Talk

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.

The Masur-Veech volume $b_{g,n}$ of  $\overline{\mathcal{M}}_{g,n}$ can be defined in two ways. Firstly, as the integral over  $\overline{\mathcal{M}}_{g,n}$ 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 $\overline{\mathcal{M}}_{g',n'}$).

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

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.