Séminaires : Séminaire d'Analyse Fonctionnelle

 For $n\ge 2$ and $p\in(1,n)$ the "best $p$-Sobolev inequality" on an open set $\Om\subset\R^n$ is identified with a family $\Phi_\Om$ of variational problems with critical volume and trace constraints. In joint work with Neumayer and Tomasetti we prove that, if $\Om$ is bounded, then (i) for every $n$ and $p$, there exist generalized minimizers of $\Phi_\Om$, having at most one boundary concentration point, and: (ii) if $n> 2\,p$, then there exist (classical) minimizers. We then establish rigidity results for the comparison theorem "balls have the worst best Sobolev inequalities", thus giving the first affirmative answers to a question raised in joint work with Villani (JGA 2005). A key ingredient in our analysis is the complete characterization of $\Phi_{Half Space}$ obtained in joint work with Neumayer (JFA 2017). Connections with the Yamabe problem are also discussed.

We discuss a non-linear version of the well-known theorem by L. Carleson on the pointwise convergence of the Fourier transform. Our methods are based on the study of resonances of a Dirac system of differential equations on a half-line. Among applications of our results is a new proof of the linear theorem.

Given two non-negative functions $f$ and $g$ such that the Radon transform of $f$ is pointwise smaller than the Radon transform of $g$, does it follow that the $L^p$-norm of $f$ is smaller than the $L^p$-norm of $g$ for a given $p>1?$ We consider this problem for the spherical and spatial Radon transforms. In both cases we point out classes of functions for which the answer is affirmative. The results are in the spirit of the solution of the Busemann-Petty problem from convex geometry, and the classes of functions that we introduce generalize the class of intersection bodies introduced by Lutwak in 1988. We also deduce slicing inequalities that are related to the well-known Oberlin-Stein type estimates for the Radon transform.

We study the asymptotic behaviour of large symmetric tensor powers of finitely dimensional normed vector spaces. More precisely, contrary to full tensor powers, we show that the distortion between the injective and projective tensor norms grows subexponentially on symmetric tensor powers of a fixed finitely dimensional complex normed vector space. We will then apply this to give a classification result for graded commutative normed algebras. If time permits, we discuss the motivations for these questions arising from complex and algebraic geometry.

Let $W$ be an $n\times n$ symmetric matrix with i.i.d centered entries and unit variance. The celebrated Wigner's theorem states that the empirical law of eigenvalues of $W/\sqrt{n}$ converges weakly to the semicircle law, a measure supported in $[-2,2]$. For the largest eigenvalue, Bai and Yin showed that it converges to $2$ if and only if the entries of $W$ have a bounded fourth moment, namely, $W$ does not have outliers. In this talk, we explore the universality and stability of Bai-Yin's result under sparsification. In other words, we consider the random matrix $X=\Sigma \circ W$, where $\Sigma$ is a deterministic matrix, and $\circ$ denotes the Hadamard product. We contribute sharp conditions for subgaussian matrices $X$ to have outliers in terms of non-structural parameters of $\Sigma$.
This is a joint work with Dylan Altschuler, Konstantin Tikhomirov, and Pierre Youssef.
 

Two popular ways of quantifying measure concentration is through the sharp constants that appear in Poincar\'e and log-Sobolev inequalities. In addition to concentration implications, these constants quantify distance to Gaussianity in a strong sense.  In this talk, I'll show how these constants satisfy a certain central limit theorem.  Namely, under repeated convolutions, the Poincar\'e constant of the convolution measure approaches that of the Gaussian measure with the same second moments.  The same is shown for the log-Sobolev constant, under mild regularity assumptions. 
 

The main focus of this talk will be to describe recent work (joint with Braverman) on the Lipschitz extension problem that obtains solutions to various natural quantitative questions by thinking about its (known) dual formulation as a question about randomly rounding an ambient metric space to its subset while preserving certain natural guarantees that are measured in terms of transportation cost. We will start by discussing the classical formulation of these old questions as well as some background and earlier results, before passing to examples of how one could reason quantitatively using the dual perspective.

We will introduce a few asymptotic smoothness properties of Banach spaces. We will define them in terms of upper $\ell_p$ estimates for weakly null trees and describe their main characterizations, in particular through the existence of asymptotically uniformly smooth renormings. Then we will survey some of their following applications: for three-space problems, nonlinear classification of Banach spaces, universality and complexity questions.

The vector balancing constant of two convex bodies $U, V \subset \R^n$, $\beta(U, V)$, is defined to be the smallest $b > 0$ such that for any $n$ vectors in $U$, some signed combination of these vectors lies in a $b$-scaled copy of $V$. This constant is the subject of the well-known Komlós conjecture, which asks whether $\beta(B_2^n, B_\infty^n)$ is bounded by a universal constant, where $B_2^n, B_\infty^n$ are the unit ball and cube, respectively, in dimension $n$. We will introduce a related parameter $\alpha(U, V) \leq \beta(U,V)$, defined via lattices, and review some known inequalities for these two parameters. In 1997, W. Banaszczyk and S. Szarek showed that (for some universal $\theta>0$) if a convex body has gaussian measure $\gamma_n(V)\geq 1/2$, then $\alpha(B_2^n, V) \leq \theta$; this yields $\alpha(B_2^n, B_\infty^n)=O(\sqrt{\log n})$ for the cube. They conjecture that a similar inequality holds for convex bodies of gaussian measure $p<1/2$, i.e., that there exists a non-increasing function $f$ (independent of $n$) such that $\beta(B_2^n, V)\leq f(\gamma_n(V))$. We answer this question in the affirmative for the parameter $\alpha$.

We show that there exists a convex body that tiles $\R^n$ under translations by $\Z^n$ whose surface area is $\tildeO(\sqrt{n})$. No asymptotic improvement over the trivial $O(n)$ (achieved by the cube $[0,1]^n$) was previously known. A non-convex body achieving surface area $O(\sqrt{n})$ was constructed by Kindler, O’Donnell, Rao, and Wigderson [FOCS 2008].

Comparer la répartition des points critiques d'un polynôme avec celle de ses racines est une question ancienne et profonde, dont une illustration élémentaire classique est le théorème de Gauss Lucas. Dans un article de 2013, R. Pemantle et I. Rivin constatent numériquement qu'en grand degré les deux répartitions sont souvent très proches, et ils émettent la conjecture suivante: si on choisit les racines complexes d'un polynôme z1,...,zn aléatoirement de manière indépendante suivant une même loi µ, alors la distribution empirique ν_n des points critiques du polynôme converge vers µ lorsque n tend vers l'infini. Ils prouvent la conjecture dans un cas particulier,  puis Z. Kabluchko prouvera le cas général dans un article de 2015, montrant que nécessairement ν_n → µ en probabilité dans l'espace des mesures (muni de la topologie faible). En collaboration avec Jürgen Angst et Guillaume Poly, nous complétons ce résultat en montrant que la convergence ν_n → µ est en fait presque sûre, tel que conjecturé par Kabluchko.
 

En topologie, pour définir l'homologie, on introduit des complexes de chaînes, i.e. des espaces vectoriels munis d'opérateurs $d$ tels que $d\circ d=0$. On peut parler du conditionnement de $d$. Il contient beaucoup de géométrie. Cela conduit à définir une distance sur l'espace des complexes de chaînes, et à donner un critère de compacité, d'une façon qui rappelle la distance de Gromov-Hausdorff entre espaces métriques.

The notion of stable norm was introduced by Krivine and Maurey in the early 80's in order to study the subspace structure of Banach spaces. The connection with nonlinear (uniform) embeddings was further studied by Raynaud. In the mid 2000's Kalton proved that every stable metric space admits a uniform and coarse embedding into a reflexive space, and he asked whether every reflexive space admits a coarse or a uniform embedding into a stable space. This problem is still open. In this talk we will discuss the relationship between two metric invariants, namely, Kalton's property Q and upper-stability (a natural relaxation of stability), and their connection with Kalton's problem. This is joint work in progress with Th. Schlumprecht (Texas A&M) and A. Zsák (Peterhouse, Cambridge).

Probabilistic representations of analytic objects have proven to be a valuable tool in analysis and geometry. I will talk about such representations which lead to sharp control on the Hessians of functions along the heat flow. These results lead to new Lipschitz properties of transport maps in Euclidean and manifold settings, and also show up in renormalization groups techniques. 

An n by n matrix with plus-minus 1 entries which acts as a scaled isometry is called Hadamard. Such matrices exist in some, but not all dimensions. Combining number-theoretic and probabilistic tools we construct matrices with plus-minus 1  entries which act as scaled  approximate isometries  for any n. More precisely, the matrices we construct have condition numbers bounded by a constant independent of the dimension.

We will also discuss an application in signal processing. A frame is an overcomplete set of vectors which allows a robust decomposition of any vector in the space as a linear combination of these vectors. Frames are used in signal processing since the loss of a fraction of coordinates does not prevent the recovery of the signal. We will discuss a question when a random frame contains a copy of a nice basis.

First introduced by Bourgain, Figiel and Milman in the context of the Ribe Program, Ramsey-type theorems for metric spaces state that compact metric spaces  contain ``large" subsets which are approximately ultrametric. They have since found applications in Metric Geometry, Online Algorithms, Approximation Algorithms, Data Structures, Probability, Descriptive Set Theory, and Geometric Measure Theory. We will discuss  the ``ultrametric skeleton" theorem behind most Ramsey-type theorems for metric spaces, and a recent regular form of the theorem. Time permitting, we will prove of a special case of the theorem for doubling spaces.

The periodic tiling conjecture asserts that any finite subset of a lattice Zd which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large d, which also implies a disproof of the corresponding conjecture for Euclidean spaces Rd. In fact, we also obtain a counterexample in a group of the form Z2×G0 for some finite abelian 2-group G0. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class         of "2-adically structured functions", in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist but are all non-periodic.

La géométrie des groupes considère les groupes à quasi-isométrie près. Ce point de vue a donné lieu à des résultats impressionnants de classifications notamment concernant les réseaux dans les groupes de Lie semisimples. Un point de vue différent consiste à considérer des "plongements" entre groupes. La notion la plus naturelle semble être celle de plongement grossier, celle-ci incluant les morphismes injectifs entre groupes de type fini. Cette notion intervient notamment de manière naturelle en géométrie Lorentzienne. Ici nous présenterons le résultat suivant: un groupe moyennable admet un plongement grossier dans un groupe hyperbolique si et seulement s'il est à croissance polynomiale. 

Soit S une partie génératrice finie de \Gamma=SL_2(Z) (ou plus généralement un groupe Fuchsien), \mu une mesure de probabilité sur S et (X_i) une suite indépendante de variables aléatoires identiquement distribuées de loi \mu. Le but de l'exposé sera de présenter une inégalité du type Hoeffding pour les déviations de 1/n log ||X_n ... X_1|| autour du grand exposant de Lyapunov, limite presque sûre de cette quantité (garantie par le théorème ergodique sous-additif de Kingman). Hormis la dépendance en la taille du support comme dans le cas classique des sommes de variables aléatoires i.i.d, le caractère non moyennable de \Gamma se reflète dans l'inégalité obtenue à travers le trou spectral de \mu dans la représentation régulière de \Gamma. Outre l'obtention immédiate d'un intervalle de confiance pour l'exposant de Lyapunov (quantité loin d'être bien comprise en général, contrairement au cas commutatif), un tel résultat est en lien direct avec plusieurs versions raffinées de l'alternative de Tits: alternative de Tits probabiliste (abondance des sous-groupes libres dans \Gamma) et alternative de Tits forte de Breuillard. Le caractère non asymptotique et effectif de ce résultat est à mettre en contraste avec des estimées asymptotiques connues dès les années 80 par Le Page, dans de cadres plus généraux. Le résultat est un cas particulier pour les marches aléatoires sur des espaces hyperboliques par isométries. Travail joint avec Cagri Sert.
 

Le premier théorème de Beurling et Malliavin est un résultat classique en analyse harmonique. Ce théorème a attiré l'attention de plusieurs mathématiciens reconnus dont J.-P. Kahane, Y. Kaznelson, L. de Branges, P. Koosis, V. Havin, F. Nazarov, N. Makarov, A. Poltoratski,  etc. Je vais commencer cet exposé par une discussion des résultats déjà connus liés à ce théorème. Après, je vais présenter la preuve d'une récente généralisation de ce théorème. Si le temps le permet,  nous discuterons aussi des pistes qui pourront mener à la démonstration de l'analogue de ce résultat en plusieurs dimensions.

 https://arxiv.org/abs/2110.12828

Let $V$ be an analytic subvariety of a domain $\Omega$ in $\C^n$. When does $V$ have the Isometric Extension Property (IEP), i.e. when does every bounded holomorphic function $f$ on $V$ has an  extension to a bounded holomorphic function on $\Omega$ with the same norm?

If $V$ is a retract, i.e. if there exists  a holomorphic $r: \Omega \to V$ so that $r|_V = {\rm id}$,  then there is an obvious isometric extension, namely  $f \circ r$. If $\Omega$ is very nice, for example the ball, this condition is also necessary. We shall discuss why convexity assumptions lead to theorems that say only retracts have the IEP.

If the convexity assumption is dropped, functional analysis can be used to show that every $V$ has the isometric extension property for some $\Omega$.  We shall discuss the proof of this theorem. This is joint work with Jim Agler and Lukasz Kosinski.

We discuss upper bounds for the Fisher information
in high dimensions in terms of the total variation and norms
in Sobolev spaces.

Let $\Gamma$ be a poly-tree, i.e. a collection of dyadic rectangles on $\mathbb{R}^n$ (Cartesian product of usual dyadic intervals on $\mathbb{R}$) with natural order by inclusion.\\

The Hardy operator and its 'adjoint' are

\begin{equation}\notag

\begin{split}

&\mathbf{I}f(R) := \sum_{R\subset Q}f(Q)\\

&\mathbf{I}^*f(Q) := \sum_{R\subset Q}f(R).

\end{split}

\end{equation}

We are investigating the action of this operator  from $L^2(\Gamma,w^{-1})$ to $L^2(\Gamma,\mu)$, or, which is the same, $\mathbf{I}^*$ from $L^2(\Gamma,\mu^{-1})$ to $L^2(\Gamma,w)$, where $w$ and $\mu$ are just collections of non-negative weights attached to the elements of $\Gamma$. If for given $\mu,w$ the Hardy operator is bounded, we call $(\mu,w)$ \textit{the trace measure-weight pair}.\par

In this talk we consider a special case -- the dimension $n$ is either 2 or 3 and the weight $w$ is a product weight (a typical case is just $w\equiv 1$). We give a couple of descriptions of such pairs in potential theoretical terms: capacitary and energy conditions. We give a short exposition of two-dimensional results, and discuss problems that arise with increasing the dimension. We also establish a connection to weighted Dirichlet spaces on the polydisc.

We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, relative entropy, and transportation cost. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes abruptly at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein-Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions.

Dans un travail récent en commun avec Bo'az Klartag, on montre que la conjecture de l'hyperplan de Bourgain ainsi que la conjecture de Kannan, Lovasz et Simonovits portant sur la constante de Poincaré des mesures log-concaves, sont toutes les deux vérifiées, à un facteur polylog en la dimension près. Dans cet exposé je donnerai les grandes lignes de la démonstration.

I shall present Rosenthal-type inequalities with sharp constants for moments of sums of independent random variables which are mixtures of a fixed distribution. I will also discuss related problems and partial results for sums of independent log-concave random variables. Based on joint work with Chasapis and Eskenazis.

In this talk I will discuss a version of an old question of Vitali Milman about almost Euclidean and well-complemented subspaces. In particular, I will introduce a notion of `$\epsilon$-good points', which allows for a convenient reformulation of the problem. Let $(X, \|\cdot\|_X)$ be a normed space. It turns out that if a linear subspace $Y\subset X$ consists entirely of $\epsilon$-good points then the restriction of the norm $\|\cdot \|_X$ to $Y$ must be approximately a multiple of the $\ell_2$ norm and the operator norm of the orthogonal projection onto $Y$ is close to 1. I will present an example of a normed space $X$ of arbitrarily high dimension, whose Banach-Mazur distance from the $\ell_2^{{\rm dim} X}$ is at most $2$, but such that non of its (even two-dimensional) subspaces consists entirely of $\epsilon$-good points. The talk is based on joint work with Timothy Gowers. 

In arxiv:1901.01339, we gave a counterexample to a Mergelyan type statement for cartesian products stated by Gamelin and Garnett in an article published in TAMS fifty years ago, in 1969. In order to correct this false statement we introduced a natural algebra $A_D(K)$, smaller than the classical  algebra $A(K)$, where $K$ is a compact subset of $\Bbb C^d$. The introduction of $A_D(K)$ explains partially why approximation theory in several variables is not so developed and opens a new road of research. We believe that several new Mergelyan type theorems will be obtained. So far we have one such theorem for products and one for graphs. Applications to universality can be found in arxiv:1909.03521 and arxiv:1910.01759. Also, a parallel theory can be developed when replacing uniform convergence by uniform on $K$ convergence of all orders derivatives (in preparation).

We will present a Logvinenko-Sereda type theorem on quantitative bounds for planar subsampling of true polyanalytic Bargmann-Fock spaces. As a side product, which deserves some interesting on its own, we show a Remez-type inequality for polyanalytic functions.
 

Je présenterai le résultat classique de Logvinenko-Sereda, et sa démonstration puis j'expliquerai comment on peut le généraliser pour des ensembles discrets.

 We will  describe Kirchberg's conjecture on tensor products of $C^*$-algebras  which, as he showed, is   equivalent to the Connes embedding problem. His conjecture is that the $C^*$-algebra of the free  groups $C^*(\mathbb F_\infty)$ (or $C^*(\mathbb F_2)$) which has the Local  Lifting  Property  (LLP in short) also has the Weak  Expectation  Property  (WEP in short). We will describe the construction of the first example of a  non-nuclear $ C^*$-algebra $A$ with both LLP and WEP.  Our algebra $A$ has the ``same" finite dimensional operator spaces  as $C^*(\mathbb F_\infty)$.  In the second part several operator space variants involving analogues of the Gurarii space will be presented.
 

Dans cet exposé (tiré de collaborations avec Hao Jia, Carlos  Kenig et Frank Merle), je présenterai certaines minorations de  l'énergie de l'équation des ondes linéaire sur l'espace euclidien, à  l'extérieur du cône d'onde. Je donnerai également des applications à  l'étude la dynamique de l'équation des ondes non-linéaires, et notamment à la démonstration récente de la résolution en solitons pour l'équations des ondes critiques en dimension impaire.
 

Si $M_f$ désigne l’opérateur maximal ``fort’’ défini par
$$
M_f(x):=\sup_R \frac{1}{|R|} \int_R |f|,
$$
où la borne supérieure est étendue à tous les parallélépipèdes $n$-dimensionnels $R$ parallèles aux axes contenant $x$, l’inégalité suivante, due indépendamment à Fava et de Guzmán, est classique:
$$
|\{M_ff>\lambda\}|\leq C\int_{\R^n} \frac{|f|}{\lambda}\left(1+\log_+ \frac{|f|}{\lambda}\right).
$$
Nous nous intéresserons, dans cet exposé, à la validité (ou non) d’inégalités de ce type pour des opérateurs maximaux associés à des familles de rectangles de $\R^2$ dont les côtés sont orientés selon un ensemble (dénombrable) de directions. Il s’agit de travaux à paraître avec E. D’Aniello et J. Rosenblatt.

The Brunn-Minkowski and Prékopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special significance. On the other hand, it was recently shown that the Brunn-Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. We show that for the subclass of log-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa-Leindler inequality. In collaboration with Peter Pivovarov.

Pour essayer de définir quand même une mesure harmonique sur des domaines dont le bord est de grande codimension,
on a été amenés à étudier des opérateurs elliptiques un peu particuliers. On s’intéresse, comme en codimension $1$,
aux relations entre régularité du domaine et propriétés de la mesure harmonique associée.

Travaux avec M. Engerstein, J. Feneuil, et S. Mayboroda.

On se propose d illustrer sur un exemple simple (NLS quintique 1d) d’équation non intégrable comment l’introduction de corrections dans les énergies d'ordre élevé permet de montrer la croissance polynomiale en temps de ces énergies par un argument élémentaire ; on abordera également la quasi-invariance des mesures gaussiennes associées. Il s’agit de travaux en collaboration avec Nikolay Tzvetkov et Nicola Visciglia.

It was shown by Ascenzi, Lyubarskii, Seip that the upper density of any complete and minimal system
of Gaussians is between $\frac{2}{\pi}$ and 1, if we know that there exists an angular density. It was an open
question whether it is true in general. We will show that there exists a complete and minimal system with
upper density $\frac{1}{\pi}$, and that the upper density of any such system is larger than $\frac{1}{3\pi}$.
The talk is based on work in collaboration with A. Borichev and A. Kuznetsov.

Every function $f$ on the $n$-dimensional discrete cube $\{-1,1\}^n$ admits a unique representation as a multilinear polynomial of total degree at most $n$, called the Walsh expansion of $f$. We will review the basics of Fourier analysis on the discrete cube and explain a duality argument (inspired by classical work of Figiel) which leads to approximation theoretic estimates for functions whose Walsh coefficients are supported on frequencies bounded above or below. These include Bernstein--Markov type inequalities and their reverses, moment comparison for vector-valued Rademacher chaos of low degree and estimates on the $\ell_p$ sum of influences of bounded functions. The talk is based on joint work with Paata Ivanisvili.

 L'espace libre F(M), pour un espace métrique M, est un espace de Banach qui contient M isométriquement et qui est tel que toute application Lipschitzienne de M dans R peut être "prolongée canoniquement" en une forme linéaire définie sur F(M). La structure linéaire de ces espaces libres reste assez peu comprise à ce jour. Pour illustrer cela, nous ne savons pas si F(R^2) et F(R^3) sont isomorphes. Cependant une manière de progresser dans la compréhension de ces espaces est d'étudier les éventuels plongements linéaires d'espaces de Banach classiques dans les espaces libres, ou vice-versa.  Nous nous intéresserons particulièrement à la relation qu'ont les espaces libres avec l1, l'espace de Banach des suites réelles sommables. Plus précisément, nous chercherons à nous rapprocher d'une caractérisation des espaces métriques M pour lesquels F(M) se plonge isométriquement dans l1. Il s'agit de travaux en collaboration avec Antonín Procházka et Ramón J. Aliaga.