## Lecture 1

## Even Galois Representations and the Fontaine-Mazur conjecture

## The Completed Cohomology of Arithmetic Groups

## Voting Paradoxes and Combinatorics

The early work of Condorcet in the eighteenth century, and that of Arrow and others in the twentieth century, revealed the complex and interesting mathematical problems that arise in the theory of social choice. In this lecture, Noga Alon, Visiting Professor in the School of Mathematics, explains how the simple process of voting leads to strikingly counter-intuitive paradoxes, focusing on several recent intriguing examples.

## Descriptions of the Grain-Growth Structure

## Approximating the Longest Increasing Subsequence in Polylogarithmic Time

Finding the longest increasing subsequence (LIS) is a classic algorithmic problem. Simple $O(n log n)$ algorithms, based on dynamic programming, are known for solving this problem exactly on arrays of length $n$.

## The Complexity of the Non-commutative Determinant

I will talk about the computational complexity of computing the noncommutative determinant. In contrast to the case of commutative algebras, we know of (virtually) no efficient algorithms to compute the determinant over non-commutative domains. Our results show that the determinant in noncommutative settings can be as hard as the permanent.

## Symplectic Homogenization

Given a Hamiltonian on $T^n\times R^n$, we shall explain how the sequence of suitably rescaled (i.e. homogenized) Hamiltonians, converges, for a suitably defined symplectic metric. We shall then explain some applications, in particular to symplectic topology and invariant measures of dynamical systems.