## Mini-courses and bonus lecture

Introduction to resurgence and applications

by I. Aniceto

Despite their vanishing radius of convergence, asymptotic expansions of observables can provide extremely accurate approximations. Nevertheless the asymptotic behaviour of these observables will differ in different sectors of the complex plane — this is a consequence of the existence of non-perturbative contributions, originally exponentially small, which can grow to dominate the system.The process of analytic continuation between these sectors, describing the appearance of the non-perturbative contributions, is given by the well-known Stokes phenomenon. The asymptotic observable will then be described by a transseries solution, including all perturbative and non-perturbative contributions.
The aim of these lectures is to introduce a systematic approach to study Stokes phenomenon of the transseries, based on Borel analysis and analytic methods of resurgence introduced by Écalle. This analysis will then be combined with accurate summation methods such as Borel summation, hyperasymptotics and transasymptotics, allowing us to study global analytic properties of the observables.

Lecture notes, Problem set

Riemann surfaces and Theta functions

by M. Bertola

We begin with definitions and examples, including the introduction to the concepts of genus and Euler characteristics. We then move to differentials and the Riemann bilinear relations, building towards the Abel map and the Jacobian variety. This leads to Theta functions, for which we discuss properties and applications. Further topics include the Fay identities, the fundamental bidifferential and the Bergman projective connection. If time permits, we will also provide hints about the Szegö kernel.

Lecture 1, Lecture 2, Lecture 3

Topological recursion

by V. Bouchard

We introduce the concept of topological recursion (TR) through the lens of Airy ideals. Airy ideals offer a straightforward and elegant formulation of the recursion relations central to this topic, making them an excellent entry point into the subject. The discussion then shifts to the classical framework of TR focusing on spectral curves with simple ramification points. The connection between the Airy ideal framework and the spectral curve formulation is established through loop equations. These equations are fundamental to the origin of TR, initially arising in the context of matrix models via Ward identities.

Lecture notes

Moduli spaces of Riemann surfaces

by A. Giacchetto and D. Lewański

We provide an introduction to the moduli space of Riemann surfaces, a fundamental concept in the theories of 2D quantum gravity, topological string theory, and matrix models. We begin by reviewing some basic results concerning the recursive boundary structure of the moduli space and the associated cohomology theory. We then present Witten's celebrated conjecture and its generalisation, framing it as a recursive computation of cohomological field theory correlators via topological recursion. Time permitting, we touch on JT gravity in relation to hyperbolic geometry and topological strings.

Lecture notes, Exercises, Solutions (coming soon)

Matrix models (bonus lecture)

by B. Eynard

Lecture notes from the Les Houches School "Stochastic Processes and Random Matrices" (2015)