Ollivier curvature, Isoperimetry, concentration, and Log-Sobolev inequalitiy

We introduce a Laplacian separation principle for the the eikonal equation on Markov chains. As application, we prove an isoperimetric concentration inequality for Markov chains with non-negative Ollivier curvature. That is, every single point from the concentration profile yields an estimate for every point of the isoperimetric estimate. Applying to exponential and Gaussian concentration, we obtain affirmative answers to two open quesions by Erbar and Fathi. Moreover, we prove that the modified log-Sobolev constant is at least the minimal Ollivier Ricci curvature, assuming non-negative Ollivier sectional curvature, i.e., the Ollivier Ricci curvature when replacing the $\ell_1$ by the $\ell_\infty$ Wasserstein distance. This settles a recent open Problem by Pedrotti. We give a simple example showing that non-negative Ollivier sectional curvature is necessary to obtain a modified log-Sobolev inequality via positive Ollivier Ricci bound. This provides a counterexample to a conjecture by Peres and Tetali

Around stability for functional inequalities

Les inégalités fonctionnelles sont des inégalités qui encodent beaucoup d'information, tant de nature probabiliste (concentration de la mesure), qu'analytique (théorie spectrale des opérateurs) ou encore géométrique (profil isopérimétrique). L'inégalité de Poincaré en est un exemple fondamental. Dans cette thèse, nous obtenons des résultats de stabilité dans le cadre d'hypothèses de normalisation de moments, ainsi que dans le cadre de conditions de courbure-dimension. Un résultat de stabilité est une façon de quantifier la différence entre deux situations dans lesquelles les mêmes inégalités fonctionnelles sont presque vérifiées. Les résultats de stabilité obtenus dans cette thèse sont en particulier basés sur la méthode de Stein, qui est une méthode en plein développement ces dernières années, provenant du domaine des statistiques et permettant d'établir des estimations quantitatives sur des résultats de convergence. Par ailleurs, une partie de cette thèse est consacrée à l'étude des constantes optimales des inégalités de Bobkov, qui sont des inégalités fonctionnelles à caractère isopérimétrique.Functional inequalities are inequalities that encode a lot of information, both of a probabilistic (the concentration of measure phenomenon), analytical (the spectral theory of operators) and geometric (isoperimetric profile) nature. The Poincaré inequality is a fundamental example. In this thesis, we obtain stability results under moment normalisation assumptions, as well as under curvature-dimension conditions. A stability result is a way to quantify the difference between two situations where almost the same functional inequalities are verified. The stability results obtained in this thesis are in particular based on the Stein method, which is a method in full development in recent years, coming from the field of statistics and allowing to establish quantitative estimates on convergence results. In addition, a part of this thesis is devoted to the study of the optimal constants of Bobkov inequalities, which are functional inequalities of isoperimetric character

Extracting features from eigenfunctions: higher Cheeger constants and sparse eigenbasis approximation

This thesis investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We obtain a new lower bound on the negative Laplace-Beltrami eigenvalues in terms of the corresponding higher Cheeger constant. The level sets of Laplace-Beltrami eigenfunctions sometimes reveal sets with small Cheeger ratio, representing well-separated features of the manifold. Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios. In a later chapter, we propose a numerical method for identifying features represented in eigenvectors arising from spectral clustering methods when those features are not cleanly represented in a single eigenvector. This method provides explicit candidates for the soft-thresholded linear combinations of eigenfunctions mentioned above. Many data clustering techniques produce collections of orthogonal vectors (e.g. eigenvectors) which contain connectivity information about the dataset. This connectivity information must be disentangled by some secondary procedure. We propose a method for finding an approximate sparse basis for the space spanned by the leading eigenvectors, by applying thresholding to linear combinations of eigenvectors. Our procedure is natural, robust and efficient, and it provides soft-thresholded linear combinations of the inputted eigenfunctions. We develop a new Weyl-inspired eigengap heuristic and heuristics based on the sparse basis vectors, suggesting how many eigenvectors to pass to our method

From large deviations to semidistances of transport and mixing: coherence analysis for finite Lagrangian data

One way to analyze complicated non-autonomous flows is through trying to understand their transport behavior. In a quantitative, set-oriented approach to transport and mixing, finite time coherent sets play an important role. These are time-parametrized families of sets with unlikely transport to and from their surroundings under small or vanishing random perturbations of the dynamics. Here we propose, as a measure of transport and mixing for purely advective (i.e., deterministic) flows, (semi)distances that arise under vanishing perturbations in the sense of large deviations. Analogously, for given finite Lagrangian trajectory data we derive a discrete-time and space semidistance that comes from the "best" approximation of the randomly perturbed process conditioned on this limited information of the deterministic flow. It can be computed as shortest path in a graph with time-dependent weights. Furthermore, we argue that coherent sets are regions of maximal farness in terms of transport and mixing, hence they occur as extremal regions on a spanning structure of the state space under this semidistance---in fact, under any distance measure arising from the physical notion of transport. Based on this notion we develop a tool to analyze the state space (or the finite trajectory data at hand) and identify coherent regions. We validate our approach on idealized prototypical examples and well-studied standard cases.Comment: J Nonlinear Sci, 201

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry