# School of Mathematics

## Bilinearized Legendrian Contact Homology

## Around the Davenport-Heilbronn Function

The Davenport-Heilbronn function (introduced by Titchmarsh) is a linear combination of the two L-functions with a complex character mod 5, with a functional equation of L-function type but for which the analogue of the Riemann hypothesis fails. In this lecture, we study the Moebius inversion for functions of this type and show how its behavior is related to the distribution of zeros in the half-plane of absolute convergence. Work in collaboration with Amit Ghosh.

## Arnold Diffusion by Variational Methods III

## Vertex Sparsification: An Introduction, Connections and Applications

The notion of exactly (or approximately) representing certain combinatorial properties of a graph $G$ on a simpler graph is ubiquitous in combinatorial optimization. In this talk, I will introduce the notion of vertex sparsification. Here we are given a graph $G = (V, E)$ and a set of terminals $K \subset V$ and our goal is to find one single graph $H = (K, E_H)$ on just the terminal set so that $H$ approximately preserves the minimum cut between every bi-partition of the terminals.

## Strong and Weak Epsilon Nets and Their Applications

I will describe the notions of strong and weak epsilon nets in range spaces, and explain briefly some of their many applications in Discrete Geometry and Combinatorics, focusing on several recent results in the investigation of the extremal questions that arise in the area, and mentioning some of the remaining open problems.

## Chow Rings, Decomposition of the Diagonal and the Topology of Families

Lecture 4: Integral Coefficients; Application to Birational Invariants

## Chow Rings, Decomposition of the Diagonal and the Topology of Families

Lecture 3: Decomposition of the Small Diagonal and the Topology of Families

## Chow Rings, Decomposition of the Diagonal and the Topology of Families

Lecture 2: On the Generalized Bloch and Hodge Conjectures for Complete Intersections