## A Constant-factor Approximation Algorithm for the Asymmetric Traveling Salesman Problem

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem. Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation.

## Modular symbols and arithmetic

## The Matching Problem in General Graphs is in Quasi-NC

We show that the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in polylogarithmic time on quasi-polynomially many processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani to obtain a Randomized NC algorithm.

## Modular symbols and arithmetic

## Edward T. Cone Concert Series: Surface Image

## Transfer operators between relative trace formulas in rank one II

I will introduce a new paradigm for comparing relative trace formulas, in order to prove instances of (relative) functoriality and relations between periods of automorphic forms.

## Sieve methods: what are they, and what are they good for?

## Transfer operators between relative trace formulas in rank one

I will introduce a new paradigm for comparing relative trace formulas, in order to prove instances of (relative) functoriality and relations between periods of automorphic forms.