The $O(n)$ model can be formulated in terms of loops living on the lattice, with n the fugacity per loop. In two dimensions, it is known to possess a rich critical behavior, involving critical exponents varying continuously with n. In this talk, we will consider the case where the model is ”coupled to 2D quantum gravity”, namely it is defined on a random map.
It has been known since the 90's that the partition function of the model can be expressed as a matrix integral, which can be evaluated exactly in the planar limit. A few years ago, together with G. Borot and E. Guitter, we revisited the problem by a combinatorial approach, which allows to relate it to the so-called Boltzmann random maps, which have no loops but faces of arbitrary (and controlled) face degrees. In particular we established that the critical points of the $O(n)$ model are closely related to the ”stable maps” introduced by Le Gall and Miermont.
After reviewing these results, I will move on to a more recent work done in collaboration with G. Borot and B. Duplantier, where we study the nesting statistics of loops. More precisely we consider loop configurations with two marked points and study the distribution of the number of loops separating them. The associated generating function can be computed exactly and, by taking asymptotics, we show that the number of separating loops grows logarithmically with the size of the maps at a (non generic) critical point, with an explicit large deviation function. Using a continuous generalization of the KPZ relation, our results are in full agreement with those of Miller, Watson and Wilson concerning nestings in Conformal Loop Ensembles.[-]

Given a representation of a reductive group, Braverman-Finkelberg-Nakajima have defined a remarkable Poisson variety called the Coulomb branch. Their construction of this space was motivated by considerations from supersymmetric gauge theories and symplectic duality. The coordinate ring of this Coulomb branch is defined as a kind of cohomological Hall algebra; thus it makes sense to develop a type of “Springer theory” to define modules over this algebra. In this talk, we will explain this BFN Springer theory and give many examples. In the toric case, we will see a beautiful combinatorics of polytopes. In the quiver case, we will see connections to the representations of quivers over power series rings. In the general case, we will explore the relations between this Springer theory and quasimap spaces.[-]

In the 40s R. Feynman invented a simple model of electron motion, which is now known as Feynman's checkers. This model is also known as the one-dimensional quantum walk or the imaginary temperature Ising model. In Feynman's checkers, a checker moves on a checkerboard by simple rules, and the result describes the quantum-mechanical behavior of an electron.
We solve mathematically a problem by R. Feynman from 1965, which was to prove that the model reproduces the usual quantum-mechanical free-particle kernel for large time, small average velocity, and small lattice step. We compute the small-lattice-step and the large-time limits, justifying heuristic derivations by J. Narlikar from 1972 and by A.Ambainis et al. from 2001. The main tools are the Fourier transform and the stationary phase method.
A more detailed description of the model can be found in Skopenkov M.& Ustinov A. Feynman checkers: towards algorithmic quantum theory. (2020) https://arxiv.org/abs/2007.12879[-]

Liouville conformal field theory (LCFT) was introduced by Polyakov in 1981 as an essential ingredient in his path integral construction of string theory. Since then Liouville theory has appeared in a wide variety of contexts ranging from random conformal geometry to 4d Yang-Mills theory with supersymmetry.
Recently, a probabilistic construction of LCFT on general Riemann surfaces was provided using the 2d Gaussian Free Field. This construction can be seen as a rigorous construction of the 2d path integral introduced in Polyakov's 1981 work. In contrast to this construction, modern conformal field theory is based on representation theory and the so-called bootstrap procedure (based on recursive techniques) introduced in 1984 by Belavin-Polyakov-Zamolodchikov. In particular, a bootstrap construction for LCFT has been proposed in the mid 90's by Dorn-Otto-Zamolodchikov-Zamolodchikov (DOZZ) on the sphere. The aim of this talk is to review a recent series of work which shows the equivalence between the probabilistic construction and the bootstrap construction of LCFT on general Riemann surfaces. In particular, the equivalence is based on showing that LCFT satisfies a set of natural geometric axioms known as Segal's axioms.
Based on joint works with F. David, C. Guillarmou, A. Kupiainen, R. Rhodes.[-]

Liouville conformal field theory is a 2 dimensional field theory introduced in physics in the 80's. Here we give a probabilistic construction of the amplitudes of Riemann surfaces with boundary for this field theory, and we prove that they satisfy the so called Segal Axioms. This allows to decompose the correlation function of the theory using the diagonalisation of a certain operator, the Hamiltonian, using scattering theory. One can then show the formulas conjectured in physics in terms of conformal blocks. The spectral analysis, although in an infinite dimensional setting, has some similarities with scattering theory on symmetric spaces. This is joint work with Kupiainen, Rhodes and Vargas.[-]