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.
A reference for this talk is Debord-Lescure, Index theory and groupoids - arXiv:0801.3617
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
Mirzakhani and Delecroix-Goujard-Zograf-Zorich). It is likely
that the Masur-Veech volumes $b_{g,n}$ satisfy TR and that the
functions $B_{g,n}$ satisfies GR. However it is not (yet) transparent
from the formulas.
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
such "quadruple ratio" and as a by-product we get an explicit formula
for the Borel regulator of K_7 in terms of the tetralogarithm function
(joint work with S. Charlton and H. Gangl).
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.
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