Gaëtan Borot
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Graduate course: Topological and geometric recursion.
Summer school on Topological recursion, Tübingen Universität
27th31st August 2018

Program of the school. I will give a 3h minicourse + 1h30 exercise session.
 In the first lecture, we will follow what I believe is the shortest route to define the topological recursion: following an approach of Kontsevich and Soibelman, it realizes a procedure of quantization of quadratic Lagrangians in symplectic vector spaces (but we will prior to that make a presentation without any reference to geometry !). We will then see the first examples of applications to 2d topological quantum field theories and cohomological field theories.
 In practice, the topological recursion computes some formal series (which we call TR amplitudes). In applications, the TR amplitudes often receive a meaning in enumerative geometry of surfaces. TR itself is defined by an induction which is reminiscent to the recursive construction of surfaces by glueing pairs of pants.
 This relation to 2d topology is not artificial, as we will see in the 2nd lecture. I will refine the formalism of TR to a formalism called geometric recursion (GR), which produces by recursive glueing of embedded pairs of pants functorial assignements for surfaces. We will see a few examples of this setting and explain how it "projects" to TR.
 The exercise session will be the occasion of performing some computations with TR and GR.
Minicourse: Topological and geometric recursion
Summer school on Geometry, Quantum Topology and asymptotics, Confucius Institute
Geneva, 2nd6th July 2018
 Program of the school. I will give a 5h minicourse.
 We will present the basic theory of the topological recursion (TR) invented by Eynard and Orantin, its recent lift to a geometric setting (GR) in a joint work with Andersen and Orantin, and some of its application to cohomological field theories and GromovWitten theory.
 In a first part, we will take the perspective recently proposed by KontsevichSoibelman of quantization of (quadratic) Lagrangians in symplectic vector spaces. This quantization is a Dmodule called "quantum Airy structure" and considered as the initial data for TR. Its space of solutions has a distinguished solution which the TR computes. As a particular case, one can handle in this way 2d TQFT partition functions, Virasoro constraints for GromovWitten theory of a point, and W_rconstraints for its rspin version. By studying the symmetries of quantum Airy structure, one can also reach Virasoro constraints attached to spectral curves and Virasoro constraints attached to semisimple cohomological field theories.
 In a second part, I will expose the GR construction, which leverage ideas underlying TR and MirzakhaniMcShane identities, and allows the construction of functorial assignments from a category of surfaces with morphisms given by isotopy classes of diffeomorphisms. In particular, from a small amount of initial data, GR builds mapping class group invariants, which we call GR amplitudes. We will see how this setting works to produce mapping class group invariant functions on Teichmuller space, and explain that integration of GR amplitudes over a fundamental domain of Teichmuller space satisfies TR. Conversely, we can lift any initial data for TR coming from a spectral curve, to an initial data for GR, such that the TR amplitudes are integrals of GR amplitudes over a fundamental domain of the Teichmuller space.
Minicourse: Topological expansions.
Workshop on Random maps, random matrices and gauge theories
ENS Lyon, 2529th June 2018

Program of the workshop. I will give a 3h minicourse.
 The same mathematical structures arise for a class of problems in each of the following themes:
 (1) large N asymptotic expansions in 1d Coulomb gases of N particles
 (2) counting surfaces of an arbitrary (but fixed) topology
 (3) studying the geometry of (families of) algebraic varieties
 (4) (toy models, discretizations, etc.) of quantum field theories
 The reason is the appearance in each of these problems of a tower of equations (SchwingerDyson equations, Tutte's equations, Ward identities, etc.) which often reduce after manipulations to the same universal form which we call "abstract loop equations". Each problem has its own set of applications (and in particular, of universality questions) driving the interest
 (1) repulsive particle systems, random tilings,
 (2) combinatorics of maps, random geometry of surfaces, enumerative geometry
 The relation between (2) and (3) is one of the avatar of mirror symmetry.
 In (4) one is interested in studying some aspects (or perhaps constructing) a QFT by first studying a welldefined toy model (or very fine discretization) for it.
 The name topological expansion which one commonly use for all 4 themes comes from the fact, via (2), that coefficients in these expansions have something to do with bordered surfaces sorted out by their topology.
 The lectures will present some of the mathematical progress of the last 15 years, which tends (in the big picture) towards a better understanding of all the relations and common structures between the above problems. Remaining downtoearth, our practical goal will be
 a) explaining a general strategy to establish the existence of allorder asymptotic expansions for the macroscopic properties of a class of models in (1), based on large deviations and the analysis of SchwingerDyson equations. As a spinoff, we can catch central limit theorems and Gaussian Free Field behaviors when applicable.
 b) explaining how to enumerate maps (possibly carrying an O(n)loop model) using Tutte's recursion, which produce a tower of equations identical to the above SchwingerDyson tower (but now in the context of formal series).
 c) explaining how in both cases these SchwingerDyson equations imply abstract loop equations, and their solution via the EynardOrantin recursion. This exploits the geometry of the spectral curve defined from the large N spectral density (in a) or the generating series of disks (in b).
Lecture series: Topological recursion and applications
Melbourne University, 11th February11th March 2018
 Lecture 1 (Feb. 15)  Topological recursion and quantum Airy structures
 Lecture 2 (Feb. 22)  Application of Airy structures to geometry
 Lecture 3 (Feb. 26)  Introduction to geometric recursion
 Lecture 4 (Mar. 1)  Relations between geometric and topological recursions
 Lecture 5 (Mar. 8)  Allorder asymptotics in Coulomb gases via topological recursion
Graduate course: Topological recursion and geometry
Preschool to workshop "Quantum topology and geometry"
Université de Toulouse, 9th13th May, 2017.
 Lecture Notes, arXiv:mathph/1705.09986
Outreach: Math en Jeans
I gave a lecture (lycée level, in French) for the Math en Jeans Congress, at Lycée Français de Düsseldorf, April 4th 2017.
Outreach: Trajectoires
I participated to Trajectoires, a radio podcast on mathematical culture, animated by Fibre Tigre (in French).
 (4) "Combiner" recorded February 24th, 2017, at IHP, at the occasion of the trimester "Combinatorics and Interactions", with Hugo Labrande, Denise Maurice, Arthur Milchior et Arnold Zéphir.
Minicourse: Introduction to random matrix theory,
Summer school on Stochastic Processes and Applications, National University of Mongolia, August 2015
 Lecture Notes, Mong. J. Math. 20 1 (2016) 552, math.PR/1710.10792.
 A brief history of mathematics in Mongolia, by Ninjbat Uuganbaatar.
Minicourse: A walk through the woods of integrability,
Hannover University, July 67th 2015
 Lecture Notes (last version: July 14th 2015)
Master course: Random matrix theory
Bonn University, Wintersemester 201415
 Lecture Notes (last version: February 15th 2015)
MiniCourse: Theory of loop equations and applications
MIT, Fall 2013
 Session 1  Abstract loop equations (pages 19 + extra notes on global initial data).
 Session 2  Enumerating planar maps.
 Session 3  Enumerating maps in any topology.
Last modified: January 7th 2018.