PROGRAM of the High Dimensional Phenomena day 2023
10h-10h30: welcome coffee in the lobby of the IHP
10h30-11h25: Sergei Treil (Brown University): The matrix $A_2$ conjecture fails, or $3/2>1$.
11h30-12h25: Justin Salez (Université Paris Dauphine): Entropy, curvature and the cutoff phenomenon.
The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergoned by certain Markov processes in the limit where the number of states tends to infinity. Discovered forty years ago in the context of card shuffling, it has since then been established in a variety of contexts, including random walks on graphs and groups, high-temperature spin glasses, or interacting particle systems. Nevertheless, a general theory is still missing, and identifying the general mechanisms underlying this mysterious phenomenon remains one of the most fundamental problems in the area of mixing times. In this talk, I will give a self-contained introduction to this fascinating question, and then present a recent approach based on entropy and curvature.
12h30-14h: lunch in the lobby of the IHP
14h-14h55: Tomasz Tkocz (Carnegie Mellon University): Resilience of cube slicing in $l_p$.
I shall present an extension of Ball's cube slicing result to $l_p$ spaces for large $p$. Based on joint work with Eskenazis and Nayar.
15h-15h55: Giovanni Conforti (Ecole Polytechnique): Invariant integrated convexity profiles for Hamilton-Jacobi-Bellman equations and applications.
It has been known for a long time that Hamilton-Jacobi-Bellman (HJB) equations preserve convexity, namely if the terminal condition is convex, the solution stays convex at all times. Equivalently, log-concavity is preserved along the heat equation, namely if one starts with a log-concavity density, then the solution stays log-concave at all times. Both these facts are a direct consequence of Prékopa-Leindler inequality. In this talk, I will argue that carrying out a second-order analysis on coupling by reflection on the characteristics of the HJB equation prompts the existence of weaker notions of convexity that propagate backward along HJB equations. More precisely, by introducing the notion of integrated convexity profile, we are able to construct families of functions that fail to be convex, but are still invariant under the action of the HJB equation. In the second part of the talk I will illustrate some applications of these invariance results to functional inequalities, in particular to the construction of Lipschitz maps, and to the exponential convergence of learning algorithms for entropic optimal transport
16h-17h: coffee in the lobby of the IHP
Sponsored by the Simone et Cino del Duca Fondation
