Short Presentations

• Hybrid methods for Regularized Optimal Transport
  Maxime SYLVESTRE
The weak optimal transport is a generalization of OT with multiple applications. However the lack of structure renders it harder to compute. We propose to study methods of computation.

• Asymptotics of random characteristic polynomials
  Quentin FRANCOIS
In this talk, I will present recent results on the convergence of the reciprocal characteristic polynomial for Girko matrices as a random holomorphic function. This result is a universality result as the limiting object only depends on the second moment of the entries of the matrix. After exposing the techniques used in the proof, I will point out a continuation of this work in the context of Elliptic Ginibre matrices.

• L2 Hypocoercivity methods for kinetic Fokker-Planck equations with factorised Gibbs states
  Luca ZIVIANI
This contribution deals with L2 hypocoercivity methods for kinetic Fokker-Planck equations with inte- grable local equilibria and a factorisation property that relates the Fokker-Planck and the transport operators. Rates of convergence in presence of a global equilibrium, or decay rates otherwise, are estimated either by the corresponding rates in the diffusion limit, or by the rates of convergence to local equilibria, under moment conditions. On the basis of the underlying functional inequalities, we establish a classification of decay and convergence rates for large times, which includes for instance sub-exponential local equilibria and sub-exponential potentials.

• Wasserstein space and entropy
  Hugo MALAMUT
I will introduce the concepts of Wasserstein distance and entropy, give some basic properties and a non-exhaustive list of their fruitful interactions.

• Multidimensional price model for intraday power markets
  Antoine LOTZ
We try to model prices on the EPEX intraday power markets.

• Gibbs Sampling with Robust Statistics
  Antoine LUCIANO
In some applied cases, access to complete data is restricted, often for privacy reasons, and only robust statistics of the data are available. These statistics are less sensitive to outliers and better protect the data thanks to a higher breakdown point. In this article, within a parametric framework, we propose a method for simulating complete data conditional on different robust statistics: the pairs (median, MAD) or (median, IQR) or one or more quantiles. Based on a Gibbs sampler, our method allows simulation according to the posterior of the parameters of certain family of distribution such as Gaussian or Cauchy families.

• Many-body localization: an approach via the renormalization group
  Lydia GIACOMIN
In this presentation, I propose to introduce a model of a disordered many-particle system for which the goal is to prove the localization of energy at large disorder. I will start by introducing a well-known single particle model for which this localization phenomenon has been proven (this is known as Anderson localization). It is possible to prove Anderson localization by using the renormalization group, and we are attempting to use the same approach to prove many-body localization.

Long Presentations

• Isentropic Euler equations: uniqueness and vacuum
  Animesh JANA
The motion of a compressible non-viscous fluid is governed by the Euler equations. In multi dimension, the well-posedness of entropy solution is an open question. We discuss the uniqueness of Hölder continuous solution in the set of weak solutions satisfying entropy conditions. To prove the uniqueness, we impose (i) 1/2 Hölder regularity and (ii) a one-sided Lipschitz bound condition on velocity. We further show the weak-strong uniqueness for the isentropic Euler system when the strong solution may contain a vacuum region. To prove the uniqueness result for the vacuum case, we assume an integrability condition on reciprocal of the density function. This talk will be based on the following two articles [1, 2].
References
[1] E. Feireisl, S. S. Ghoshal and A. Jana, On uniqueness of dissipative solutions to the isentropic Euler system. Comm. Partial Differential Equations, 44, no. 12, 1285–1298, (2019).
[2] S. S. Ghoshal, A. Jana and E. Wiedemann, Weak-Strong Uniqueness for the Isentropic Euler Equations with Possible Vacuum. Partial Differ. Equ. Appl. 3 (2022), no. 4, Paper No. 54, 21 pp.

• Vicsek-Kuramoto third order system in collective dynamics and their macroscopic equations
  Carmela MOSCHELLA (University of Vienna)
Joint work with Sara Merino Aceituno.
In this talk we consider an Individual-Based Model for self-propelled particles interacting through local alignment and investigate its macroscopic limit. This model has been first introduced by C. Chen et al. to describe the behaviour of dense colony of flagellated bacteria (e.g. Escherichia coli) which self organize into robust collective oscillatory motion. In view of this, a ’Vicsek- Kuramoto style’ model has been used. The agent based model combines both Kuramoto and Vicsek dynamics - it takes from Kuramoto model the way in which agents try to syncronize the phase of their rotational movement and from the Vicsek model the fact that the syncronization is spacially local. We study the mean-field kinetic and hydrodynamic limits of this system. Due to the lack of energy and momentum preservation in the system, we use the notion of gen- eralized collisional invariant to obtain a closed set of macroscopic equation for the model. The final macroscopic model involves a continuity equation for the total density of particles, a non conservative equation for angular momentum density and a non conservative equation for the direction of the mean velocity.

• An Introduction to Self-Supervised Learning
  Nicolas MAKAROFF
The presentation is a comprehensive introduction to Self-Supervised Learning (SSL), focusing on its key methods: Contrastive Learning, Self-Distillation, and Correlation Methods. The growing importance of SSL in leveraging unlabeled data for representation learning is underscored. We explore Contrastive Learning's theoretical underpinnings, distinguishing between positive and negative samples, and its successful implementations. Self-Distillation's knowledge propagation between teacher-student networks, even without label information, is examined, alongside its current applications and theoretical foundations. Next, we investigate Correlation Methods, which use statistical relationships within data for learning, providing a theoretical analysis of their effectiveness and limitations. In conclusion, the presentation aims to provide a introductory theoretical understanding of SSL.

• Multi-Marginal Entropic Martingale Optimal Transport and applications to the calibration of Stochastic Processes
  Guillaume CHAZAREIX
Martingale Optimal Transport has found extensive applications in various financial contexts, particularly in the calibration of stochastic processes. The numerical solutions for this problem involve tackling a variational problem that entails non-linear partial differential equations. In this presentation, we delve into a discretization approach for the continuous problem and introduce a relaxation technique by incorporating an entropy term. This relaxation enables the utilization of algorithms analogous to those employed in classical entropic optimal transport. Moreover, we outline potential methodologies for implementing this algorithm on a GPU platform, thereby enhancing computational efficiency.

• Non-decreasing martingale coupling
  Kexin SHAO
The traditional optimal transport problem (OT) consists in minimizing the expected cost $\mathbb{E}[c(X_1, X_2)]$ by considering the joint distribution $(\mu,\nu)$ where the marginal distributions of the random variables $X_1 \sim \mu$ and $X_2 \sim \nu $ are fixed. Motivated by financial applications, martingale optimal transport is considered adding an additional martingale constraint $\mathbb{E}[ X_2 \vert X_1] = X_1$ on top of the OT problem. Hobson and Neuberger first studied the problem with the specific cost function $c(x, y) = −\vert y – x\vert$, Juillet and Beiglbock proved the uniqueness of the associated optimizer $\pi^{\rm HN}$ when $\mu$ is continuous. We observe numerically that the $\pi^{\rm HN}$ is still a maximizer for $\rho\in(0,2)$ and a minimizer for $\rho>2$. We investigate the theoretical validity of this numerical observation and give rather restrictive sufficient conditions for the property to hold. We also exhibit couples $(\mu,\nu)$ such that it does not hold. $\pi^{\rm HN}$ is known to satisfy some monotonicity property which we call non-decreasing. We check that the non-decreasing property is preserved for maximizers when $\rho\in(0,1]$. In general, there exist distinct non-decreasing martingale couplings, and we find some decomposition of $\nu$ which is in one-to-one correspondence with couplings in a non-decreasing sense.

• A varitional approach to reforesting
  João PINTO ANASTACIO MACHADO
We will discuss a variational model for the growth of a tree consisting of the maximization of a harvesting term, modelling the nutrients the plant can get from the soil, minus the cost of constructing the roots, modelled by the branched transportation distance between a Dirac measure (base of the trunk) and a measure giving the distribution of micro-filaments of the roots. The nutrients on the soil satisfy a quasilinear elliptic PDE with measure coefficients. Our first step is to prove wellposedness of the model in the form an existence and uniqueness result for this PDE. In the sequel use this model to define a N-player game where each tree is an agent competing for resources selfishly, and a master player has the agency of controlling the distribution of nutrients on the soil and the positions where the trees are planted.
Joint work with Idriss Mazari and Paul Pegon.

• Regularisation by noise of singular SDEs and the selection problem
  Łukasz MĄDRY
We will consider the following class of equations $- \mathrm{d}X = b(X) \mathrm{d}t + \mathrm{d}W$, where b is a singular function, possibly a distribution and W is an irregular noise. The well-posedness of these equations is by now a well-understood phenomenon, which is called regularisation by noise. We will outline recent developments of this theory - in particular, its non-Markovian aspects which allow to investigate the influence of the noise of different regularity than Brownian motion.
Time permitting, we will also discuss new results on the selection problem, i.e. the limit of $\mathrm{d}X = b(X) \mathrm{d}t + \varepsilon \mathrm{d}W$ where $\varepsilon$ tends to zero.

• Geometry and optimal transportation: Applications to visual generative models
  Changqing FU
In this talk, we takle the problem of adding geometrical constraints into neural networks for visual generative models. First we reformulate standard neural networks under a variational perspective and introduce the concept of geometry and invariance. Then we review major methods including Wasserstein Generative Adversarial Networks and Diffusion Models, and the constraints to control the generation process, including style, text, or other forms of signals. Finally we propose several practical solutions using geometry and optimal transportation, involving convolution, activation, perceptual distance, etc. Along the way, we put emphasis on the description of the attention funcion which is a recent breakthrough in machine learning.

• Closed-form MLE for multivariate/multiparameter regressions models with categorical explanatory variables
  Antoine BURG
The maximum likelihood estimator (MLE) remains the most frequently used method to estimate the parameters of generalized linear models (GLM). But even for distributions within the exponential family, MLEs are not always tractable and need to be computed with numerical methods like Newton-Rapshon-type algorithm. Alternative closed-form estimator have been found in case of categorical explanatory variables for univariate random variables of one-parameter exponential type. Extensions of these estimators in case of multivariate and/or multi-parameter exponential type distributions are proposed, and some of their properties are studied. Those estimators are illustrated with simulated data. Actuarial applications involving causes-of-deaths mortality modelling are also presented.

• On existence and uniqueness of stationary equilibria in macroeconomics models with heterogeneous agents
  Diego Alejandro MURILLO TABORDA
Macroeconomics models with heterogeneous agents can be used to answer to several of the most classical questions in economics. These models usually are represented as mean field games with equilibria given by solutions to nonlinear systems of couple partial differential equations (PDE), completely different to usual mean field games, where the nonlinearity between the value function and the distribution of the state variables is separable. Then there are some open mathematical problems about the existence and the uniqueness of the equilibria in the macroeconomic models with heterogeneous agents, unlike the same problem in usual mean field games, which have been debeloped several years ago. In this talk, I will present some own results about existence or uniqueness of stationary (without time dependence) equilibria in macroeconomics models with heterogeneous agents.

• Some remarks on the long time behviour of Stochastic McKean-Vlasov Equations
  Raphael MAILLET
In this talk, we explore the long-term behavior of solutions to a nonlinear McKean-Vlasov equation with common noise. This equation arises naturally when studying the collective behavior of interacting particles driven by both individual and shared noise. Our main focus is to understand how the presence of common noise affects the stability of the system. We will first present the results for the case without common noise and then discuss how the introduction of common noise can enhance stability in certain respects.

• Diffusion Limit and other Process Approximations for Control problems
  Lorenzo CROISSANT
Control theory is the study of the best way to interact with a dynamical system to drive it to maximise a criterion. This is a functional optimization problem, which can be solved by resorting to a non-linear Partial Integro-Differential Equation equation. In some situations, this equation can be difficult to analyse or solve numerically. In such cases, we would like to be able to use a different system as an approximation. In this talk we will go over one methodology to swap two systems in control problems with approximation guarantees. After giving a general introduction to control theory, we will focus the presentation on the diffusion limit of high-frequency pure-jump systems and their various limit regimes, then talk about the method from a general perspective including fluid limits and numerical schemes.