I am currently a postdoctoral researcher at Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) under the responsibility of Rémi Bardenet. Beforehand, I worked from 2020 to 2022 at Institut de Recherche Mathématique Avancée (IRMA) of Université de Strasbourg, for the project “Geometry of quantum Hall states” of USIAS directed by Semyon Klevtsov. From 2016 to 2020, I was a PhD student in Laboratoire de Probabilités, Statistique et Modélisation (LPSM) of Sorbonne Université, under the direction of Thierry Lévy. A pdf version of my resume is available here.
Keywords: random matrices, point processes, random partitions, Yang–Mills theory, quantum Hall effect, numerical integration.
My work primarily falls within the field of mathematical physics, which involves developing rigorous mathematical tools for physical models. To achieve this, I draw from various mathematical theories: probability, algebra, geometry, combinatorics, and analysis.
I focus on the asymptotic aspects of the Yang–Mills measure on compact surfaces with a compact matrix group of large rank as the structure group. This involves a nontrivial model of random matrices that incorporates couplings of Brownian motions on the group. I also plan to study the interactions between this model and random surfaces.
The integer quantum Hall effect corresponds to a model of free fermions subjected to a confining potential. When these fermions live on a complex variety, the model can be expressed as a determinantal point process whose kernel is the Bergman kernel associated with a line bundle. I investigate the limiting behavior of the model as the number of particles tends to infinity. I also hope to find a probabilistic model of the fractional quantum Hall effect.
Determinantal processes on complex manifolds, initially studied for the quantum Hall effect, prove to be a valuable tool for performing Monte Carlo methods. The theoretical tools are well-known and rely on existing results from complex geometry and pluripotential theory, but practical implementation remains a major challenge.
My PhD thesis was devoted to study asymptotic aspects of two-dimensional Yang-Mills theory. More precisely, considering the Yang–Mills measure on a compact orientable surface of genus greater or equal to 1, or a compact nonorientable surface of genus greater or equal to 2, I proved the convergence of its partition function with structure group U(N) or SU(N), using the character expansion of the heat kernel. In order to establish this convergence, I highlighted a class of highest weights that already appeared in works from Gross and Taylor (94) that I named ‘almost flat highest weights’, or AFHW, and that help getting a fine approximation of the Laplace operator on U(N) or SU(N) when N goes to infinity. I have later used these almost flat highest weights to compute the large N limit of Wilson loops for contractible simple loops on the underlying surface.
Office : Bâtiment ESPRIT, Avenue Paul Langevin, 59650 Villeneuve-d’Ascq (France), office S1.58 Mail : thibaut.lemoine((AT))univ-lille.fr