The absolute Galois group of the rational numbers acts on the various flavours (profinite, prounipotent, pro-$\ell$) of the fundamental group of a smooth projective curve over the rationals. The image of the corresponding homomorphism normalizes the image of the profinite mapping class group in the automorphism group of the geometric fundamental group of the curve. The image of the Galois action modulo these “geometric automorphisms” is independent of the curve. A basic problem is to determine this image. This talk is a report on a joint project with Francis Brown whose goal is to understand the image mod geometric automorphisms in the prounipotent case. Standard arguments reduce the problem to one in genus 1, where one can approach the problem by studying the periods of iterated integrals of modular forms and their relation to multiple zeta values.[-]

I will talk about joint work during the recent years with Amin Gholampour, Richard Thomas and Yukinobu Toda, on proving the modularity property of the generating series of certain DT invariants of torsion sheaves with two dimensional support in ambient threefolds. More specifically, I will talk about algebraic-geometric proof of S-duality conjecture in superstring theory made formerly by physicists: Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hibert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson for absolute Hilbert schemes. These intersection numbers, together with the generating series of Noether-Lefschetz numbers, will provide the ingrediants to prove modularity of the above DT invariants over the quintic threefold.[-]

We will overview some conjectures on the mixed Hodge structure of character varieties in the framework of non-abelian Hodge theory on a Riemann surface. Then we introduce and study toric analogues of these spaces, in particular we prove that the toric character variety retracts to its core, the zero fiber of the toric Hitchin map, that its cohomology is Hodge-Tate and satisfies curious Hard Lefschetz, as well as the purity conjecture. We will indicate how these shed light on the $P=W$ conjecture in the toric case as well as for general character varieties. This is based on joint work with Nick Proudfoot.[-]

The general principally polarized abelian variety of dimension at most five is known to be a Prym variety. This reduces the study of abelian varieties of small dimension to the beautifully concrete theory of algebraic curves. I will discuss recent breakthrough on finding a structure theorem for principally polarized abelian varieties of dimension six as Prym-Tyurin varieties associated to covers with $E_6$-monodromy, and the implications this uniformization result has on the geometry of the moduli space $A_6$. This is joint work with Alexeev, Donagi, Izadi and Ortega.[-]