A Diophantine upper bound on the dimensions of certain spaces of holonomic functions was the main ingredient in our proof with Calegari and Tang of the 'unbounded denominators conjecture' (presented by Tang in last year's number theory seminar) from...

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Joint IAS/PU Number Theory

This lecture will partly survey branching laws for real and p-adic groups which often is related to period integrals of automorphic representations, discuss some of the more recent developments, focusing attention on homological aspects and the...

Half-integral weight modular forms are classical objects with many important arithmetic applications. In terms of automorphic representations, these correspond to objects on the metaplectic double cover of SL(2). In this talk, I will outline a...

In this three-part lecture series, we will present a series of works by Bedrossian, Blumenthal and Punshon-Smith on the chaotic mixing and enhanced dissipation properties of a passive tracer subject to the motion of an ergodic Markovian flow of...

I will discuss recent work with Maksym Radziwill in which we show that for any fixed tempered cuspidal representation π of GL(4) over the rationals, there exist infinitely many primitive characters χ such that the twisted L-function L(s,π×χ) is non...

We determine the average size of the 3-torsion in class groups of G-extensions of a number field when G is any transitive 2-group containing a transposition, for example, D4. It follows from the Cohen--Lenstra--Martinet heuristics that the average...

The black hole information paradox — whether information escapes an evaporating black hole or not — remains one of the greatest unsolved mysteries of theoretical physics. The apparent conflict between validity of semiclassical gravity at low...

Let N and p greater than or equal to 5 be primes such that p divides N−1. In his landmark paper on the Eisenstein ideal, Mazur proved the p-part of the BSD conjecture for the p-Eisenstein quotient J(p) of J0(N) over Q. Using recent results and...

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier...

Wiles’s proof of the modularity of (semistable) elliptic curves over the rationals and Fermat’s Last Theorem relied on his invention of a modularity lifting method. There were two strands to the method:

(i) A numerical criterion to for a map...