Joint IAS/PU Number Theory

Absolute vs. relative Gromov-Witten invariants

Mohammad Tehrani
Stonybrook University
December 18, 2015
We compare absolute and relative Gromov-Witten invariants with the basic contact vector for very positive divisors. For such divisors, one might expect that these invariants are the same up to a natural multiple. We show that this is indeed the case outside of a narrow range of the dimension of the target and the genus of the domain. We provide explicit examples to show that these invariants are generally different inside of this range. This is joint work with A. Zinger.

Decoupling in harmonic analysis and the Vinogradov mean value theorem

Jean Bourgain
IBM von Neumann Professor; School of Mathematics
December 17, 2015
Based on a new decoupling inequality for curves in $\mathbb R^d$, we obtain the essentially optimal form of Vinogradov's mean value theorem in all dimensions (the case $d = 3$ is due to T. Wooley). Various consequences will be mentioned and we will also indicate the main elements in the proof (joint work with C. Demeter and L. Guth).

Modularity and potential modularity theorems in the function field setting

Michael Harris
Columbia University
December 17, 2015
Let $G$ be a reductive group over a global field of positive characteristic. In a major breakthrough, Vincent Lafforgue has recently shown how to assign a Langlands parameter to a cuspidal automorphic representation of $G$. The parameter is a homomorphism of the global Galois group into the Langlands $L$-group $^LG$ of $G$. I will report on my joint work in progress with Böckle, Khare, and Thorne on the Taylor-Wiles-Kisin method in the setting of Lafforgue's correspondence.

Geometric deformations of orthogonal and symplectic Galois representations

Jeremy Booher
Stanford University
November 19, 2015
For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we would like to know if there exists a lift to characteristic zero with nice properties. In particular, we would like it to be geometric in the sense of the Fontaine-Mazur conjecture: ramified at finitely many primes and potentially semistable at $p$. For two-dimensional representations, Ramakrishna proved that under technical assumptions, odd representations admit geometric lifts.

Adjoint Selmer groups for polarized automorphic Galois representations

Patrick Allen
University of Illinois, Urbana-Champaign
October 15, 2015
Given the $p$-adic Galois representation associated to a regular algebraic polarized cuspidal automorphic representation, one naturally obtains a pure weight zero representation called its adjoint representation. Because it has weight zero, a conjecture of Bloch and Kato says that the only de Rham extension of the trivial representation by this adjoint representation is the split extension. We will discuss a proof of this case of their conjecture, under an assumption on the residual representation.

Motivic cohomology actions and the geometry of eigenvarieties

David Hansen
Columbia University
October 1, 2015
Venkatesh has recently proposed a fascinating conjecture relating motivic cohomology with automorphic forms and the cohomology of arithmetic groups. I'll describe this conjecture, and discuss its connections with the local geometry of eigenvarieties and nonabelian analogues of the Leopoldt conjecture. This is joint work with Jack Thorne.