can be viewed as a prototype of a Galois theory for certain transcendental numbers, and then explain how they were used to prove a conjecture due to Hoffman giving a generating family for multiple zeta values, and a conjecture of Deligne and Ihara on the fundamental group of the projective line minus 3 points.

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.

Arithmetic Fake Compact Hermitian Symmetric Spaces

Gopal Prasad
University of Michigan
February 16, 2012
A fake projective plane is a smooth complex projective algebraic surface whose Betti numbers are same as those of the complex projective plane but which is not the complex projective plane. The first fake projective plane was constructed by David Mumford in 1978 using p-adic uniformization. This construction is so indirect that it is hard to obtain geometric properties of the fake projective plane. A major problem in the theory of complex algebraic surfaces was to find all fake projective planes in a way which allows us to discover their geometric properties.