Jussieu

I will present a definition of symplectic homology groups for pairs of Liouville cobordisms with fillings, and explain how these fit into a formalism of homology theory similar to that of Eilenberg and Steenrod. This construction allows to understand form a unified point of view many structural results involving Floer homology groups, and yields new applications. Joint work with Kai Cieliebak.

Motivated by a conjecture of Alday, Gaiotto and Tachikawa (AGT

conjecture), we construct an action of

a suitable $W$-algebra on the equivariant cohomology of the moduli

space $M_r$ of rank r instantons on $A^2$ (i.e.

on the moduli space of rank $r$ torsion free sheaves on $P^2$,

trivialized at the line at infinity). We show that

the resulting $W$-module is identified with a Verma module, and the

characteristic class of $M_r$ is the Whittaker vector

of that Verma module. One of the main ingredients of our construction

is the so-called cohomological Hall algebra of the

commuting variety, which is a certain associative algebra structure on

the direct sum of equivariant cohomology spaces

of the commuting varieties of $gl(r)$, for all $r$. Joint work with E. Vasserot.