The goal of this talk is to illustrate the hydrodynamic limit theory by examples. First, we study the Donsker’s theorem (about the convergence of the rescaled simple symmetric random walk on Z to the standard Brownian motion) with this point of view. Then, I will show the particle system on which I’m working: these particles are described by their position and their velocity. The hydrodynamic limit of this system should be the solution of a heat equation with free boundary: the space domain of the equation depends on the time. This free boundary is the main difficulty of this system: I will end the talk explaining how we manage to show the convergence, using relative entropy of the system.

How much should you pay today for a contract whose value depends on an uncertain future? This talk introduces the pricing problem in mathematical finance from scratch. Starting from a concrete hedging problem — an airline protecting itself against a spike in oil prices — we build up the key ideas: derivatives and options, self-financing replicating portfolios, and the principle of no arbitrage. We show how this single economic axiom forces the existence of a risk-neutral measure under which prices are simply discounted expectations, first in the one-period binomial model and then in continuous time via the Fundamental Theorems of Asset Pricing. We close with Girsanov's theorem and the Feynman–Kac link to PDEs, recovering the celebrated Black–Scholes formula.

In this presentation, I will motivate and present the non-convex low-dimensional relaxation of the orthogonal synchronization problem. As a non-convex optimization problem, it can have spurious local minima but we will explore conditions — on the Laplacian matrix of the problem and on the signal to noise ration of the model — under which all local minima are global minima.



 

In finite dimension, self-adjoint operators can be reduced to diagonal matrices by changing the basis to a set of eigenvectors. In infinite dimension, the spectral theorem tells us how to generalize this behavior. First we need to expand our definition of the spectrum to include numbers that are only quasi-eigenvalues, due to their Weyl sequences not converging. Then it turns out that an operator can be isometrically associated with a multiplier on an L^2 space with a specific « spectral » measure. The spectrum of an operator can then be broken into discrete and essential components, which are very useful in the study of its perturbations. 

Quantum mechanics aims to describe all kinds of phenomena that occur at the microscopic scale. In particular, one of the first and most remarkable result of this theory consists in providing an incredibly accurate description of atoms, molecules and their periodic features. Despite the simplicity of the physical systems considered, the mathematical background used to develop these models combines important theorems of Spectral Theory and Calculus of Variations. After a brief introduction to the mathematical formalism behind the quantum atomic model, we will see the so-called ionization conjecture, an open problem in mathematics linked to one most important periodic property of atoms.


 

In this talk I will survey the stability theory for the damped wave equation, focusing on the two extreme situations of strong stability and exponential (or uniform) stability. This problem is motivated by inverse problem (guessing the shape of a domain by measuring sound wave propagating along it) and sensor/room design (e.g. minimize/maximize the reverberation). The main part of the talk is to connect these two stability properties to controllability properties, which are themselves equivalent to observability properties of the free wave equation. Strong stability is seen as a rather weak property (any decent damping achieves it) while it is not evident at first glance how to impose exponential stability. The complete characterization of exponential stability by the geometric control condition will be sketched without proof, mainly with drawings.

Randomized clinical trials remain the gold standard for evaluating drug effects, but they are costly, not always feasible, and often poorly suited to studying long-term outcomes in large populations. Real-world data can help address these limitations, while raising important methodological challenges related to bias, confounding, and causal interpretation.
The French Système National des Données de Santé is a nationwide medico-administrative database containing health insurance claims for almost the entire French population, including detailed drug reimbursement data. However, these data were collected for administrative purposes rather than research, and must be carefully transformed before analysis.
In this presentation, I will focus on the interpretation of drug reimbursements as proxies for drug consumption or exposure. I will present three examples from doctoral projects conducted in the HeKA team, showing how similar SNDS reimbursement data can be transformed in different ways to answer distinct pharmacoepidemiological questions.

Classical causal inference largely targets the average treatment effect (ATE), a single scalar summary of how an intervention shifts an outcome's mean. Yet two treatment arms can share an identical ATE while differing dramatically in variance, modality, or tail behavior: differences that carry real consequences for risk and heterogeneity but are invisible to the mean alone. This talk argues for shifting the object of interest from the ATE to the full interventional density of the outcome, identified through the backdoor formula under standard unconfoundedness assumptions. We then show that the natural plug-in estimator of this density inherits a first-order bias whenever flexible machine learning models are used for the conditional density, and we derive a one-step, influence-function-based correction that restores root-n efficiency and double robustness, extending the classical AIPW construction from means to entire densities.

We will present the problem of proving Donsker-type theorem for an empirical process indexed by a class of functions. For this, we will present chaining, a technique developed to control the supremum of a stochastic process, and show how it can be used to prove Donsker type theorems. This talk is based on the books « Weak Convergence and Empirical Processes » by van der Vaart and Wellner and « The Generic Chaining » by Talagrand.

In this talk, we explore the interplay between the geometry (convexity) and the regularity (continuity, differentiability) of real-valued functions defined on R^d. First, we review the standard properties of convex functions on R^d: continuity, as well as first- and second-order differentiability almost everywhere. Finally, we prove our recent improvement of a lemma that quantifies the Lipschitz continuity (in an averaged sense) of the gradient of a convex function.