Joint IAS/PU Number Theory

Low-lying, fundamental, reciprocal geodesics

Alex Kontorovich
Rutgers University; Member, School of Mathematics
March 24, 2016
Markoff numbers give rise to extremely low-lying reciprocal geodesics on the modular surface, but it is unknown whether infinitely many of these are fundamental, that is, the corresponding binary quadratic form has fundamental discriminant. In joint work with Jean Bourgain, we unconditionally produce infinitely many low-lying (though not "extremely" so), reciprocal geodesics on the modular surface, settling a question of Einsiedler-Lindenstrauss-Michel-Venkatesh.

Iwasawa theory for the symmetric square of a modular form

David Loeffler
University of Warwick
March 10, 2016
Iwasawa theory is a powerful technique for understanding the link between the special values of L-functions and arithmetic objects (such as class groups of number fields, or Mordell-Weil groups of elliptic curves). In this talk I'll discuss what Iwasawa theory predicts for the symmetric square L-function attached to a modular form; and I'll describe some recent results (joint with Sarah Zerbes) confirming some of these conjectures, using the method of Euler systems.

Vanishing cycles and bilinear forms

Will Sawin
February 18, 2016
In joint work with Emmanuel Kowalski and Philippe Michel, we prove two different estimates on sums of coefficients of modular forms---one related to L-functions and another to the level of distribution. A key step in the argument is a careful analysis of vanishing cycles, a tool originally developed by Lefschetz to study the topology of algebraic varieties. We will explain why this is helpful for these problems.

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.