# School of Mathematics

## Legendrian Invariants in Rational Homology Spheres

## Existence of Small Families of t-wise Independent Permutations and t-Designs Via Local Limit Theorems

We show existence of rigid combinatorial objects that previously were not known to exist. Specifically, we consider two families of objects:

1. A family of permutations on n elements is t-wise independent if it acts uniformly on tuples of t elements. Constructions of small families of t-wise independent permutations are known only for \( t=1,2,3 \) . We show that there exist small families of t-wise independent permutations for all t , whose size is \( n^{O(t)} \) .

## Workshop on Sheaf-Theoretic Methods in Symplectic Topology

## 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.

## 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.

## 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.

## 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.