9h30-10h15 Alexandros Eskenazis "The dimensional Brunn-Minkowski conjecture"
10h15-10h35 Joao Machado "On the effective sharp stability of the Faber-Krahn inequality"
10h35-11h Coffee break
11h-11h45 Quentin Mérigot
11h45-12h05 Pierre Bizeul "Gaussian-rate polynomial approximation for strongly log-concave measures"
12h05-12h25 Anna Kazeykina "Stability and exponential convergence of Sinkhorn for entropic martingale optimal transport"
12h30-13h45 Lunch break
13h45-14h30 Katharina Eichinger "Regularity and stability of diffusion transport maps"
14h30-14h50 Eli Putterman "Dimension-free stability estimates for the (B)-theorem"
14h50-15h10 Jordan Serres "On generalizations of Caffarelli's contraction theorem"
15h10-15h40 : Coffee break
15h40-16h25 Dario Cordero "Moment measures of log-concave measures after Klartag and Chen-Klartag"
16h25-16h45 Omer Friedland "Sur le rayon de Banach–Mazur de (\ell_\infty^n)"
16h45-17h30 Discussions
Speaker: Alexandros Eskenazis
Title: The dimensional Brunn-Minkowski conjecture
Abstract: I will present the latest developments on the dimensional Brunn-Minkowski conjecture for log-concave measures. A main goal of the talk is to draw parallels between the geometric, functional and entropic formulations of the problem.
Speaker: Joao Machado
Title: On the effective sharp stability of the Faber-Krahn inequality
Abstract: The Faber–Krahn inequality states that, among sets of fixed volume, balls minimize the first Dirichlet eigenvalue. Its sharp quantitative form controls the square of the Fraenkel asymmetry by the spectral deficit. Brasco, De Philippis and Velichkov proved this estimate through an argument using compactness, a contradiction step, and the selection principle of Cicalese-Leonardi; consequently, their stability constant is not computable. In this talk, we will discuss a proof with a computable dimensional constant, resolving Open Problem 2 in the survey of Brasco and De Philippis. The argument passes through the Saint–Venant inequality, which states that the torsion functional is maximised by balls. Its two main ingredients are a new level-set characterisation of the torsional deficit and a differential estimate for the distance of the level sets from a moving ball. These ingredients are connected by the strong quantitative isoperimetric inequality of Fusco and Julin. The Kohler–Jobin inequality then transfers the effective sharp stability of the Saint-Venant inequality back to Faber–Krahn. This is joint work with André Guerra (Cambridge) and João P. G. Ramos (IMPA).
Speaker: Pierre Bizeul
Title: Gaussian-rate polynomial approximation for strongly log-concave measures
Abstract: It is well known that the standard Gaussian measure satisfies the Poincaré and logarithmic Sobolev inequalities with constant 1. Caffarelli’s contraction theorem allows one to transfer both inequalities to measures that are log-concave with respect to the Gaussian measure, commonly referred to as strongly log-concave measures.
Another property of Gaussian space is that functions with bounded Sobolev norm can be approximated by polynomials of degree at rate , which may be viewed as a family of higher-order Poincaré inequalities. Using the heat semigroup, we extend this result to strongly log-concave measures. Finally, we explain that, within the class of log-concave measures, this polynomial approximation property implies the logarithmic Sobolev inequality.
Joint work with Bo’az Klartag.
Speaker: Anna Kazeykina
Title: Stability and exponential convergence of Sinkhorn for entropic martingale optimal transport
Abstract: We prove exponential convergence of the Sinkhorn algorithm for entropic martingale optimal transport. Our proof is based on a stability result for the EMOT. We first establish uniform boundedness and Lipschitz estimates for the dual potentials, up to affine gauges. We then construct a nearby martingale coupling with perturbed marginals, with quantitative control in terms of the change in the marginals. The construction relies on the invertibility of a suitable Fredholm operator.
Speaker: Katharina Eichinger
Title: Regularity and stability of diffusion transport maps
Abstract: Finding regular transport maps between measures is an important task in generative modelling and a useful tool to transfer functional inequalities. The most well-known result in this field is Caffarelli’s contraction theorem, which shows that the optimal transport map from a Gaussian to a uniformly log-concave measure is globally Lipschitz. Note that for our purposes optimality of the transport map does not play a role. This is why several works investigate other transport maps, such as those derived from diffusion processes, as introduced by Kim and Milman. Here, we establish a lower bound on the log-semiconcavity along the heat flow for a class of what we call asymptotically log-concave measures. We will see that this implies Lipschitz bounds for the heat flow map introduced by Kim and Milman. We will also show that these log-semiconcavity bounds are sufficient for stability of these maps in entropy and Wasserstein distance. The proofs of the stability are based on an interplay between certain functional inequalities along the heat flow, such as the Log-Sobolev inequality and Wang’s Harnack inequality.
Based on a joint work with Louis-Pierre Chaintron and Giovanni Conforti, and a joint work with Sinho Chewi and Aram-Alexandre Pooladian.
Speaker: Eli Putterman
Title: Dimension-free stability estimates for the (B)-theorem
Abstract: The (B)-theorem of Cordero-Erausquin, Fradelizi and Maurey states that if $\gamma_n$ denotes the standard Gaussian in dimension $n$, $K$ is an origin-symmetric convex set, one has for any $t$ that $\gamma_n(e^t K)^2 \geq \gamma_n(K)\gamma_n(e^{2t}K)$; this strengthens the Prékopa-Leindler inequality. Herscovici et al. (2023) proved a stability version of this result, showing that if one has equality up to a factor $1+\delta$ then the inradius of $K$ must be either "very large" or "very small", where the bounds depend on $\delta$ and on $n$. We will present a new, sharp, stability estimate which is dimension-free and also yields more precise information about bodies which are near-optimizers: every principal component of the covariance matrix of the measure obtained by restricting the Gaussian to $K$ must either be at least $1-O(\delta)$ or at most $O(\delta)$.
Speaker: Jordan Serres
Title: On generalizations of Caffarelli's contraction theorem
Abstract: Emanuel Milman's conjecture asked whether any manifold diffeomorphic to a sphere and having Ricci curvature bounded below by a positive constant is the contractive volume-preserving image of a round sphere. The conjecture has been completely solved this year: it is only true in dimension 2. In this brief talk, I will attempt to describe its resolution and discuss some open problems it leaves.
Speaker: Dario Cordero Erausquin
Title: Moment measures of log-concave measures after Klartag and Chen-Klartag
Abstract: We report on the recent proof by Chen and Klartag of the variance conjecture. It is a good occasion to review some properties of moment measures of log-concave measures.
Speaker: Omer Friedland
Title: Sur le rayon de Banach–Mazur de $\ell_\infty^n$
Abstract: Nous montrons que tout espace normé de dimension $n$ est à distance de Banach–Mazur $O(n^{2/3})$ de $\ell_\infty^n$.