# School of Mathematics

## An asymptotic for the growth of Markoff-Hurwitz tuples

\[ x_1^2 + x_2^2 + \cdots + x_n^2 = ax_1x_2 \cdots x_n + k.\]

## Integral points and curves on moduli of local systems

## Spectral gaps without frustration

## Short proofs are hard to find (joint work w/ Toni Pitassi and Hao Wei)

## Motivic correlators and locally symmetric spaces IV

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.

## Automorphy for coherent cohomology of Shimura varieties

## General strong polarization

## 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.