Monday, 26 February 2024

A history and survey of the subject.

Everything about the thrice-punctured sphere.

In this talk, we discuss background materials and main questions surrounding anabelian algebraic geometry starting from the arithmetic fundamental group of $\mathbb{P}^1-{0,1,\infty}$.

We will present and define the Grothendieck-Teichmüller group and explain Grothendieck's conception of the Teichmüller tower and the two-level principle.

Tuesday, 27 February 2024

I will overview basic properties of the KZ associator (aka. the Drinfeld associator).

We review the construction of the scheme of associators. We explain its relation with formality isomorphisms for the braid groups on the plane. This leads to the definition of the prounipotent GT group and to the proof of the torsor structure of the scheme of associators. Time permitting, we will discuss the interpretation of the grt Lie algebra in terms of a tower of outer derivation Lie algebras (Ihara)

In this talk, we discuss Grothendieck's conjecture on anabelian geometry initiated by his letter to Faltings (1983) and results developed by other authors afterwards.

Wednesday, 28 February 2024

One of the main themes of Grothendieck’s "Esquisse" is about giving a combinatorial/topological description of absolute Galois groups. The first major developments concerning this were the “children’s drawings” and the definition/introduction/study of the Grothendieck-Teichmueller group (GT), followed by the Ihara question, respectively the Oda-Matsumoto conjecture

(I/OM). I plan to explain how I/OM relates to GT (the latter object being thoroughly discussed at this workshop) and how this fits into the bigger picture about the initial question above. Finally, I plan to present a recent result (collaboration with Adam Topaz) concerning a line/hyperplane variant of I/OM and/or GT which is both (i) closer in nature to GT than I/OM is; (ii) giving a topological description of absolute Galois, e.g. that of Q.

The main conjectures play a central role in Iwasawa theory and relate an arithmetic and analytic invariant associated to certain modules that arise naturally in the study of arithmetic of elliptic curves and Galois representations, in general. We shall give an overview of the different main conjectures in this talk.

We review the family of double shuffle relations between multiple zeta values (MZVs; Ihara-Kaneko-Zagier, Racinet, Ecalle), the construction of the scheme attached to this family of relations, and Racinet's theorem according to which it is a torsor under the action of a certain pro-unipotent group. Time permitting, we will discuss the intepretation (obtained jointly with Furusho) of this group and scheme in terms of stabilizers.

TBA

Thursday, 29 February 2024

The two-level principle in four versions.

TBA

The confluence relation introduced by Hirose and Sato is a relation among multiple zeta values. I will explain that it is equivalent to the associator relation.

Will discuss analogs and contrasts in the number field - function field situation of the Grothendieck program.

Friday, 01 March 2024

We will explain how formality isomorphisms for the braid groups on the 2-dimensional torus can be obtained out of a particular family of compatible flat connections on the tower of configurations spaces of an elliptic curve, called the "universal Knizhnik-Zamolodchikov-Bernard (KZB) connection". This gives rise to a family of isomorphisms between towers of "Betti" and "de Rham" groupoids associated with the torus, which are counterparts of the "genus 0" groupoid towers from the theory of associators, and compatible with the isomorphism between the latter towers arising from the KZ associator. We define an elliptic associator to be a pair of compatible isomorphisms between the genus 0 and genus 1 towers, and show the set of elliptic associators to be a bitorsor over two isomorphic groups, the elliptic GT group and its graded version. We start discussing the relations between the torsors of associators and elliptic associators.

Completed curve complexes.