Abstract: CFT can be defined in an axiomatic way, by the "bootstrap axioms". Topological recursion provides a natural model for these axioms, it automatically satisfies OPE and Ward identities. The less trivial axiom is 'crossing symmetry', and proving or verifying that this axiom is also satisfied in TR, is a challenge.

**Bertrand Eynard**(CEA Saclay):__Conformal field Theory, Topological Recursion, and Geometry__Abstract: CFT can be defined in an axiomatic way, by the "bootstrap axioms". Topological recursion provides a natural model for these axioms, it automatically satisfies OPE and Ward identities. The less trivial axiom is 'crossing symmetry', and proving or verifying that this axiom is also satisfied in TR, is a challenge.

**Malin Palö Forsström**(Gothenburg):

__Recent results on lattice gauge theories with a finite gauge group__Abstract: Lattice gauge theories are lattice approximations of the Yang-Mills theory in physics. The corresponding models are similar to the Ising model and various plaquette percolation models but have a slightly more complicated structure. The abelian lattice Higgs model is one of the simplest examples of a lattice gauge theory interacting with an external field. In this talk, I will review and compare recent results on lattice gauge theories and the abelian lattice Higgs model.

**Davide Gaiotto**(Perimeter Institute):

*I will review recent work on algebraic structures in twisted supersymmetric gauge theories. As an example, I will discuss a potential application to the quantization of Poisson vertex algebras, in the same spirit as Kontsevich deformation quantization formulae.*

__Quantization of Poisson vertex algebras__

**Elba Garcia-Failde**(Sorbonne):

__Quantization of spectral curves via topological recursion__Abstract: This talk will follow the introduction to topological recursion of Nicolas Orantin. Topological recursion associates to some initial data, called spectral curve, a doubly indexed family of multi-differentials on the curve, which often encode important enumerative geometric information. The quantum curve conjecture claims that one can associate to a spectral curve a differential equation whose solution can be reconstructed by the topological recursion applied to the original spectral curve. I will explain how starting from loop equations, one can construct a system of KZ equations whose solutions are vectors of wave functions built from topological recursion. These equations can often be interpreted as PDEs with respect to the moduli of the family of initial data and can be transformed into a Lax system that shares the same pole structure as the initial spectral curve. This procedure solves the conjecture affirmatively for a large class of spectral curves. The talk will be an introduction to this topic, but I will also comment on the technicalities that arise when attacking this conjecture for generic algebraic spectral curves, the solutions we proposed (in joint work with B. Eynard, N. Orantin and O. Marchal) and what remains to be done.

**Alba Grassi**(Geneva):

__Modern developments in (q-) Painlevé equations from Topological String Theory__

Abstract: In the first part, I will give an overview of the recent exact results obtained in the literature on the quantization of JT gravity. In the second part, we will compare these to Liouville gravity results from two perspectives: firstly from a generalization of the underlying group theory, and secondly from a gravitational point of view as a dilaton gravity model with a modified dilaton potential.

**Thomas Mertens**(Ghent):__JT gravity and its structural relation to Liouville gravity__Abstract: In the first part, I will give an overview of the recent exact results obtained in the literature on the quantization of JT gravity. In the second part, we will compare these to Liouville gravity results from two perspectives: firstly from a generalization of the underlying group theory, and secondly from a gravitational point of view as a dilaton gravity model with a modified dilaton potential.

**Nicolas Orantin**(Geneva):

__An introduction to topological recursion and its applications__Abstract: The topological recursion is a formalism which was first developed in order to solve combinatorial applications of random matrices such as the Ising model on a random lattice. It later turned out to be a much more general and was found to solve many other problems in physics and mathematics including the enumeration of random discrete or continuous surfaces, the computation of Gromov-Witten invariants, a quantization of algebraic curves as well as the construction of isomonodromic tau functions. In addition, it is conjecturally linked to the computation of tau functions of integrable hierarchies, of the asymptotics of some knot invariants as well as some correlation functions in conformal field theories.

This talk will be an introduction to the topological recursion formalism starting from the original example of the enumeration of maps, i.e. discrete surfaces built from polygons glued along their edges. I will explain how this combinatorial problem can be turned into a problem of computation of residues on an associated Riemann surface thanks to the definition of appropriate generating series.

I will then explain how this example can be generalised to give rise to a universal inductive procedure taking as input a Riemann surface and building an infinite family of multi-differentials on it. We will see different examples showing that, whenever the Riemann surface considered encodes some enumeration of discs, the multi-differentials built by the topological recursion enumerate the corresponding surfaces with more complicated topology.

This introductory talk will be followed by lectures by Garcia-Failde and Eynard showing other applications of this formalism to quantization of algebraic curves and conformal field theories.

Abstract: Liouville quantum gravity (LQG) is a theory of random surfaces that originated from string theory. Schramm Loewner evolution (SLE) is a family of random planar curves describing scaling limits of many 2D lattice models at their criticality. Before the rigorous study via LQG and SLE in probability, random surfaces and scaling limits of lattice models have been studied via another approach in theoretical physics called conformal field theory (CFT) since the 1980s. In this talk, I will demonstrate how a combination of ideas from LQG/SLE and CFT can be used to rigorously prove several long standing predictions in physics regarding random surfaces and planar lattice models. I will also present some conjectures which further illustrate the deep and rich interaction between LQG/SLE and CFT. Based on joint works with Ang, Holden, Remy, Xu, and Zhuang.

**Xin Sun**(U. Penn.):__Random surface, planar lattice model, and conformal field theory__Abstract: Liouville quantum gravity (LQG) is a theory of random surfaces that originated from string theory. Schramm Loewner evolution (SLE) is a family of random planar curves describing scaling limits of many 2D lattice models at their criticality. Before the rigorous study via LQG and SLE in probability, random surfaces and scaling limits of lattice models have been studied via another approach in theoretical physics called conformal field theory (CFT) since the 1980s. In this talk, I will demonstrate how a combination of ideas from LQG/SLE and CFT can be used to rigorously prove several long standing predictions in physics regarding random surfaces and planar lattice models. I will also present some conjectures which further illustrate the deep and rich interaction between LQG/SLE and CFT. Based on joint works with Ang, Holden, Remy, Xu, and Zhuang.