Workshop: Kontsevich formality theory and the Duflo isomorphism

Banach Center, 06-09 April 2009

Introduction

M. Kontsevich proved that every Poisson manifold admits a canonical
formal quantisation up to equivalence. This theorem has opened up new horizons in numerous domains of mathematics,
namely Lie theory.
Indeed, in the particular case where the considered manifold is the dual of a finite dimensionnal Lie algebra
Kontsevich shows that the cohomological isomorphism H^\star(g, S(g))\mapsto H^\star(g,U(g)) (induced by the formality)
can be identified in degree 0 with the Duflo isomorphism. This Duflo isomorphism can also be viewed as a consequence of the
Kashiwara-Vergne conjecture. This conjecture has been a longstanding issue. A proof, based again on Kontsevich methods, has been given
by Alekseev and Meinrenken.
An alternative proof, based on the the theory of Drinfeld associators, has been recently written by Alekseev and Torossian. The main idea of this
proof is to link the solutions of the Kashiwara-Vergne conjecture to the Drinfeld associators.
Program

In this workshop, we want to study Kontsevich
formality theorem and its application to Duflo isomorphism and the
Kashiwara-Vergne conjecture.
First we will introduce the deformation quantization theory after Kontsevich.
Then we'll see how it induces the Duflo isomorphism (Kontsevich, Manchon, Pevzner, Torossian) and the
Kashiwara-Vergne conjecture through Alekseev, Meinrenken and Torossian works.
We will tackle the recent work of Alekseev, Enriquez and Torossian on the link between the solutions of the Kashiwara-Vergne
conjecture and Drinfeld associators.

Talks will be given by the participants of the
workshop on a level of a seminar that can be followed in detail by graduate students.
Participants

This workshop is addressed to non-specialists
in the field who want to learn more about this part
of mathematics - either to broaden their mathematical horizon or in relation to their own work.

If you want to participate in this workshop, please e-mail as soon as possible to bcworkshop09 [at] gmail.com .
Support

The Banach Center can provide accommodation for those who volunteer to give a talk.
Organisers

This workshop is organised by Emily Burgunder (Warsaw, Poland), Damien Calaque (Lyon, France)
and Maria Ronco (Valparaiso, Chile).