Here are my full research statement and my papers on Arxiv.
I have developed in the following three fields of mathematics.
Optimization
Optimization is the study of algorithms that allow the computer to find the configurations of minimal or maximal energy for a given system. The field is about as broad as is its field of application, which ranges from the training of neural networks for artificial intelligence to planning logistics to solving differential equations.
- With Milan Korda, we studied the gap between a Lagrangian optimization problem and its occupation measure relaxation, with applications to algorithms solving PDEs, optimal control and transport problems.
- With Jérome Bolte and Edouard Pauwels, we studied the long-term dynamics of the subgradient descent method in nonsmooth and nonconvex optimization problems using a novel methodology based on closed measures.
- Found some pathological examples of subgradient descent dynamics.
- With Pascal Bianchi, generalized the closed measure methodology to understand stochastic dynamics driven by subdifferential inclusions with applications to stochastic first-order optimization methods and game theory.
Geometric Measure Theory
A field born from the connection of bubbles and minimal surfaces, it now has connections with all sorts of physical models and mathematical theories. My personal interest in it stems from the possibility of using it to quantize space-time, a very hard project that still needs work — in the meantime, here are some fruits of my research.
- A characterization of the distributions that arise as derivatives of families of probability measures and found a way to make sense of their direction of transport when it does not correspond to a vector field.
- A characterization of minimizable Lagrangian actions beyond quasiconvexity and the direct method, finally pinpointing the connection to the holonomicity constraint.
- Solved the long-standing problem of variationally characterizing closed measures invariant under the Euler-Lagrange flow.
- Developed a categorical formulation of weak KAM theory, and uncovered a version of it that exists in a space of cobordisms.
Representation Theory
This field focuses on understanding symmetry, ranging from the simplest situations in which one may rotate a shape or exchange some components of a complex object, to very complicated circumstances involving infinite-dimensional symmetry group actions.
- My PhD thesis, written under the direction of Andrei Okounkov, focused on leveraging the symmetries of almost-involutive permutations to understand how many different surfaces one can construct by gluing a number of squares of equal size. This shed light on the volumes of the strata of moduli of quadratic differentials and was closely connected to billiard dynamics. It was published as a paper.
- I also wrote a pedagogical introduction to the half-infinite wedge, a sort of zero-dimensional quantum field theory with theoretical importance in algebraic geometry.