# Chaire de la Vallée Poussin 2019

Louvain-La-Neuve

ANTON ALEKSEEV

Professeur à l'Université de Genève

Low dimensional topology, Lie theory and number theory :
from Duflo isomorphism to multiple zeta values

•  Leçon inaugurale - mardi 30/04/2019 à 16h15

"Casimir elements, the Baker-Campbell-Hausdorff series and Bernoulli numbers"

The Duflo Theorem states that the center of the universal enveloping algebra is isomorphic to the ring of invariant polynomials. This isomorphism is given by an explicit formula which is expressed in terms of Bernoulli numbers.

The Kashiwara-Vergne (KV) problem is a way to prove the Duflo isomorphism using the properties of the Baker-Campbell-Hausdorff series. Surprisingly, it turns out that the KV problem is related to several other fields of mathematics : to 3-dimensional topology via braid groups and Drinfeld associators, to 2-dimensional topology via Goldman brackets and to number theory via multiple zeta values.

In this lecture, we will show the unexpected relations between the different fields of mathematics mentioned above and we will state several open problems.

La leçon inaugurale sera suivie d’un cocktail dînatoire.

• Leçons 2 et 3 - Jeudi 2/05/2019 de 10h45 à 12h45

"Goldman brackets, moduli of flat connections and Duflo isomorphism for quadratic Lie algebras"

The vector space spanned by homotopy classes of free loops on an oriented 2-manifold carries a canonical Lie bracket defined in terms of intersections of curves (the Goldman bracket). This Lie bracket is intimately related to the Atiyah-Bott symplectic structures on moduli spaces of flat connections. The Goldman Lie algebra carries a canonical filtration, and its associated graded is the necklace Lie algebra which admits an easy combinatorial description.

In this lecture, we will explain the results of Kawazumi-Kuno and Massuyeau-Turaev which show that the Goldman Lie algebra is isomorphic to its associated graded algebra. As an application, we will construct a Darboux chart on the moduli of flat connections, and we’ll show an elementary proof of the Duflo isomorphism theorem for Lie algebras which admit an invariant scalar product (for instance, the Killing form for semisimple Lie algebras).

• Leçons 4 et 5 - vendredi 3/05/2019 de 14h00 à16h00

"Non-commutative differential calculus, Turaev cobrackets and the Kashiwara-Vergne problem"

Differential calculus on polynomial algebras admits a generalization to free associative algebras.

In particular, there is a very interesting construction of divergence which resembles divergence of vector fields in differential geometry. In this lecture, after the introduction into the topic, we will show how this notion is related to the Turaev cobracket in 2-dimensional topology defined in terms of self-intersections of curves.

We will also introduce the Kashiwara-Vergne problem in Lie theory and formulate a number of open problems.

Toutes les leçons seront données en l’auditoire Charles de la Vallée Poussin (CYCL 01) du bâtiment Marc de Hemptinne, chemin du Cyclotron, 2 à Louvain-la-Neuve, Belgique.
Renseignements : 010 47 33 12 ou carine.baras@uclouvain.be