Colloquium in September 2022

24th September 2022,  Saturday 15:00 Hrs IST (GMT + 5:30) 

Title- "Serre's Modularity Conjecture".  (in-person and offline talk at BP, Pune)

Speaker - Eknath P. Ghate, TIFR Mumbai.

Chair for the Talk: S. A. Katre, BP Pune. 

 Abstract:  A deep conjecture of Serre says that every odd irreducible two-dimensional representation of the Galois group of the rational number field over the algebraic closure of a finite field of characteristic p is modular.  We shall explain the precise statement of the conjecture and mention some ingredients that led to its proof (by Khare-Wintenberger and Kisin). 

We then ask a related question: if one twists such a representation which arises from a modular form of slope v by the mod p cyclotomic character, does the new representation arise from a modular form of slope v+1? We mention some recent work with A. Kumar which gives conditions under which this question may be answered in the affirmative.

            Colloquium in October 2022

10th October 2022,  Monday 14:00 Hrs IST (GMT + 5:30) 

Title- "Dessins and their Fields of Definition". (Here are slides and here is Video)

Speaker - Jeroen Sijsling, Ulm University, Ulm.

Chair for the Talk: Anand Deopurkar, Australian National University Canberra.

 Abstract:  A dessin d'enfant is a bipartite graph on a compact topological surface. It can be interpreted as a pair (X, f), where X is a smooth algebraic curve over CC and where f is a Belyi map, that is, a morphism X --> PP^1 whose branch locus is contained in { 0, 1, infty }. 

It turns out that Belyi maps are always defined over the algebraic closure of QQ. A question is whether one can describe the possible fields of definition of (X, f), and especially whether it is defined over its field of moduli, which is the smallest field over which (X, f) could a priori be defined (defined as the fixed field of the automorphisms of CC that fix the isomorphism class of (X, f)). 

It is possible to decide effectively whether RR is a field of definition for a Belyi map, but the corresponding topological criterion already turns out to be fairly subtle, and there are other obstruction when trying to descend all the way to the field of moduli. 

In this talk, we will give precise definitions of the field of moduli and fields of definition of a given dessin, and we will study the corresponding questions in detail, developing a range of methods that decide whether or not the field of moduli is a field of definition. As we will see, the smallest degree of a dessin for which the field of moduli is in fact not a field of definition equals 16, but there are infinitely many dessins for which this issue occurs.

            Colloquium in November 2022

Date and Time: 16th November, Wednesday at 19:00 Hrs (IST).

Title- "A Topological Approach to Grothendieck-Teichmüller Theory". (Here are slides and here is Video)

Speaker - Pierre Lochak, IMJ-PRG Paris.

Chair for the Talk: William Chen, Rutgers University.

 Abstract:  Beyond "dessins d'enfant", the theory nowadays referred to as Grothendieck-Teichmüller theory (Galois-Teichmüller in Grothendieck's manuscripts) may well represent the main new theme in the Esquisse (as confirmed in the Promenade à travers une œuvre). Simpliflying a great deal one may say that Grothendieck's main ideas were taken up especially by Y.Ihara, V.Drinfeld et P.Deligne in the mid and late eighties. An important bifurcation occured in Deligne's paper on Le plan projectif privé de trois points (1990), between the prounipotent (motivic) genus 0 variant, which is now largely prevalent (in particular in deformation theory) and the original profinite, all genus, version, which stays much closer to the original ideas of the Esquisse. These versions can also be termed "linear" and "nonlinear". Only the second one conjecturally recovers the full Galois group Gal(Q). 

In this lecture I intend to first recall some aspects of this mathematical history, then introduce the so-called "curve complexes" (and variants thereof) which represent a vast generalization of the dessins d'enfants and have been much used by topologists in a different context. I will then explain some of the main issues and results of joint work with Leila Schneps (dating back from the end of the last century!) before hopefully moving to a more recent approach involving in particular profinite geometry and the completions of these complexes.

            Colloquium in December 2022

There was no Colloquium in December due to  other  offline Activities

 Colloquium in  2023

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