### Seminars/GdT - Working groups or reading groups to which I participate.

#### Variétés abéliennes et théorème de Honda-Tate (Fall 2016, organiser Giuseppe Ancona)

Here are liveTeX notes (french) that I took (last update 25 Jan 2017). I'm responsible for all the errors.

## Research interests

I did my PhD with Olivier Schiffmann at Universite Paris 7 and I defended it in July 2014. My research area is geometric representation theory.

I've worked on the Hall algebra of an elliptic curve and on the spherical Eisenstein sheaves for an elliptic curve.

- Geometric Langlands correspondence and applications
- Eisenstein sheaves, elliptic character sheaves
- Hall algebras for quivers/curves
- Automorphic forms for functions fields
- Principal bundles and their moduli stack/space
- Categorification and applications to combinatorics and representation theory (e.g. KLR algebras, cluster algebras)
- Diagrammatics, generators and relations
- Representation theory of algebraic groups

### Papers

Disclaimer and warning: the pdf linked to are not identical to the published version.- Revisiting the moduli space of semistable G-bundles over an elliptic curve
- On the stack of semistable G-bundles on an elliptic curve

Published version Math. Annalen, 2016, 365:401–421. - Addendum to Olivier Schiffmann:"On the Drinfeld realization of the elliptic Hall algebra"

Published version Journal of Algebraic Combinatorics March 2012, Volume 35, Issue 2, pp 263-267 - Cusp eigenforms and the Hall algebra of an elliptic curve

Published version Compositio Math. Vol 149 Issue 06 June 2013, pp 914-958 - Crossed product of cyclic groups

joint with Ana-Loredana Agore

Published version Czecholslovak Mathematical Journal December 2010, Volume 60, Issue 4, pp 889-901

The main result says that we can describe the moduli space (not stack) of G-bundles of arbitrary degree on an elliptic curve in terms of line bundles and a certain Weyl group. The idea is to use the previous result that describes the Jordan-Hölder "series" of a G-bundle on an elliptic curve and some cohomological computations. The proof is algebraic and works in arbitrary characteristic. Important work has been previously done by Laszlo and Friedman-Morgan (and here). Our method is close in spirit to Laszlo's.

This paper grew from an attempt to understand the simple summands of spherical Eisenstein sheaves for an elliptic curve. The main result is that for every component of the stack of G-bundles on an elliptic curve there is a unique smallest parabolic subgroup such that the induction map is a small map and the deck group is proved to be a certain Weyl group. This extends a result of Ben-Zvi and Nadler.

This is a short note that is not self contained at all in which I prove that the presentation of the Drinfeld double of the elliptic Hall algebra from the paper in the title actually holds for the half, i.e. for the elliptic Hall algebra itself. The proof is just some combinatorial trick and abstract non-sense of the Drinfeld double construction. The result can be interpreted as a description of higher (more than quadratic) functional equations of the Eisenstein series for an elliptic function field.

I give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field using the theory of Hall algebras and the Langlands correspondence for function fields and GL(n). As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal elliptic Hall algebra defined by Burban and Schiffmann.

We describe explicitly by generators and relations all the crossed products of two cyclic groups. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given. This can be viewed as an extension of the Chinese Remainder Theorem for non coprime integers.