Geometric Aspects of Functional Analysis Israel Seminar (GAFA) 2017-2019 Volume I
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| Main Authors | , |
|---|---|
| Format | eBook |
| Language | English |
| Published |
Cham
Springer International Publishing AG
2020
Springer International Publishing |
| Edition | 1 |
| Series | Lecture Notes in Mathematics |
| Subjects | |
| Online Access | Get full text |
| ISBN | 9783030360191 3030360199 |
| ISSN | 0075-8434 1617-9692 |
| DOI | 10.1007/978-3-030-36020-7 |
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Table of Contents:
- 6 Three Applications of the Siegel Mass Formula -- 6.1 Background on the Siegel Mass Formula -- 6.2 Uneven Parsell-Vinogradov Sums -- 6.3 Non-congruent Tetrahedra in the Truncated Lattice [0,q]3Z3 -- 6.4 Distribution of Lattice Points on Caps -- References -- 7 Decouplings for Real Analytic Surfaces of Revolution -- 7.1 Background and the Main Result -- 7.2 A Case Analysis Based on Principal Curvatures -- 7.3 The Case of the Quasi-Torus -- 7.4 The Perturbed Cone -- 7.5 Final Remarks -- References -- 8 On Discrete Hardy-Littlewood Maximal Functions over the Balls in Zd: Dimension-Free Estimates -- 8.1 Introduction -- 8.1.1 Motivations and Statement of the Results -- 8.1.2 The Large-Scale Case -- 8.1.3 The Intermediate-Scale Case -- 8.1.4 The Small-Scale Case -- 8.1.5 Notation -- 8.2 Estimates for the Dyadic Maximal Function: Intermediate Scales -- 8.2.1 Some Preparatory Estimates -- 8.2.2 Analysis on Permutation Groups -- 8.2.3 A Decrease Dimension Trick -- 8.2.4 All Together -- 8.3 Estimates for the Dyadic Maximal Function: Small Scales -- 8.3.1 Some Preparatory Estimates -- 8.3.2 Analysis Exploiting the Krawtchouk Polynomials -- 8.3.3 All Together -- References -- 9 On the Poincaré Constant of Log-Concave Measures -- 9.1 Introduction and Overview -- 9.2 Revisiting E. Milman's Results -- 9.2.1 (p,q) Poincaré Inequalities -- 9.2.2 Log-Concave Probability Measures -- 9.2.3 Some Variations and Some Immediate Consequences -- 9.2.3.1 Immediate Consequences -- 9.2.3.2 Variations: The L2 Framework -- 9.2.3.3 Other Consequences: Reducing the Support -- 9.3 Transference of the Poincaré Inequality -- 9.3.1 Transference via Absolute Continuity -- 9.3.2 Transference Using Distances -- 9.4 Mollifying the Measure -- 9.4.1 Mollifying Using Transportation -- 9.4.1.1 Convolution with the Uniform Distribution -- 9.4.1.2 Convolution with the Gaussian Distribution
- 9.4.2 Mollifying with Gaussian Convolution: A Stochastic Approach -- 9.5 Probability Metrics and Log-Concavity -- 9.5.1 Comparing Bounded Lipschitz and Total Variation Distances -- 9.5.2 Comparing Total Variation and W1 -- 9.5.3 Comparison with Other Metrics on Probability Measures -- References -- 10 On Poincaré and Logarithmic Sobolev Inequalities for a Class of Singular Gibbs Measures -- 10.1 Introduction -- 10.1.1 Functional Inequalities and Concentration of Measure -- 10.1.2 Dynamics -- 10.1.3 Hermite-Lassalle Orthogonal Polynomials -- 10.1.4 Comments and Open Questions -- 10.2 Useful or Beautiful Facts -- 10.2.1 Random Matrices, GUE, and Beta Hermite Ensemble -- 10.2.2 Isotropy of Beta Hermite Ensembles -- 10.2.3 Log-Concavity and Curvature -- 10.2.4 Factorization by Projection -- 10.3 Proofs -- 10.3.1 Proof of Theorems 10.1.1 and 10.1.2 -- 10.3.2 Proof of Theorem 10.1.3 and Comments on Optimality -- 10.3.3 Proof of Corollary 10.1.4 and Comments on Concentration -- 10.3.4 Proof of Theorem 10.1.5 -- References -- 11 Several Results Regarding the (B)-Conjecture -- 11.1 Introduction -- 11.2 A Gaussian Counter-Example -- 11.3 A Functional (B)-Conjecture -- References -- 12 A Dimension-Free Reverse Logarithmic Sobolev Inequalityfor Low-Complexity Functions in Gaussian Space -- 12.1 A Reverse Logarithmic Sobolev Inequality -- 12.2 Stochastic Calculus and the Föllmer Process -- References -- 13 Information and Dimensionality of Anisotropic Random Geometric Graphs -- 13.1 Introduction -- 13.1.1 Previous Work -- 13.1.2 Main Results and Ideas -- 13.2 Preliminaries -- 13.3 Estimates for a Triangle in a Random Geometric Graph -- 13.3.1 Lower Bound: The Case p=12 -- Step 1-The Integral is Bounded from Below on B1 = {x R3 : || x||2 ≤1||α||22 }, the Ball of Radius 1|| α||2
- Step 2-The Integrand is Positive on B2 = {x R3 : || x|| 2 ≤1|| α||222/12 }, the Ball of Radius 1|| α||211/12 -- Step 3-The Absolute Value of the Integrand Is Negligible on the Spherical Shell B B2 Where B Is the Unit Ball in R3 -- Step 4-The Integral Is Negligible Outside of B -- Final Step-R3 -Im(φ(a,b,c))abcdadbdc ≥1100(||α||3||α||2)3 -- 13.3.2 Arbitrary 0 < -- p < -- 1 -- 13.3.3 Upper Bound -- 13.4 Proof of Theorem 13.3 -- 13.5 Proof of the Lower Bound -- References -- 14 On the Ekeland-Hofer-Zehnder Capacity of Difference Body -- 14.1 Proof of Theorem 14.1 -- 14.2 Preliminaries for the proof of Theorem 14.2 -- 14.3 Proof of Theorem 14.2 -- References
- Intro -- Preface -- Jean Bourgain: In Memoriam -- Contents Overview for Volume II -- Contents -- 1 Gromov's Waist of Non-radial Gaussian Measures and Radial Non-Gaussian Measures -- 1.1 Introduction -- 1.2 Nonradial Gaussian Measures: Proof of Theorem 1.1.1 -- 1.3 Existence of Waist Theorems for Some Radial Measures -- 1.4 Waist for Radial Measures and Odd Maps -- 1.5 Neighborhoods of Critical Submanifolds -- 1.5.1 Neighborhoods of Submanifolds in the Euclidean Space -- 1.5.2 Neighborhoods in the Sphere and the Complex Projective Space -- 1.6 Appendix: Explanation of the Caffarelli Theorem -- 1.7 Appendix: Explanation of the Pancake Decomposition -- 1.7.1 General Version of the Equipartition Argument -- 1.7.2 Modification of the Equipartition Argument for the Spherical Waist Theorem -- 1.7.3 Simplified Version of the Equipartition Argument -- 1.7.4 Proof that the Parts Are Pancakes -- References -- 2 Zhang's Inequality for Log-Concave Functions -- 2.1 Introduction -- 2.2 Notation and Preliminaries -- 2.3 Proof of the Inequality in Theorem 2.1.1 -- 2.4 Characterization of the Equality in Theorem 2.1.1 -- References -- 3 Bobkov's Inequality via Optimal Control Theory -- 3.1 Bobkov's Inequality -- 3.2 Bellman Principle -- 3.3 The Proof -- 3.3.1 Hamilton-Jacobi-Bellman PDE -- 3.3.2 Deriving Bobkov's Inequality -- 3.3.3 Optimizers -- References -- 4 Arithmetic Progressions in the Trace of Brownian Motionin Space -- 4.1 Introduction -- 4.2 Proofs -- Reference -- 5 Edgeworth Corrections in Randomized Central Limit Theorems -- 5.1 Introduction -- 5.2 Construction of Asymptotic Expansions -- 5.3 Moments and Deviations of Lyapunov Coefficients -- 5.4 Upper Bounds on Characteristic Functions -- 5.5 Proof of Theorem 5.1.1 -- 5.6 General Lower Bounds -- 5.7 Approximation by Mean Characteristic Functions -- 5.8 Proof of Theorem 5.1.2 -- References