>> Lien sur les conférences de la Chaire Aisenstadt

Du 28 septembre au 6 novembre 2015

ADRESSE AU CRM: Tous les mini-cours au CRM se dérouleront à l’Université de Montréal, Pavillon André-Aisenstadt, 2920, Chemin de la tour, salle 4336.

ADRESSE À CONCORDIA: Tous les mini-cours à Concordia se dérouleront au Library Building, Concordia University, 1400 Boul. de Maisonneuve Ouest, Math Help Center.
ATTENTION: salle 921.04 LES MERCREDIS/salle 912.00 LES JEUDIS.

MINI-COURS I

(29 septembre - 15 octobre 2015)

Introduction to topological recursion

This series is an introduction to topological recursion, both by treating examples, and explaining the general formalism. The goal is to arrive at the proof that Gromov-Witten invariants satisfy the topological recursion.

Topological recursion is an ubiquitous and universal recursive relationship that has appeared in various domains of mathematics and physics: volumes of moduli spaces, coefficients of asymptotic expansions in random matrix theory, Hurwitz numbers and many other combinatorial objects, Gromov-Witten invariants, all mysteriously satisfy the same relation. Moreover, this recursion relation is effective: it allows an actual computation of all functions, provided that one knows the 1st one, called the spectral curve.

Présentation 1: Mardi 29 septembre

CRM, 15:30 - 16:30, Séminaire de Physique Mathématique
Introduction to topological recursion: examples

Résumé : Hurwitz numbers and Mirzakhani's recursion

Présentation 2: Jeudi 1er octobre

Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
General topological recursion

Résumé : Notion of spectral curves, basic algebraic geometry of plane curves. (Ref: Fay's lectures on "Theta functions on Riemann surfaces")

Présentation 3: Mardi 6 octobre

CRM, 15:30 - 16:30, Séminaire de Physique Mathématique .
Diagrammatic computation, link to Givental's formalism, properties

Résumé : Diagrammatic computation, symplectic invariance, modular invariance, singular limits, form-cycle duality.

Présentation 4: Mercredi 7 octobre

Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics .
Topological recursion: links to integrable systems

Résumé : Notion of tau functions, Baker-Akhiezer functions, Sato relations.

Présentation 5: Jeudi 8 octobre

Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
Moduli spaces and Gromov-Witten theory

Présentation 6: Jeudi 15 octobre

Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
Topological recursion and asymptotics

Résumé : Topological recursion and asymptotics. Link to knot theory.

MINI-COURS II

Integrable systems, random matrices, Hitchin systems and CFTs

Random matrices are a prototype of most integrable systems. We will use them to illustrate many features of integrable systems. The goal is to arrive at a general formalism for all integrable systems.

Présentation 1: Mercredi 21 octobre

Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
Introduction to integrable systems and solutions

Résumé : The Lax formalism, isospectral systems, algebro-geometric solutions (Baker Akhiezer functions), prime forms, and theta functions. Hitchin systems.

Présentation 2: Jeudi 22 octobre

Concordia, 16:00 - 18:00, Working Seminar in Mathematical Physics
Random matrices, orthogonal polynomials isospectral flows

Résumé : Random matrices and orthogonal polynomials, expectation values of resultants, and expectation values of characteristic polynomials. Isospectral systems from the ODEs satisfied by orthogonal polynomials Notion of Miwa-Jimbo Tau function, Fay identities, Hirota equations, Sato relations.

Présentation 3: Jeudi 29 octobre

Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
Introduction to Hitchin systems and CFT's

Résumé : Liouville theory 4-point function is related to the Painlevé VI tau function. We propose a systematic construction of CFT amplitudes from an arbitrary Hitchin system.

Présentation 4: Mardi 3 novembre

CRM, 15:30 - 16:30, Séminaire de Physique Mathématique
Hitchin systems and CFT's

Résumé : This Liouville theory 4-point function is related to the Painlevé VI tau function.