Working seminar,
2009/2010.

Organizers: Serguei Barannikov & Claude Sabbah

Topic:
Towards non-commutative Hodge theory

Contacts:

When?: One or two wednesdays each month, 14h-17h

Where?: ENS Paris, 45 rue d'Ulm, Room Info 2 (Level -1 under the "new" library)

Hodge theoretic aspects of Mirror symmetry by L. Katzarkov, M. Kontsevich & T. Pantev, From Hodge theory to integrability and TQFT: tt*-geometry, R. Donagi et K. Wendland ed., Proc. Symposia in Pure Math. vol. 78, Amer. Math. Soc. 2008, p. 87-174, arXiv: 0806.0107.

Program:

Wednesday, March 24, salle des Résistants (1rst floor) (Be careful, NOT the usual room)

14h Alexandre Polishchuk (Oregon University and IHÉS):
Introduction to curved dg-algebras pdf

15h30 Alexander Givental (UC Berkeley and IHÉS):
Hirzebruch-Riemann-Roch for
K-theoretic Gromov-Witten invariants in genus 0 pdf

Abstract: K-theoretic Gromov-Witten invariants are holomorphic Euler characteristics of various interesting vector bundles over Kontsevich's moduli spaces of stable maps. The problem of computing these invariants is well-motivated by examples of flag manifolds, where "quantum K-theory" turned out to be related to quantum groups and finite-difference versions of Toda lattices (pretty much the same way as quantum cohomology theory of flag manifolds is related to semisimple Lie groups and differential Toda lattices). Although it seems natural to express K-theoretic Gromov-Witten invariants in terms of the usual (cohomological) ones by means of the formula of Riemann-Roch-Hirzebruch, there has been little success in doing so, mostly because moduli spaces of stable maps behave like orbifolds (rather than manifolds). The talk will be an introduction, based on the examples of CP^n as target spaces, into an emerging new theory which seems to resolve this decade-old problem in a rather elegant way. This is a joint work in progress of the speaker and Valentin Tonita.

Wednesday, February 17, salle des Résistants (1rst floor) (Be careful, NOT the usual room)

14h Bertrand Toen (Montpellier):
Dégénérescence de Hodge vers de Rham (d'après Kaledin)

15h30 Bertrand Toen (Montpellier):
K-théorie topologique des dg-algèbres

Wednesday, January 6

14h Christian Sevenheck (Mannheim):
Landau-Ginzburg models for toric Fano manifolds pdf

15h30 Étienne Mann (Montpellier):
Mirror symmetry in terms of nc Hodge structures pdf

Wednesday, December 16

14h Serguei Barannikov:
dg BV-algebras and variations of non commutative Hodge structures

15h30 Étienne Mann (Montpellier):
Non-commutative Hodge structure on quantum cohomology pdf

[Abstract: We discuss some of the axioms of nc-Hodge structure for manifolds and may see that they are natural for toric Fano manifolds]

Wednesday, December 2, salle des Résistants (1rst floor) (Be careful, NOT the usual room)

14h Maxim Kontsevich (IHÉS):
Mixed exponential motives

15h30 Maxim Kontsevich (IHÉS):
Mixed motives in the non-commutative setting

Wednesday, November 18

14h Serguei Barannikov:
Deformations of nc spaces nc and gluing data

15h30 Alexander Shannon (Cambridge):
Examples of non commutative Gauss-Manin connections pdf

Tuesday, November 10, salle des actes (1rst floor, Stairs A) (Be careful, NOT a Wednesday and NOT the usual room)

14h Giovanni Morando (Padua):
Données de recollement (transformation de Laplace)

15h30 Claude Sabbah:
Structure de Hodge non commutative pour le miroir
de l'espace projectif (II) pdf

Wednesday, October 21

14h Serguei Barannikov:
Structure de Hodge non commutative sur l'homologie
cyclique périodique

15h30 Jean-Baptiste Teyssier:
Structure de Hodge non commutative pour le miroir
de l'espace projectif (I) pdf

Wednesday, October 7

14h Claude Sabbah: Introduction aux structures de Hodge non commutatives,
d'après Katzarkov, Kontsevich et Pantev pdf

15h30 Jean-Baptiste Teyssier:
Structures de Hodge et structures de Hodge non
commutatives pdf