Wednesday, Sep 17, 15:00-16:00, Zoom: Link

Speaker: Titouan Vayer, Inria
Title: Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein
Abstract: Unsupervised learning aims to capture the underlying structure of potentially large and high-dimensional datasets. Traditionally, this involves using dimensionality reduction (DR) methods to project data onto lower-dimensional spaces or organizing points into meaningful clusters (clustering). In this work, we revisit these approaches under the lens of optimal transport and exhibit relationships with the Gromov-Wasserstein problem. This unveils a new general framework, called distributional reduction, that recovers DR and clustering as special cases and allows addressing them jointly within a single optimization problem. We empirically demonstrate its relevance to the identification of low-dimensional prototypes representing data at different scales, across multiple image and genomic datasets.
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Wednesday, Sep 24, 15:00-16:00, Zoom: Link

Speaker: Arunima Bhattacharya, UNC
Title: Lagrangian mean curvature equations and flows
Abstract: In this talk, we will introduce the special Lagrangian equation and the Lagrangian mean curvature flow. We will discuss interior Hessian estimates for shrinkers and expanders of the Lagrangian mean curvature flow, and further extend this result to a broader class of Lagrangian mean curvature type equations. We assume the Lagrangian phase to be hypercritical, which results in the convexity of the potential of the initial Lagrangian submanifold. Convex solutions of the second boundary value problem for certain such equations were constructed by Brendle-Warren 2010, Huang 2015, and Wang-Huang-Bao 2023. We will also briefly introduce the fourth-order Hamiltonian stationary equation and mention some recent results on the regularity of solutions of certain fourth-order PDEs, which are critical points of variational integrals of the Hessian of a scalar function. Examples include volume functionals on Lagrangian submanifolds. This is partially based on joint work with Jeremy Wall.
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Wednesday, Oct 01, 15:00-16:00, SAS 4201

Speaker: Toan Nguyen, Penn State University
Title: Landau damping below survival threshold
Abstract: In the collisionless Vlasov theory of excited electrons, plasmas oscillations arise due to their long-range meanfield interaction, and the classical Landau damping concerns decay of such an oscillation. While the damping mechanism is exponentially fast by phase mixing for short-wave perturbations, it’s extremely slow or not available for long-wave perturbations (i.e. oscillations may survive and remain as a Klein-Gordon dispersive wave). This talk aims to present recent results on nonlinear Landau damping, focusing on the plasma oscillation regime.
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Wednesday, Oct 08, 15:00-16:00, SAS 4201

Speaker: Sangmin Park, California Institute of Technology
Title: Dissipative Hamiltonian structure of the Vlasov-Fokker-Planck equation
Abstract: The Vlasov-Fokker-Planck equation (VFP) describes the evolution of the probability density of the position and velocity of particles under the influence of external confinement, interaction, friction, and stochastic force. It is well-known that this equation can be formally seen as a dissipative Hamiltonian system in the Wasserstein space of probability measures. Moreover, the geometric structure has possible connections to the conjectured optimal convergence rates of underdamped Langevin Monte Carlo (ULMC), a sampling algorithm known to empirically outperform the (standard) Langevin Monte Carlo. 


This talk will focus on a time-discrete variational scheme for VFP which we introduce to understand the geometric structure of the equation more rigorously. We will begin by introducing the optimal transport problem and the Wasserstein distance, as well as the techniques of gradient flows which form the basis of our variational scheme. After highlighting the connections to ULMC, we will discuss how the proposed variational scheme is (i) consistent with the dissipative Hamiltonian structure, and (ii) (geodesically)-convex at each iteration.
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Wednesday, Oct 15, 15:00-16:00, Zoom: Link

Speaker: Marius Tucsnak, University of Bordeaux
Title: Relaxation enhancement by controlled incompressible fluid flows
Abstract: We propose a PDE-controllability based approach to the enhancement of diffusive mixing for passive scalar fields. Unlike in the existing literature, our relaxation enhancing fields are not prescribed ab initio at every time and at every point of the spatial domain. Instead, we prove that time-dependent relaxation enhancing vector fields can be obtained as state trajectories of control systems described by the  incompressible Euler equations either driven by finite-dimensional controls or by controls localized in space. The main ingredient of our proof is a new approximate controllability theorem for the incompressible Euler equations on the two dimensional torus, ensuring the approximate tracking of the full state all over the considered time interval. Combining this with a continuous dependence result  yields enhanced relaxation for the passive scalar field. Another essential tool in our analysis is the exact controllability of the incompressible Euler system driven by spatially localized forces.
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Wednesday, Oct 22, 15:00-16:00, Zoom: Link

Speaker: Pietro Zanotti, University of Milan, Italy
Title: On a technique for the analysis and the discretization of poroelastic models 
Abstract: Poroelastic models describe the flow of a fluid inside an elastic porous medium. Such models have attracted growing attention in recent years because of their wide range of applications. The most well-established techniques for both the analysis and the discretization of these problems build either on the so-called Faedo-Galerkin (Rothe’s) method or on the theory of implicit evolution equations. The talk introduces and motivates an alternative technique, recently developed in collaboration with A. Khan (IIT Roorkee) and C. Kreuzer (TU Dortmund). The proposed approach, in contrast to previous ones, aims at establishing an isomorphism between the solution and the data spaces, which is a desirable property for the design and the analysis of discretization schemes. Advantages and current limitations of this approach will be illustrated by means of new results and open problems. 
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Wednesday, Oct 29, 15:00-16:00, Zoom: Link

Speaker: Marianne Akian, Inria and CMAP, École polytechnique,
Title: The competitive spectral radius of families of nonexpansive mappings
Abstract: We introduce a new class of perfect information repeated zero-sum games in which the payoff of one player is the escape rate of a switched dynamical system which evolves according to a nonexpansive nonlinear operator depending on the actions of both players. This is motivated by applications to population dynamics (growth maximization and minimization). Considering order preserving finite dimensional linear operators over the positive cone endowed with Hilbert’s projective (semi-)metric or the Funk hemi-metric, we recover subclasses of Matrix multiplication games, a 2-player extension of the joint spectral radius of sets of nonnegative matrices, introduced by Asarin and coauthors (2016). We prove that escape rate games have a uniform value, which is characterized by a nonlinear eigenproblem. Then, we discuss the continuity and approximability of the value of the game with respect to the parameters. We show in particular that the competitive spectral radius of positive matrices can be approximated up to any accuracy.

This is joint works with Stéphane Gaubert, Loïc Marchesini, and Ian Morris.
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Wednesday, Nov 05, 15:00-16:00, Zoom: Link

Speaker: Rida Laraki, Moroccan Center for Game Theory, UM6P (Rabat, Morocco)
Title: On the Relationship Between Strategic Properties of Nash Equilibria and Their Index
Abstract: This talk investigates the relation between some strategic features of mixed Nash equilibria and their fixed point index in finite games. We present new results that deepen our understanding of how equilibrium structure relates to index theory:

1. A mixed Nash equilibrium x is isolated with index +1 if and only if it can be made the unique equilibrium of a larger game, constructed by adding strategies that are strictly inferior responses to x. This settles an open question posed explicitly by Hofbauer (2003) and implicitly by Myerson (1996).

2. A Nash component admits an equilibrium of index +1 in its neighborhood under every perturbation of any strategically equivalent game if and only if the component itself has a positive index.

3. For any finite game, any selection of equilibria from each Nash component, and any assignment of indices ±1 to these equilibria such that their sum equals the index of the component, there exists a perturbation of a strategically equivalent game whose equilibria approximate the selected ones and preserve the assigned indices.

These results bridge equilibrium refinement, index theory, and robustness to strategic perturbations, offering new insights into the structure and stability of Nash equilibria.
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Wednesday, Nov 12, 15:00-16:00, Zoom: Link

Speaker: Mateo Diaz Diaz, Johns Hopkins University
Title: TBA
Abstract: TBA
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Wednesday, Nov 19, 15:00-16:00, Zoom: Link

Speaker: Filipo Santambrogio, Université Claude Bernard – Lyon 1
Title: The JKO scheme for evolution equations with a gradient flow structure in the Wasserstein space 
Abstract: The Jordan-Kinderleherer-Otto scheme has been proposed 25 years ago to provide a variational structure to some diffusion PDEs such as the porous medium or the Fokker-Planck equations using tools from optimal transport. It consists in iterating a proximal scheme in the space of probability measures endowed with the distance W_2 induced by the optimal transport problem with quadratic cost.
In the talk, I will recall the basis of this theory and the main variants, in order to show which PDEs, most often of parabolic type, can be attacked through this scheme. Then, I will show how some estimates can be obtained in the scheme (for instance: it is well-known that all norms decay along the heat flow, and the heat equation is also a Wasserstein gradient flow, is it true that the same norms also decay along the discrete steps of the JKO scheme?). This will recover some well-known estimates and sometimes show some new ones, which are easier to observe in this discrete setting.
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Wednesday, Feb 25, 15:00-16:00, SAS 4201

Speaker: Paul Manns, TU Dortmund University
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Wednesday, Mar 04, 15:00-16:00, SAS 4201

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Wednesday, Mar 11, 15:00-16:00, SAS 4201

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Wednesday, Mar 25, 15:00-16:00, SAS 4201

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Wednesday, April 01, 15:00-16:00, SAS 4201

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Wednesday, April 08, 15:00-16:00, SAS 4201

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Wednesday, April 15, 15:00-16:00, SAS 4201

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Wednesday, April 22, 15:00-16:00, SAS 4201

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Wednesday, April 29, 15:00-16:00, SAS 4201

Speaker: Franco Rampazzo, University of Padova
Title: TBA
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