Abstract: Okounkov's conjecture, now a theorem due to Botta-Davison and Schiffmann-Vasserot, establishes a relation between an enumerative invariant, the Kac polynomial of a quiver, and a Lie-theoretic structure, the so-called Maulik-Okounkov Lie algebra. Okounkov's conjecture generalizes an earlier conjecture of Kac concerning the constant term of this polynomial. Schiffmann's conjecture is a sheaf-theoretic analogue of this statement, in which the enumerative invariant is the Kac-Schiffmann polynomial, counting absolutely indecomposable vector bundles on a smooth projective curve over a finite field. This invariant also appears in Schiffmann's computation of the Betti numbers of the Hitchin moduli space of stable Higgs bundles on a smooth curve. In the first part of the talk, I will introduce the relevant notions needed to state Okounkov's conjecture, as well as the theory of cohomological Hall algebras of quivers, which plays a key role in its proof. The second part of the talk will be devoted to Schiffmann's conjecture and its proof via cohomological Hall algebras of Higgs sheaves over the moduli stack of smooth curves. This is work in progress with Lucien Hennecart, Mauro Porta, and Olivier Schiffmann.
Il seminario si svolgerà in presenza nell'aula M1. Per ulteriori informazioni, si può contattare gli organizzatori all'email amos.turchet@uniroma3.it.
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