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Pour la troisième année consécutive les doctorants du CEREMADE ont organisé les Journées Jeunes Chercheurs 2025 qui se sont déroulées du 3 au 5 juin.
L'objectif de cette rencontre est de réunir les jeunes membres de la communauté, en les encourageant à participer, échanger et partager. Les interactions scientifiques et humaines sont favorisées par ces moments, et jouent un rôle important dans le début de leur carrière (académique ou non). Les domaines de recherche des étudiants couvrent un large spectre de sujets en mathématiques, dans le cadre du laboratoire.
Organisateurs : Lucas Davron, Brune Massoulié, Amal Omrani et Thaddeus Roussigné.
Au programme de cette année :
An intuitive proof of the Cwikel-Lieb-Rozenbljum inequality |
| The spectrum of the Schrodinger operator describes the possible states for a quantum particle under the effect of a given potential V, hence a core question in quantum mechanics is to measure how many eigenvalues a specific Hamiltonian might have. The CLR bound is a subcase of the celebrated Lieb-Thirring inequalities, and states that the number of negative eigenvalues must be bounded by the d/2 Lebesgue norm of the potential. The result has many derivations, we here present an intuitive and robust approach similar to Rozenbljum’s original proof from 1972, correspondingly the constant it yields is an order of magnitude away from being sharp. This talk is based on Rupert Frank’s recent seminar on Lieb-Thirring inequalities. |
Particle systems and relative entropy |
| We consider a particle system where particles evolve along a line at constant speed. This system has already been studied with some boundary conditions (particles on a torus or confined between two walls), but we propose here an unsolved problem, where the particles are evolving between a wall on one side and some pressure on the other side. As this system is not easy to understand, we add a thermalization phenomenon, ie some probabilistic term in the equation (here it will be Langevin dynamics). The goal is to compute the hydrodynamic limit of this system, ie the behaviour of the system when the time and number of particles goes to infinity — the idea is to describe the macroscopic behaviour from the microscopic equations. The goal of this talk is to define the relative entropy in this context and to show how entropy inequality helps in the study of such systems. |
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Bayesian statistics: posterior contraction |
| In this talk, I will present an important result by Ghosal, Ghosh and van der Vaart (2000) in Bayesian statistics. This result gives sufficient conditions to obtain (optimal) rate of convergence of the posterior measure in non-parametric models. The proof is not too difficult from a technical point of view and so should be accessible to everyone! |
Exponential Estimates for Multitype Poissonian Branching Processes. |
| In this talk I will discuss Poisson branching processes: Galton Watson trees and cluster processes. We will look at the multitype case and give estimates of the Laplace transform of the typed cardinal of these processes. We will see how branching translates into fixed point equations and how we can study them. |
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| This presentation is meant to be a pedestrian introduction to partial differential equations. We will begin by some historical context to understand what motivated our ancestors to study these equations. A (modern) challenge is to formalize what it mean for a PDE to be well-posed, and I will give classical examples of ill-posed (although very simple!) problems. We will then solve the Dirichlet problem for the Poisson equation following Hilbert's program, and incidentally discover the Sobolev space H^1_0(\Omega). If time still allows for it I will present the method of separation of the variables for evolution problems, and more generally the semigroup theory. |
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Stochastic Optimal Control under Constraints with Deep Learning |
| This talk revisits an example of stochastic optimal control applied to finance and aims to introduce a numerical resolution for this problem using Deep Learning. More specifically, the example looks at the case where both the function to be optimized and the constraint function takes the form of an expected value, which motivates the usage of a martingale representation for the constraint condition. As for the numerical resolution, I will present the basis of training a neural network and demonstrate a few steps of the algorithm in more detail. |
A flocking Cucker-Smale model with a continuum of steady states. |
| I will first present the most well-known flocking models, namely the Vicsek and Cucker-Smale models. The model I am studying is very similar to the Vicsek model and also exhibits a phase transition. We will focus on the low-temperature regime, which corresponds to the case where the equation admits a continuum of stable, non-isotropic equilibria. In this case, we will show that the solution indeed converges to a stable equilibrium state, which is a new result, and we obtain an exponential rate of convergence. |  |
Geometric action of the Lax-Oleinik semi-group and its regularizing effect |
 | In the setting of a Tonelli Hamiltonian, Arnaud (2011) unveiled a beautiful geometric explanation of a regularization property of the Lax-Oleinik semi-group for small enough time, introduced by Bernard (2007). In the first part of this talk, we aim to explain — in an informal way — the main ideas of this result using examples. Afterwards, we discuss how a similar statement can be obtained for "arbitrarily long time", while restricting the regularization property to a "local" version. |
Existence and uniqueness of stationary equilibria in the Aiyagari model with finite individual states space |
The Aiyagari (1994), Hugget (1993), and Bewley (1983) model is one of the workhorse models of modern macroeconomics. It is used for the study of financial wealth inequality. Although of its importance for economics, this model is a case of a mean field game, which is not covered by the existing theory about existence and uniqueness of equilibria in mean field games, creating new open mathematical problems. In this presentation, I will present own results about existence and uniqueness of stationary equilibria in the Aiyagari model when a finite number of possible financial wealth and income are considered, as well the numerical methods to compute the stationary equilibria.
Super-Hedging under Transaction Costs |
| In financial markets, buying and selling assets involves costs that impact investment strategies. My research explores mathematical methods to solve the hedging problem in the presence of transaction costs. Unlike traditional approaches that rely on strong assumptions, my work focuses on exploiting real market data—specifically, the conditional support of relative prices. For a fixed sequence of transaction costs and a fixed number of revision dates, we do not assume a specific price model. The approach is based on a key economic principle called Absence of Immediate Profit (AIP) and uses mathematical optimization tools such as Legendre-Fenchel conjugates. In this talk, I will present a backward-forward computational scheme that determines the optimal super-hedging price and the associated strategy at each date, given a specific form of the payoff at maturity—a form that is preserved throughout the recursive pricing scheme. |
Probabilistic View on the Signature Method |
| This talk explores how path signatures can effectively capture path dependence and serve as a foundation for modeling stochastic processes. We begin by analyzing a stochastic exponential $S$ whose volatility process is given by a linear functional of the signature of a time-extended Brownian motion. Excluding trivial cases, we show that $S$ is a true martingale if and only if the order of the linear functional is odd and a correlation parameter is negative. The proof involves a fine analysis of the explosion time of a signature-driven stochastic differential equation. Once martingality is established, we further characterize the existence of higher moments of $S$ via a condition on the same correlation parameter. We then introduce the fading-memory (FM) signature, a time-invariant transformation of infinite paths that plays the role of a mean-reverting analogue to the classical path signature. The FM-signature retains many desirable algebraic properties, including a modified version of Chen’s identity and a linearization property. Unlike the classical signature, it provides a "stationarized" representation of paths, making it particularly well-suited for time series modeling and signal processing. For the FM-signature of time-extended Brownian motion, we establish stationarity, exponential ergodicity in Wasserstein distance, and derive an explicit formula à la Fawcett for its expected value. Finally, we briefly discuss how tools from Malliavin calculus can be applied to the signature of Brownian motion. |
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Symmetry of Optimizers of Functional Inequalities |
| Energy functionals are central in physics, with ground states (their minimizers) providing key insights into the behavior of physical systems. In certain cases, particularly for quantum one-particle systems, finding ground states becomes equivalent to solving variational problems arising from functional inequalities. Functional inequalities, in turn, are a cornerstone of modern analysis and appear across various areas in mathematics. A fundamental and challenging question in this context is whether the minimizers of such inequalities, if they exist, retain the symmetries of the functional — that is, whether they are invariant under the same transformations. In this talk we will present some strategies to prove symmetry and symmetry breaking illustrated on the example of the Caffarelli-Kohn-Nirenberg inequality. Furthermore analytic and numerical results on the symmetry problem in a spinorial Caffarelli-Kohn-Nirenberg inequality are presented. For dimension 2, this has deep connections with non-relativistic quantum mechanics in an Aharonov-Bohm magnetic field. |
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