Geometric Aspects of Functional Analysis

Continuing the theme of the previous volume, these seminar notes reflect general trends in the study of Geometric Aspects of Functional Analysis, understood in a broad sense. Two classical topics represented are the Concentration of Measure Phenomenon in the Local Theory of Banach Spaces, which has recently had triumphs in Random Matrix Theory, and the Central Limit Theorem, one of the earliest examples of regularity and order in high dimensions. Central to the text is the study of the Poincaré and log-Sobolev functional inequalities, their reverses, and other inequalities, in which a crucial role is often played by convexity assumptions such as Log-Concavity. The concept and properties of Entropy form an important subject, with Bourgains slicing problem and its variants drawing much attention. Constructions related to Convexity Theory are proposed and revisited, as well as inequalities that go beyond the Brunn–Minkowski theory. One of the major current research directions addressed is the identification of lower-dimensional structures with remarkable properties in rather arbitrary high-dimensional objects. In addition to functional analytic results, connections to Computer Science and to Differential Geometry are also discussed.

- Jean Bourgain: In Memoriam. - Gromov’sWaist of Non-radial Gaussian Measures and Radial Non-Gaussian Measures. - Zhang’s Inequality for Log-Concave Functions. - Bobkov’s Inequality via Optimal Control Theory. - Arithmetic Progressions in the Trace of Brownian Motion in Space. - Edgeworth Corrections in Randomized Central Limit Theorems. - Three Applications of the Siegel Mass Formula. - Decouplings for Real Analytic Surfaces of Revolution. - On Discrete Hardy–Littlewood Maximal Functions over the Balls in Zd : Dimension-Free Estimates. - On the Poincaré Constant of Log-Concave Measures. - On Poincaré and Logarithmic Sobolev Inequalities for a Class of Singular Gibbs Measures. - Several Results Regarding the (B)-Conjecture. - A Dimension-Free Reverse Logarithmic Sobolev Inequality for Low-Complexity Functions in Gaussian Space. - Information and Dimensionality of Anisotropic Random Geometric Graphs. - On the Ekeland-Hofer-Zehnder Capacity of Difference Body.