## Research interests

My main interest lies in optimal transport and the geometry of the Wasserstein space, convex analysis and optimization, calculus of variations and Bayesian statistics. Below is a list of publications organized by research themes. The years indicated refer to the first submission of the work.

- Optimal curves and mappings valued in the space of measures.
- Numerical solution to the dynamical optimal transport problem.
- Optimal transport in Bayesian statistics.
- Optimal transport and calculus of variations in data science.

### Optimal curves and mappings valued in the space of measures

Optimal transport provides a natural way to interpolate between probability distributions, generating geodesics (that is, shortest paths) in the so-called Wasserstein space. I have studied some variations of this problem where one not only looks for the shortest path, but adds additional terms in the objective functional. Penalization of congestion yields model of fluid dynamics and is connected to Mean Field Games; allowing for growth terms leads to unbalanced optimal transport.

An extension of the geodesic problem is to consider not curves valued in the Wasserstein space, but mappings: that is, probability measures depending on more than one parameter. I have worked on a definition of Dirichlet energy for such mappings proposed by Brenier, studied its minimizers (the harmonic mappings) and with collaborators we proposed an application to the theory of elasticy. This is related to analysis in non smooth spaces and the problem of harmonic metric valued mappings.

- The Riemannian geometry of Sinkhorn divergences (slides). With Jonas Luckhardt, Gilles Mordant, Bernhard Schmitzer and Luca Tamanini (2024).
*Submitted paper*. - Lifting functionals defined on maps to measure-valued maps via optimal transport (slides). 2023.
*Submitted paper*. - Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps. HL, Léonard Monsaingeon, Luca Tamanini and Dmitry Vorotnikov (2024).
*Accepted paper: Calculus of Variations and PDEs*. - Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion (slides). With Aymeric Baradat (2021).
*Accepted paper: Astérisque*. - Hidden convexity in a problem of nonlinear elasticity (slides). With Nassif Ghoussoub, Young-Heon Kim, and Aaron Zeff Palmer (2021).
*Accepted paper: SIAM Journal on Mathematical Analysis*. - New estimates on the regularity of the pressure in density-constrained Mean Field Games (slides). With Filippo Santambrogio (2019).
*Accepted paper: Journal of the London Mathematical Society*. - Harmonic mappings valued in the Wasserstein space (slides). 2019.
*Accepted paper: Journal of Functional Analysis*. - Optimal density evolution with congestion: L infinity bounds via flow interchange techniques and applications to variational Mean Field Games (slides). With Filippo Santambrogio (2018).
*Accepted paper: Communications in PDEs*. - Time-convexity of the entropy in the multiphasic formulation of the incompressible Euler equation. 2017.
*Accepted paper: Calculus of Variations and PDEs*.

### Numerical solution to the dynamical optimal transport problem

I have worked on the dynamical optimal transport problem (a.k.a. Benamou-Brenier formulation) and its numerical solution. I have proposed a new discretization of the problem on surfaces, and also done the analysis of the convergence of solutions to the discrete problems to the continuous one when the spatial and temporal step size vanish.

- Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation. With Sadashige Ishida (2023).
*Accepted paper: Foundations of Computational Mathematics*. - Unconditional convergence for discretizations of dynamical optimal transport (slides). 2021.
*Accepted paper: Mathematics of Computations*. - Dynamical Optimal Transport on Discrete Surfaces (supplemental, code, slides). With Sebastian Claici, Edward Chien and Justin Solomon (2018).
*Accepted paper: SIGGRAPH Asia 2018*.

### Optimal transport in Bayesian statistics

With collaborators in Bayesian statistics, we have been using optimal transport to define distances between Completely Random Measures. With the help of such distances, we are able to define a tractable index of dependence between Completely Random Vectors and evaluate the impact of the prior in a nonparametric setting. I have also used optimal transport and variational approaches to study the sampling problem with coordinate-wise methods, such as Coordinate Ascent Variational Inference and Gibbs sampling.

- Entropy contraction of the Gibbs sampler under log-concavity. With Filippo Ascolani and Giacomo Zanella (2024).
*Submitted paper*. - Convergence rate of random scan Coordinate Ascent
Variational Inference under log-concavity. With Giacomo Zanella (2024).
*Accepted paper: SIAM Journal on Optimization*. - Hierarchical Integral Probability Metrics: A distance on random probability measures with low sample complexity (slides). With Marta Catalano (2024).
*Accepted paper: ICML 2024*. - Merging Rate of Opinions via Optimal Transport on Random Measures (slides). With Marta Catalano (2023).
*Submitted paper*. - A Wasserstein index of dependence for random measures (slides: version for statisticans, version for analysts). With Marta Catalano, Antonio Lijoi and Igor Prünster (2024).
*Accepted paper: Journal of the American Statistical Association*.

### Optimal transport and calculus of variations in data science

Below is a list of projects (unrelated one to the other) involving optimal transport and calculus of variations in data science, understood in a broad sense. In particular I have worked on the trajectory inference problem (reconstructing the law of a stochastic process from samples of its temporal marginals) and connected it to entropic optimal transport and the Schrödinger problem.

- The flow map of the Fokker-Planck equation does not provide optimal transport. With Filippo Santambrogio (2022).
*Accepted paper: Applied Mathematics Letters*. - Towards a mathematical theory of trajectory inference (slides). With Stephen Zhang, Young-Heon Kim and Geoffrey Schiebinger (2024).
*Accepted paper: Annals of Applied Probability*. - Total Variation Isoperimetric Profiles. With Daryl DeFord, Zachary Schutzman and Justin Solomon (2019).
*Accepted paper: SIAM Journal on Applied Algebra and Geometry*.

## PhD

Between 2016 and 2019, I worked under the supervision of Filippo Santambrogio to complete a PhD. My PhD manuscript can be found here and the slides of my defense here.

## Undergraduate works

You can find below works that I did before starting my PhD.

- My master thesis in history of science (September 2016, in French). It was realized under the supervision of Laurent Mazliak and was about the history of the teaching of probability calculus and statistics in France.
- My master thesis in mathematics (September 2015, in French). It was realized under the supervision of Filippo Santambrogio.
- My internship report of Master 1 (November 2014). It summarizes the work done during an intership at CalTech (United States of America) under the supervision of Oscar Bruno and Edwin Jimenez about stability problems in numerical analysis.
- The research conducted during my internship at CEA Saclay (June and July 2013) under the supervision of Grégoire De Loubens led to a publication: Mechanical magnetometry of Cobalt nanospheres deposited by focused electron beam at the tip of ultra-soft cantilevers. HL, Vladimir Naletov, Olivier Klein, Grégoire De Loubens, Laura Casado, and José Maria De Teresa. 2014.
*Nanofabrication*, 1(1).