Mini-cours de Bertrand Eynard
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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
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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
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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
(21 octobre au 3 novembre 2015)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