# School of Mathematics

## Microlocal theory of sheaves and link with symplectic geometry

## Serre's Conjectures on the Number of Rational Points of Bounded Height

**JOINT IAS/PU NUMBER THEORY SEMINAR**

We give a survey of recent results on conjectures of Heath-Brown and Serre on the asymptotic density of rational points of bounded height. The main tool in the proofs is a new global determinant method inspired by the local real and p-adic determinant methods of Bombieri-Pila and Heath-Brown.

## On the Comparison of Trace Formulas

**GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR**

We shall recall the spectral terms from the trace formula for G and its stabilaization, as well as corresponding terms from the twisted trace formula for GL(N). We shall then discuss aspects of the proof of the theorems stated in the first talk that are related to the comparison of these formulas.

## Classification of Representations

**GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR**

Suppose that G is a connected, quasisplit, orthogonal or symplectic group over a field F of characteristic 0. We shall describe a classification of the irreducible representations of G(F) if F is local, and the automorphic representations of G in the discrete spectrum if F is global. The classification is by harmonic analysis and endoscopic transfer, which ultimately ties the representations of G to those of general linear groups.

## Computer Science and Homotopy Theory

## Quadratic Goldreich-Levin Theorems

Decompositions in theorems in classical Fourier analysis which decompose a function into large Fourier coefficients and a part that is pseudorandom

## Learning and Testing k-Model Distributions

A k-modal probability distribution over the domain {1,...,N} is one whose histogram has at most k "peaks" and "valleys". Such distributions are a natural generalization of the well-studied class of monotone increasing (or monotone decreasing) probability distributions.

## Pseudorandomness in Mathematics and Computer Science Mini-Workshop

In math, one often studies random aspects of deterministic systems and structures. In CS, one often tries to efficiently create structures and systems with specific random-like properties. Recent work has shown many connections between these two approaches through the concept of "pseudorandomness". This workshop highlights these connections, aimed at a joint audience of mathematicians and computer scientists.

## On Zaremba's Conjecture

Inspired by the theory of good lattice points in numerical integration, Zaremba conjectured in 1972 that for every denominator q, there is some coprime numerator p, such that the continued fraction expansion of p/q has uniformly bounded quotients. We will present recent progress on this problem, joint with Jean Bourgain.