# School of Mathematics

## Algebraic combinatorics: applications to statistical mechanics and complexity theory

## Open Gromov-Witten theory of $(\mathbb{CP}^1,\mathbb{RP}^1)$ in all genera and Gromov-Witten Hurwitz correspondence

In joint work with Buryak, Pandharipande and Tessler (in preparation), we define equivariant stationary descendent integrals on the moduli of stable maps from surfaces with boundary to $(\mathbb{CP}^1,\mathbb{RP}^1)$. For stable maps of the disk, the definition is geometric and we prove a fixed-point formula involving contributions from all the corner strata. We use this fixed-point formula to give a closed formula for the integrals in this case.

## Locally symmetric spaces: $p$-adic aspects

## Nonuniqueness of weak solutions to the Navier-Stokes equation

## Lattices: from geometry to cryptography

## Geometric complexity theory from a combinatorial viewpoint

## Automorphic forms and motivic cohomology III

In the lectures I will formulate a conjecture asserting that there is a hidden action of certain motivic cohomology groups on the cohomology of arithmetic groups. One can construct this action, tensored with $\mathbb C$, using differential forms. Also one can construct it, tensored with $\mathbb Q_p$, by using a derived version of the Hecke algebra (or a derived version of the Galois deformation rings).