Joint IAS/PU Number Theory

In this talk, I will focus on simultaneous non-vanishing results for Dirichlet L-functions at the central point 1/2. Specifically, I will describe non-vanishing results (conditional on GRH) for two L-functions associated to Dirichlet characters in...

The Poisson summation conjecture of Braverman-Kazhdan, L. Lafforgue, Ngo, and Sakellaridis is an ambitious proposal to prove analytic properties of quite general Langlands L-functions using vast generalizations of the Poisson summation formula. In...

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. In a series of papers, Venkatesh and his collaborators proposed an arithmetic reason for...

In characteristic zero, Castelnuovo proved that every unirational surface is rational. In positive characteristic, this fails dramatically: there exist many non-rational, often even general-type, surfaces that are nevertheless unirational. In 1977...

Let K be the function field of a smooth projective curve B over the complex numbers and let g be a positive integer. The uniform boundedness conjecture predicts that there exists a constant N, depending only on g and K, such that for any g...

We propose and study a new arithmetic invariant of non-hyperelliptic genus-4 curves: a canonical “quadratic” point on the Jacobian, defined by the two natural degree-2 maps to projective lines. Building on Xue’s result, that this point is...

I will give an overview of recent progress in random multiplicative functions (random models for multiplicative functions) and their connection to the study of the Fyodorov-Hiary-Keating conjecture and Polya's question on nonnegative character sums...

Central predictions of arithmetic quantum chaos such as the Quantum Unique Ergodicity conjecture and the sup-norm problem ask about the mass distribution of automorphic forms, most classically in terms of their weight or Laplace eigenvalue (for...

For an elliptic curve E defined over Q, the Mordell-Weil group E(Q) is a finitely generated abelian group. We prove that there are infinitely many elliptic curves E over Q for which E(Q) has rank 2. Our elliptic curves will be given by explicit...

How many algebraic integers of bounded height have a minimal polynomial with a given Galois group? One approach to this problem is via Malle's conjecture. In this talk we will discuss an alternative approach using a construction with the Galois...