# Mini-courses by Bertrand Eynard

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>> Link on the Aisenstadt Chair series of lectures

# September 28 to November 6, 2015

**ADDRESS AT CRM: All mini-course lectures at the CRM will take place on the campus of the Université de Montréal, Pavillon André-Aisenstadt, 2920, Chemin de la tour, Room 4336. **

**ADDRESS AT CONCORDIA: All mini-course lectures at Concordia will take place at the Library Building, Concordia University, 1400 de Maisonneuve Blvd. West, Math Help Center.**

NOTE: room 921.04 ON WEDNESDAYS/room 912.00 ON THURSDAYS.

## MINI-COURSE I

September 29 - October 15, 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.

*Presentation 1: * Tuesday, September 29

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

**Abstract** : Hurwitz numbers and Mirzakhani's recursion

*Presentation 2:* Thursday, October 1

*Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
*

**General topological recursion**

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

*Presentation 3:* Tuesday, October 6

* CRM, 15:30 - 16:30, Séminaire de Physique Mathématique
*.

** Diagrammatic computation, link to Givental's formalism, properties**

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

*Presentation 4:*Wednesday, October 7

* Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
*.

** Topological recursion: links to integrable systems**

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

*Presentation 5:* Thursday, October 8

* Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
*

**Moduli spaces and Gromov-Witten theory**

*Presentation 6:* Thursday, October 15

* Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
*

**Topological recursion and asymptotics**

**Abstract: Topological recursion and asymptotics. Link to knot theory.**

## MINI-COURSE II

(October 21 to November 3, 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.**

*Presentation 1:*Wednesday, October 21

* Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
*

**Introduction to integrable systems and solutions**

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

*Presentation 2:* Thursday, October 22

* Concordia, 16:00 - 18:00, Working Seminar in Mathematical Physics
*

**Random matrices, orthogonal polynomials isospectral flows**

**Abstract : 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. **

*Presentation 3:* Thursday, October 29

* Concordia, 16:00 - 17:00, Working Seminar in Mathematical Physics
*

**Introduction to Hitchin systems and CFT's**

**Abstract : 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. **

*Presentation 4:* Tuesday, November 3

* CRM, 15:30 - 16:30, Séminaire de Physique Mathématique
*

**Hitchin systems and CFT's**