Mini-conference "Moduli of flat connections"

18.10.2021 – 19.10.2021

Monday, October 18
Room 1-15

13:00 - 14:15 Anton Mellit, Tautological classes and Lefschetz properties I
14:45 - 16:00 Vladimir Fock, Higher laminations I

Tuesday, October 19
Room 1-07

14:00 - 15:15 Anton Mellit, Tautological classes and Lefschetz properties II
15:45 - 17:00 Vladimir Fock, Higher laminations II

Zoom data for hybrid will be provided one day before the event(s)

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Anton Mellit

Title: Tautological classes and Lefschetz properties.

Abstract:

I will introduce a class of varieties attached to surfaces with certain markings. This includes usual character varieties, as well as so-called braid varieties whose cohomology is related to the Khovanov-Rozansky homologies of knots. The fact that these varieties are attached to surfaces allows us to construct certain tautological classes in their cohomology. One of them is a certain 2-form which turns out to always satisfy a Lefschetz property. As applications we deduce q-t symmetries in two settings: in the context of character varieties conjectured by Hausel-Letellier-Rodriguez-Villegas, in the context of knot invariants conjectured by Dunfield-Gukov-Rasmussen (joint work with Gorsky and Hogancamp). The tautological classes together with the Lefschetz operator generate an algebra, conjecturally containing the Lie algebra of Hamiltonian vector fields on the plane, which in its turn contains the positive half of the Virasoro algebra. We will discuss how this may help to understand the cohomology ring (ongoing project with Tamas Hausel).

Vladimir Fock

Title: Higher laminations

Abstract:

According to the duality conjecture there exists a canonical basis of functions on the space of flat connections enumerated by the points of the tropical limit of the space of flat connections for the dual group (generalizing the pairing between the multiplicative group and integers). We will describe the geometric and combinatorial meaning of this tropical limit and its relation to affine Weyl groups, spectral networks, counting Lagrangian coverings, Hecke algebras, Satake correspondence etc. (joint with A.Thomas and V.Tatischeff).

Lieu

Bâtiment: Conseil Général 7-9

Organisé par

Section de mathématiques

Intervenants

Anton Mellit, University of Vienna
Vladimir Fock, IRMA, Strasbourg

entrée libre

Classement

Catégorie: Conférence

Mots clés: flat connections, tautological classes, higher laminations, moduli