## Invariants of Hyperbolic Three Manifolds via Ideal Triangulations

## Nonabelian level structures for elliptic curves and noncongruence modular forms

We will begin by describing the notion of nonabelian level structures for elliptic curves, which are roughly given by a G-Galois cover of the curve, ramified only above the origin. Afterwards, we will describe some questions surrounding this notion and applications to the arithmetic of the Fourier coefficients of noncongruence modular forms.

## Lie groupoids and index theory - a brief outline

In this introductory talk we will discuss Lie groupoids and give examples of how they relate to several objects of classical differential geometry such as principal bundles, Lie group actions and foliations.

We will also see some of the ways in which groupoids can appear in the study of singular (i.e. not smooth) spaces such as orbit spaces of Lie group actions, leaf spaces of foliations, and pseudomanifolds with conical singularities.

Finally, we will see a brief outline of how groupoids can be useful when proving index theorems.

## Geometric recursion for Masur-Veech volumes?

The Masur-Veech volume $b_{g,n}$ of $M_{g,n}$ can be defined in

two ways. Firstly, as the integral over $M_{g,n}$ of a certain

analytic function $B_{g,n}$. Secondly as the coefficient of the

leading term of some enumerative problem (here a certain type

of quadrangulations in genus $g$). There is a formula for the

Masur-Veech volume $b_{g,n}$ as a polynomial of integrals of

psi-classes (on various $M_{g',n'}$). This formula can be derived

from any of the two definitions (that was respectively done by

## Lifting Topological Recursion to Geometric Recursion

In previous talks, Geometric Recursion (GR) was introduced, as well as the principle of Topological Recursion (TR), for which some examples were furnished. Both procedures allow to produce, from initial data, an infinite family of "invariants". The question we tackle here is the relation between those families of invariants and between the initial data. The aim of this talk is to show that, starting from TR with a certain class of spectral curves, one can reproduce the same invariants by GR.

## An explicit quadruple ratio

In his celebrated proof of Zagier's polylogarithm conjecture

for weight 3 Goncharov introduced a "triple ratio", a projective

invariant akin to the classical cross-ratio. He has also conjectured

the existence of "higher ratios" that should play an important role

for Zagier's conjecture in higher weights. Recently, Goncharov and

Rudenko proved the weight 4 case of Zagier's conjecture with a

somewhat indirect method where they avoided the need to define a

corresponding "quadruple ratio". We propose an explicit candidate for

## Isomorphism between grt and the degree 0 cohomology of the graph complex

## What is $\overline{M}_{0,n}$?

## What is the concentration of measure phenomenon?

## The Surface Quasi-Geostrophic Equation on the sphere

The SQG equation models the evolution of buoyancy, or potential temperature, on the 2D horizontal boundaries of general 3D quasi- geostrophic equations. In this talk, we will present some regularity results for its dissipative analogue in the critical regime for the two dimensional sphere. The main techniques build over the work of Caffarelli-Vasseur based on the De Giorgi method and the nonlinear maximum principles due to Constantin-Vicol.

## Topological recursion and enumerative problems

In this talk, we introduce the notion of topological recursion. Moreover, we relate it to several enumerative problems, such as generalized Catalan numbers and variants of Hurwitz numbers.

## New guests at the MPIM

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