# School of Mathematics

## Functoriality and algebraic cycles

Abstract: I will discuss the following question: is Langlands functoriality given by algebraic cycles? After a survey of some examples of interest, the talk will focus mostly on one case, namely that of inner forms GL(2) over a totally real field. In this case, we can show that functoriality is given by something close to an absolute Hodge cycle; moreover, there is some hope of doing even better. (Joint work with Atsushi Ichino.)

## Morse-Bott cohomology from homological perturbation

## Language edit distance, $(\min,+)$-matrix multiplication & beyond

The language edit distance is a significant generalization of two basic problems in computer science: parsing and string edit distance computation. Given any context free grammar, it computes the minimum number of insertions, deletions and substitutions required to convert a given input string into a valid member of the language. In 1972, Aho and Peterson gave a dynamic programming algorithm that solves this problem in time cubic in the string length. Despite its vast number of applications, in forty years there has been no improvement over this running time.

## Introduction to works of Takuro Mochizuki

## On the notion of genus for division algebras and algebraic groups

## Two-bubble dynamics for the equivariant wave maps equation

## Structure theorems for intertwining wave operators

## Nonlinear descent on moduli of local systems

## Motivic correlators and locally symmetric spaces III

According to Langlands, pure motives are related to a certain class of automorphic representations.

Can one see mixed motives in the automorphic set-up? For examples, can one see periods of mixed motives in entirely automorphic terms? The goal of this and the next lecture is to supply some examples.

We define motivic correlators describing the structure of the motivic fundamental group $\pi_1^{\mathcal M}(X)$ of a curve. Their relevance to the questions raised above is explained by the following examples.