School of Mathematics

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

Shachar Lovett
Institute for Advanced Study
September 20, 2011

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 \( n^{O(t)} \) .

Classification of Representations

Jim Arthur
University of Toronto
April 28, 2011

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

Jim Arthur
University of Toronto
April 28, 2011

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

Per Salberger
Chalmers University of Technology
April 28, 2011

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.