## 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.

## Morse-Bott cohomology from homological perturbation

## 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.)

## $p$-adic etale cohomology of $p$-adic symmetric spaces

## The mod $p$ derived spherical Hecke algebra: structure and applications

Abstract: I will introduce the mod p derived spherical Hecke algebra of a p-adic group, and discuss its structure via a derived version of the Satake homomorphism. Then, I will survey some speculations about its action on the cohomology of arithmetic manifolds.

## Algorithms for the topology of arithmetic groups and Hecke actions

Abstract: We will describe new algorithms to compute an explicit finite simplicial model for compact, congruence locally symmetric spaces and Hecke actions thereon. Joint work with Aurel Page.

## Pseudorandom generators for unordered branching programs

We present an explicit pseudorandom generator with seed length $\tilde{O}((\log n)^{w+1})$ for read-once, oblivious, width $w$ branching programs that can read their input bits in any order. This improves upon the work of Impaggliazzo, Meka and Zuckerman where they required seed length $n^{1/2+o(1)}$.

## Higher Hida theory

## Modularity lifting theorems for non-regular symplectic representations

Abstract: We prove an ordinary modularity lifting theorem for certain non-regular 4-dimensional symplectic representations over totally real fields. The argument uses both higher Hida theory and the Calegari-Geraghty version of the Taylor-Wiles method. We also present some applications of these theorems to abelian surfaces. (Joint work with F. Calegari, T. Gee, and V. Pilloni.)