Congruences between motives and congruences between values of $L$-functions

Olivier Fouquet
Université Paris-Sud
April 13, 2017
If two motives are congruent, is it the case that the special values of their respective $L$-functions are congruent? More precisely, can the formula predicting special values of motivic $L$-functions be interpolated in $p$-adic families of motives? I will explain how the formalism of the Weight-Monodromy filtration for $p$-adic families of Galois representations sheds light on this question (and suggests a perhaps surprising answer).

Algebraic proofs of degenerations of Hodge-de Rham complexes

Andrei Căldăraru
University of Wisconsin, Madison
April 12, 2017

In the first half of the talk I shall present a new algebraic proof of a result of Deligne-Illusie about the degeneration of the Hodge-de Rham spectral sequence. The idea is to reduce the main technical point of their proof to a question about the formality of a derived intersection in an Azumaya space.

Soliton resolution for energy critical wave and wave map equations

Hao Jia
Member, School of Mathematics
April 12, 2017
It is widely believed that the generic dynamics of nonlinear dispersive equations in the whole space is described by solitary waves and linear dispersions. More precisely, over large times, solutions tend to de-couple into solitary waves plus radiation. It remains an open problem to rigorously establish such a description for most dispersive equations. For energy critical wave equations in the radial case, we have better understanding, using tools such as ``channel of energy inequalities" firstly introduced by Duyckaerts-Kenig-Merle, and monotonicity formulae.

Mirror symmetry for moduli of flat bundles and non-abelian Hodge theory

Tony Pantev
University of Pennsylvania
April 12, 2017
I will discuss a construction of the homological mirror correspondence on algebraic integrable systems arising as moduli of flat bundles on curves. The focus will be on non-abelian Hodge theory as a tool for implementing hyper Kaehler rotations of objects in the Fukaya category. I will discuss in detail a specific example of the construction building automorphic sheaves on the moduli space of rank two bundles on the projective line with parabolic structure at five points. This is a joint work with Ron Donagi.

Noncommutative probability for computer scientists

Adam Marcus
Princeton University; von Neumann Fellow, School of Mathematics
April 11, 2017
I will give an introduction to computational techniques in noncommutative probability for people (like myself) that are less interested in the details of the theory and more interested using results in the theory to help understand the behavior of things related to random matrices (asymptotically as the size of the matrices get large). This will include introducing the concept of free independence and seeing how it compares to classical independence. No prior knowledge will be assumed.

In pursuit of obfuscation

Allison Bishop
Columbia University
April 10, 2017
This talk will survey some recent advances in research on program obfuscation, an area of theoretical cryptography that has seen unprecedented levels of activity over the last four years. We will cover material starting from the basics, and no prior knowledge on obfuscation will be assumed. We will end with some open directions that should be accessible to theoreticians outside of cryptography.

Two rigid algebras and a heat kernel

Amitai Zernik
Member, School of Mathematics
April 7, 2017
Consider the Fukaya A8 algebra $E$ of $RP^{2m}$ in $CP^{2m}$ (with bulk and equivariant deformations, over the Novikov ring). On the one hand, elementary algebraic considerations show that E admis a rigid cyclic minimal model, whose structure constants encode the associated open Gromov-Witten invariants. On the other hand, in a recent paper another rigid minimal model was computed explicitly, using fixed-point localization for A8 algebras. In this talk I'll discuss these two models and explain how to use the heat kernel on $RP^{2m}$ to relate them.

Basic loci of Shimura varieties

Xuhua He
University of Maryland; von Neumann Fellow, School of Mathematics
April 6, 2017

In mod-$p$ reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-$p$ reductions of more general Shimura varieties, there is a "Newton stratification" decomposing the reduction into finitely many locally closed subsets, of which exactly one is closed. This closed set is called the basic locus; it recovers the supersingular locus in the classical case of modular curves.