Domaine de la tour in Saint-Pierre-Canivet in the Calvados region of Normandy.
February 27 - March 04, 2022.
Program:
Sunday, February 27.
12:30pm. Departure from Paris-Dauphine University.
7:00pm. Dinner.
Monday, February 28. Optimal Transport day.
7:30am-8:15am. Breakfast.
8:30am-9:30am. Giovanni Brigati,
Course 1: Time-continuous formulations of optimal transport & Beyond.
Optimal transport has become one major tool in mathematics and applied sciences, providing a “horizontal” distance between probability distributions. As the research progresses, some advanced techniques and structures are being developed for both theoretical and applied interests.
The goal of this talk is giving an account of a selection of recents results in the area, focusing on ideas rather than technicalities. We directly build on the classic theory (presented by Katharina Eichinger in the Young Researchers’ Seminar of CEREMADE), with the help of the variational instruments exposed by João Machado in the same seminar series. Therefore, our program is the following.
1. The geometry of Wasserstein space. Infinite-dimensional Riemann structure. Riemannian metric, geodesics, tangent, cotangent and normal space. The origin of the Benamou-Brenier fluid-dynamics representation of optimal transport and the continuity equation.
2. A Lagrangian vision of gradient flows in the Wasserstein space. Maximal descending slope. Tangent dissipation of the energy. Gradient of regular functionals. Application to local and non-local diffusion equations.
3. Bonus tracks. Metric measure spaces. The displacement convexity of the entropy. The Bakry-Emery curvature-dimension condition. RCD (K, ∞) spaces. The metric heat flow and the Wasserstein flow of the entropy: Kuwada’s duality. The tangent of a metric space. Metric Sobolev and BV spaces via integration by parts.
9:30am-10:15am. Coffee Break.
10:30am-11:30am. Théo Bertrand,
Course 2: Numeric of Optimal Transport & Applications.
The development of tractable algorithms for the computation of Optimal Transport (OT) plans and distances allows researchers from other fields to rely on the properties of OT in various applicative domains. In this presentation we will recall numerical methods used for the computation of OT plans and review some of the interesting properties of the related distances and their applications. First, we will show that an approximation of the OT problem can be solved using Sinkhorn algorithm, and that it allows to create interesting algorithms for generative models, spike deconvolution or solving PDEs.
12:00pm-1:45pm. Lunch.
Free afternoon.
7:00pm. Dinner.
Tuesday, March 01. Mathematical & Statistical Physics day.
7:30am-8:15am. Breakfast.
8:30am-9:30am. Michal Jex,
Course 3: Building Quantum Mechanics from the Ground Up.
In this lecture we present the standard construction of quantum theory from the elementary building blocks. From this point of view, we can understand quantum mechanics as a mathematical theory standing on few postulates. From these basic principles we derive the whole theory using tools from functional analysis, especially analysis of Hilbert-space operators.
9:30am-10:15am. Coffee Break.
10:30am-11:30am. Ivailo Hartarsky,
Course 4: Introduction to dynamics of interacting particle systems.
We will provide a broad overview of dynamics of interacting particle systems. This is a very general class of Markov processes usually living on a lattice. Our approach will be via numerous examples. We will discuss models as diverse as the exclusion process, Glauber dynamics, first passage percolation and the contact process. This will lead us to explore various phenomena such as hydrodynamic limits, phase transitions, metastability, cutoff, limit shapes, universal fluctuations. To keep analysts awake we will mention a few connections of interacting particle systems to differential equations.
Fortunately, I am not an expert on any of these themes, so the talk is guaranteed to remain very accessible. Proofs will be kept to a bare minimum, since our main goal is to give a taste of the what the field is about as whole.
12:00pm-1:45pm. Lunch.
2:00pm-2:30pm. Réka Szabó,
Peierls bounds from Toom contours.
We review and extend Toom's classical result about stability of trajectories of cellular automata, with the aim of deriving explicit bounds for monotone Markov processes, both in discrete and continuous time. This leads, among other things, to rigorous bounds for a two-dimensional interacting particle system with cooperative branching and deaths. Our results can be applied to derive bounds for other monotone systems as well.
Joint work with Cristina Toninelli and Jan Swart.
2:30pm-3:00pm. Adéchola Kouande,
Phase Transition in the Peierls model for polyacetylene.
In this talk, we will provide the results obtained by studying the Peierls model for Polyacetylene with temperature. More precisely we will consider the Peierls model for polyacetylene in presence of temperature, we will prove the existence of a critical temperature below which the chain is dimerized or kink-like according to its parity, and above which the system is 1-periodic. The chain behaves like an insulator below the critical temperature and like a metal above it. We will characterize the critical temperature in the thermodynamic limit model and study the bifurcation around it.
Joint work with David Gontier and Éric Séré.
3:00pm-3:20pm. Coffee break.
3:30pm-3:50pm. Umberto Morellini,
The Bogoliubov-Dirac-Fock equation coupled with a classical nuclear dynamic.
Mean-field models in quantum electrodynamics involve the Dirac operator, which is not bounded from below, causing difficulties from both a physical and a computational point of view. The Bogoliubov-Dirac-Fock (BDF)
model was introduced by Chaix and Iracane in 1989 by considering the contribution of quantum vacuum
in order to overcome these problems.
In this short talk, we will consider a BDF equation coupled with a Newton equation describing the interaction between the vacuum and a finite number of classical nuclei. This system of differential equations represents an attempt to generalise the work of Cancès and Le Bris (1999) to the relativistic case.
3:50pm-4:10pm. Jean-Paul Adogbo,
Existence of global strong solution of Navier-Stokes-Korteweg system in one dimension.
One classical model used to study mixtures of two or more compressible fluids with different densities is the Navier-Stokes-Korteweg (NSK) system. In this talk, we will try to give some results of existence of global strong solution for this system, based on maximum principle, when the parabolic behavior, governed by the viscosity tensor dominates the dispersive effects that are induced by the capillarity tensor. The opposite case is the object to my thesis.
4:10pm-4:30pm. Camilo Gòmez Araya,
Edge states for one-dimensional Dirac operator.
The aim of this talk is to present a condition for the existence of edge states in a simple continuous one-dimensional model for Dirac equation in the context of junctions of two different potentials. The proof is based on the symplectic structure of the Dirac operator, where we show that these edge states of the Dirac operator come from the crossing of some Lagrangian planes of the associated symplectic space.
4:30pm-4:50pm. Coffee break.
5:00pm-6:00pm. Conviviality and exchange moment between participants.
7:00pm. Dinner.
Wednesday, March 02.
7:30am-8:15am. Breakfast.
8:30am-9:00am. Claudia Fonte Sanchez,
Long time behavior of an age and leaky memory-structured neuronal population equation.
Starting from the classical model of the elapsed time neuron network, we introduce a generalization that allows us to represent a mechanism of slow fatigue, such as peak rate adaptation or short-term synaptic depression. We arrive at a two-dimensional nonlinear model and study conditions for the existence and stability of stationary solutions. More in detail, we show that when the connections between neurons are weak, the distribution of their activity tends to a steady state with exponential speed. To do this, we make use of the Doeblin-Harris theory in conjunction with a perturbation argument.
9:00am-9:30am. Romain Petit,
An Arzelà-Ascoli type theorem for sequences of sets.
I will present a well-known result in geometric measure theory asserting that, if a sequence of sets converges in measure and has an "equi-regularity" property, then it converges for a much stronger notion of convergence: eventually, the elements of the sequence can be written as a perturbation of the limit set, with the "size" of the perturbation converging to zero. If time permits, I will briefly explain how this result can be used to obtain strong recovery guarantees for an inverse problem in imaging.
We consider differential equations with reflection, i.e. which are constrained to remain in a given domain. We obtain "regularization by noise" results in this context, namely we show that adding a (fractional) noise term allows to restore well-posedness for equations driven by singular (possibly distributional) drift vector fields. In order to obtain well-posedness under optimal regularity assumptions on the vector fields we show that suitable perturbations of fBm have similar pathwise regularizing properties as fBm.
10:00am-10:45am. Coffee Break.
11:00am-11:20am. David Lurie,
Algorithmic Aspects in Stochastic Games.
Stochastic games model dynamic interactions where the environment changes in response to player actions. The standard model for one-player stochastic games with signals is Partially Observable Markov Decision Processes (POMDPs); at each stage, the player chooses an action, obtains the stage payoff, and receives a signal about the new state. POMDPs have various applications, including robot planning and reinforcement learning. We will begin by discussing POMDPs and the motivations for studying them. Then we will look at the algorithms that have been developed to deal with this model, and we will review an important class of strategies, finite-memory strategies. Finally, we will explore how this model might be used to predict therapeutic adherence.
11:20pm-11:40pm. Diego Alejandro Murillo Taborda,
Mean field games in macroeconomic models with heterogeneous agents.
Heterogeneous agents models are a powerful and versatile tool to understand how the optimal choices of the heterogeneous economic agents determine the dynamics of the economic aggregates and their distribution across time, answering to some of the most important questions in economics. However, these models require a high technical and mathematical cost: heterogeneous agents models in continuous time are particular cases of mean field games, as defined in Lasry and Lions (2007), in which the equilibria are characterized by a couple of nonlinear partial differential equations (PDE), which usually are hard to solve: the Hamilton Jacobi Bellman (HJB) equation characterizing the optimal choices of the agents, and the Kolmogorov forward equation, which characterizes the evolution of the distribution of heterogeneous agents given their optimal choices, so this class of models can be interesting for well experimented mathematicians. During this talk, I will give a brief introduction to this class of models, including some recent results in the area, and some open mathematical open problems, which are currently at the center of an intense research agenda.
11:40pm-12:00pm. Antoine Burg,
Impact of economic or sanitary crisis on mortality trends.
Thanks to several scientific and technological progress, life expectancy has considerably increased over the past decades. Recent trends have been driven by improved medical treatments, e.g. for cancer. It then appears that the analysis of mortality by causes of death enables us to better understand and explain the evolution of mortality. In addition, the whole world has been experiencing a major health crisis since 2020 due to Covid19. This new disease has huge direct and indirect consequences on the population which might drastically change mortality patterns and trends.
The objective is to improve the knowledge and modelling of cause-of-death mortality, with a focus on interpretability and ability to reproduce Covid19-like shocks.
12:00pm-1:45pm. Lunch.
2:00pm-2:30pm. Camilla Brizzi,
Entropic approximation of ∞-Optimal Transport problems.
I will open my talk presenting the Optimal Transport problem in a L^∞ setting, commenting on some applications and mentioning some results, such as existence of a minimizer, extension of the concept of C-cyclical monotone set and existence of an optimal map.
In the second part I will briefly introduce the Entropic approximation of such a problem, and I will show the gamma convergence of the approximated functionals to the one related to the ∞-Optimal Transport problem. The strength of this approach is that regularization plays a key role in enabling efficient algorithms.
2:30pm-3:00pm. Katharina Eichinger,
Wasserstein barycenters and beyond.
In this talk I will present a variational formulation for interpolating several measures with respect to the geometry induced by the Wasserstein distance, the distance which arises naturally in optimal transport theory. This so called Wasserstein barycenter can either be seen as a generalization of barycenters in Euclidean spaces or a notion of generalized mean in metric spaces. It has first been introduced around 10 years ago by M. Agueh and G. Carlier.
After convincing you that it has useful applications in imaging analysis and data science in general, I will point out its main properties and difficulties. Finally, I will introduce a regularization of this problem which solves some of the numerical as well as analytical issues.
3:00pm-3:20pm. Coffee break.
3:30pm-4:00pm. Changqing Fu,
Geometric Deformation on Objects: Unsupervised Image Manipulation via Conjugation.
A novel two-stage approach is proposed for image manipulation and generation. User-interactive image deformation is performed through editing of contours. This is performed in the latent edge space with both color and gradient information. The output of editing is then fed into a multi-scale representation of the image to recover quality output. The model is flexible in terms of transferability and training efficiency.
4:00pm-4:20pm. Daniele De Gennaro,
Long time behaviour of the discrete volume-preserving mean curvature flow in the flat torus.
In this talk we will focus on the study of the asymptotic behaviour of the discrete-in-time approximation of mean curvature flow under volume constraint.We will first recall some aspect of the discrete flow and the issues arising from the volume constraint. Then, the study of the limit configuration will allow us to find suitable candidates for the asymptotic. Finally, we will explain the main steps of the proof of the result, combined with an estimate on the rate of convergence.
Martingale optimal transport is considering adding an additional martingale constraint to the traditional optimal transport problem, from a practical point of view it provides robust option price bounds for exotic options.
In this short talk, we will introduce Martingale Wasserstein inequality for probability measures in the convex order and the relaxation of the marginal constraints. We will also talk about the martingale coupling that attain the infimum and supremum of the Martingale Wasserstein distance.
5:20pm-5:40pm. João Miguel Machado,
A shape optimization approach to approximation of measures.
In this talk we introduce a variational problem designed to approximate a given measure among a class of measures whose support is a set with Hausdorff dimension 1. The proposed approach consists in minimizing the Wasserstein distance between the measures with a regularization of the length (Hausdorff measure) of the support of the solution. We discuss when this problem doesn’t have solutions and propose a relaxation.
7:00pm. Dinner.
Thursday, March 03. Statistics and Probabilities day.
7:30am-8:15am. Breakfast.
8:30am-9:30am. Adrien Hairault,
Course 5: Introduction to Bayesian statistics.
In this short course, we introduce the fundamentals of Bayesian statistics: the likelihood function, prior and posterior distributions, Bayesian testing procedures etc. The aim is to challenge the ‘frequentist’ intuition of the participants through numerous toy examples. Practical applications of the Bayesian paradigm to domains like physics are also presented.
9:30am-10:15am. Coffee Break.
10:30am-11:30am. Charly Andral,
Course 6: Simulation and Monte Carlo.
For this short lecture, we will cover quickly the main topics of simulation and Monte Carlo. First, we will quickly introduce the random variable generation, starting with the uniform distribution on (0,1) and some other classic distributions. Then we will focus on Monte Carlo methods and see how to use these methods to simulate according to an arbitrary density (e.g., rejection, importance sampling…). The ideas behind quasi Monte Carlo and random quasi Monte Carlo will be discussed. We will finish with the presentation of Markov Chain Monte Carlo (MCMC) methods with a special attention to the Metropolis-Hasting algorithm, and why MCMC may be useful in a Bayesian setting.
12:00pm-1:45pm. Lunch.
2:00pm-2:30pm. Luke Joseph Kelly,
Couplings Markov chains sampling phylogenetic trees and convergence of Bayesian phylogenetic inference.
A phylogeny is a graph depicting ancestral relationships between species derived from a common ancestor. In a typical phylogenetic inference problem, we attempt to infer a tree from data gathered at its leaves. Markov chain Monte Carlo is the primary tool for Bayesian phylogenetic inference but the theoretical properties of Markov chains on spaces of trees, branch lengths and other parameters are unknown except in simple cases. We build on recent work using couplings of Markov chains to estimate total variation bounds convergence by constructing couplings of Markov kernels in phylogenetic problems. Using our framework, we derive a scheme for sampling maximal couplings of proposal operators on trees. In the second part of the talk, I’ll describe some recent work on proving consistency of Bayesian phylogenetic inference.
2:30pm-3:00pm. Elena Bortolato,
Alternatives for approximation in Likelihood free inference.
For some statistical models, the likelihood function could be intractable, or a considerable amount of time could be necessary for it to be evaluated. Indirect Inference and Approximate Bayesian Computation methods aim at avoiding the problem of evaluating the likelihood by comparing actual to some simulated data according to distances.
In this framework, we propose algorithms that avoid the choice of a distance and tuning parameters, aiming at reducing the arbitrariness of the approximation. The acceptance rule implicitly makes use of a pseudo-likelihood that inherits some desirable properties from exact confidence distributions, such as median unbiasedness.
-Finally, I will consider some alternative strategies to guarantee unbiasedness in simulation based inference, relying on manifold-MCMC sampling. - (if it is not too much)
3:00pm-3:20pm. Coffee break.
3:30pm-4:00pm. Ruihua Ruan,
Modelling Bid-Ask Spread Dynamics using Hawkes processes.
Hawkes processes are a class of self- and cross-exciting point processes which are characterised by stochastic intensities. In this talk, I will give the basic definitions and properties of Hawkes processes, and some of their applications in finance. To conclude I will try to present an extension of Hawkes processes - “State Dependent Hawkes Processes", in order to model Bid-Ask spread dynamics.
4:00pm-4:30pm. Hong Quan Tran,
Mixing times and Zero-Range process.
It is generally known that a finite Markov chain will converge to equilibrium under certain conditions (irreducibility, aperiodicity). The speed of convergence is quantified by the mixing time, namely the time it takes for the system to get close to equilibrium under certain target distance. The modern theory of Markov processes studies how the mixing time grows as the state space becomes large. In this talk, we present Zero-Range process, a model of interacting random walks, and determine its mixing time. We also establish a sharp phase transition of the system around the mixing time, a phenomenon usually known as cutoff.
4:30pm-5:00pm. Yueh-Sheng Hsu,
Construction of 2d Landau Hamiltonian with constant magnetic field and white noise electric potential.
This talk is about an ongoing project studying the two-dimensional Landau Hamiltonian. The Landau Hamiltonian is a differential operator dictating the system of a charged particle moving in a magnetic environment. On the whole space, this operator is known to have purely essential spectrum composed of isolated eigenvalues with infinite multiplicity, called the Landau levels.
We are interested in how would an additional white noise electric potential affect the spectrum of Landau Hamiltonian. The first problem one faces is the construction of such operator, which is non-trivial due to the irregularity of white noise. Such construction usually involves advanced SPDE theories. In fact, there exists an elementary trick allowing us to bypass highly technical tools. The objective of this talk is to introduce this method and discuss possible future development of the subject.
5:00pm-5:30pm. Coffee.
7:00pm. Dinner.
Friday, March 04.
7:30am-8:15am. Breakfast.
8:30am-9:00am. Lorenzo Croissant,
Diffusion limit control of high-frequency pure-jump processes.
Real world controlled dynamic systems tend to involve small interactions at very high frequencies, e.g. anything computerised. It is well understood that the process of such (fundamentally finite) systems tend to a limiting behavior corresponding to a diffusion. This approximation is ubiquitous in practice, but what is the approximation order in terms of the associated control problem? In this talk we will answer this question through analysis of the Hamilton-Jacobi-Bellman equation and the limiting behaviour of the generator of the process.
9:00am-9:30am. Adrien Seguret,
A mean field control approach for smart charging.
The increase in electrical vehicles (EV for short) will raise some challenges in the management of the equilibrium production/consumption of the electrical network. To avoid congestion effects, the charging of large fleets of EVs must be optimized. In this talk, I will introduce a mathematical model to manage the consumption of such a fleet, using the mean fields limit assumption. Optimality conditions of the associated optimization problem will be presented, as well as numerical results.
9:30am-10:00am. Duc-Thinh Vu,
Computable prices in financial markets with transaction costs.
How to compute (super) hedging costs in rather general financial market models with transaction costs in discrete time? Despite the huge literature on this topic, most of results are characterizations of the super-hedging prices while it remains difficult to deduce numerical procedure to estimate them.
In this talk, I would like to present an introduction to pricing of contingent claims in financial markets with transaction cost. The new idea is to establish the equivalence between two types of optimization: optimization over a family of random variables and optimization in ω-wise sense.
We study a general mean-field optimization problem with entropicregularizer. It is known that practical problems, including notably the convergence of deep neural networks, can be modelized by the mean-field approach. Residual neural networks have been studied in the previous works of Zhenjie Ren et al and the convergence of the gradient descent, arguably the most common training method in practice, is obtained. Motivated by the classical fictitious play from game theory, we construct a novel dynamics whose convergence to the optimal solution is shown. Additionally, the rate of convergence of this dynamics is obtained in the convex case, which is absent in the previous results of gradient flow.
Gregoire Szymanski,
Estimation of the Hurst parameter from discrete noisy data.
The main goal of this talk is to present an optimal statistical procedure for the estimation of the Hurst parameter H of a fractional Brownian motion under high-frequency observation. This non-standard scheme consists in the observation of a single trajectory of our process of interest, at discrete times only and on a bounded time interval. Moreover, the observations suffer from an experimental noise. This scheme is relevant in many concrete applications, especially in finance, and leads to the non-standard convergence rate n^(-(4H+2)) where n is the number of observations.