Graduate topics course in Concentration of Measure Phenomenon and Convexity
Georgia Institute of Technology, Fall 2023
Contact: glivshyts6@math.gatech.edu or stop by Skiles 108C.
Location and time: Mondays/Wednesdays 9:30-10:45am at Skiles 006
August 21 through December 4.
As a back up (for someone who gets sick, for example), the lectures will be broadcast via Zoom, but students are generally expected to attend in person.
Office hours: Monday at 3-5pm online via Zoom or in-person by appointment.
Description: In this course, we shall learn about the Concentration of Measure Phenomenon in High Dimensions, often stemming from Convexity and Isoperimetry. Students are encouraged to provide input and suggest further/alternative topics, in addition to the ones listed here:
Grading: The course grade is based on completion of 5 points out of the before the end of the semester (this file shall be updated throughout the semester, and more and more problems will be added. Each problem is weighed with several points.) Due to the students' busy schedule, this requirement is essentially nominal. However, those interested are encouraged to complete more problems. Turn home work in anytime during the semester, any number of times, in pdf via Canvas.
For more details, see the course syllabus.
A list of a few relevant books (some of which are mentioned on the schedule already):
1. Artstein-Avidan, Giannopoulos, Milman, Asymptotic Geometric Analysis part 1, 2015.
2. Bakry, Ledoux, Gentil, Analysis and Geometry of Markov Diffusion Operators, 2014.
3. Artstein-Avidan, Giannopoulos, Milman, Asymptotic Geometric Analysis part 2, 2022.
4. Vershynin, High-Dimensional Probability, 2018.
5. Bogachev, Gaussian measures, 1998.
6. Milman, Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, 1986.
7. Bobkov, Chistyakov, Gotze, Concentration and Gaussian Approximation for randomized sums, 2023.
8. Pisier, The Volume of Convex Bodies and Banach Space Geometry, 1989.
9. Brazitikos, Giannopoulos, Valettas, Vritsiou, Geometry of Isotropic Convex Bodies, 2014.
10. Ledoux, The Concentration of Measure Phenomenon, 2001.
11. Villani, Topics in Optimal Transportation, 2003.
(a very rough file, to be significantly proofread, corrected and updated until at least January 2024)
Class whiteboard:
Lectures 1-2: A few high-dimensional phenomena riddles, volume computations, high-dimensional ball and concentration, asymptotic integrals and Laplace method, convex sets and functions, hyperplanes, Minkowski functional, support function of a convex body
Lectures 3-4: Basic convexity, Khan-Banach, support hyperplanes, sections, projections, polarity and its properties, first mention of Legendre transform, Brunn-Minkowski inequality (additive and multiplicative forms), concept of log-concavity, proof of the isoperimetric inequality, Lyusternik’s proof of the Brunn-Minkowski, Steiner symmetrizations, Steiner-style proof of Brunn-Minkowski for convex sets.
Lectures 5, 6, 7: Mixed volumes, Minkowski's first inequality, Brunn's concavity principle, Log-concavity and Borell's theorem about log-concave measures, Prekopa-Leindler inequality, Log-concavity of marginals and convolutions, Legendre transform and its properties, idea of linearizations, Generalized Log-Sobolev inequality (in the footsteps of Bobkov and Ledoux).
Lectures 8, 9: Differentiating Legendre transform of an interpolation, Generalized Log-Sobolev inequality and its reformulation and corollaries, the classical Lebesgue and Gaussian Log-Sobolev inequalities, Sobolev inequalities, Beckner inequalities, the Laplace operator associated to a log-concave measure, integration by parts, a few words about eigenvalues, from Generalized Log-Sobolev to Brascamp-Lieb, some corollaries of the Brascamp-Lieb inequality, Gaussian Poincare Inequality.
Lectures 10, 11: A short proof of the integration by parts (via linearization!), a word about Payne-Weinberger, Poincare inequality on the circle via Fourier, a remark on Colesanti's inequality via Brascamp-Lieb, dimensional extension of the Brascamp-Lieb inequality via Borell-Brascamp-Lieb, an extension of Borell-Brascamp-Lieb and a Convex Sobolev inequality, volume product and the Blaschke-Santalo inequality, discussion of the Mahler conjecture, Hanner polytopes, Volume products for non-symmetric convex bodies, Santalo point, Lehec’s result, Gozlan, Fradelizi, Sadovsky, Zugmeyer’s result, Klartag’s relation to slicing, proof of the Blaschke-Santalo inequality via Steiner symmetrizations, the open problem due to Cordero-Erasquin regarding Blaschke-Santalo for log-concave measures.
Lectures 12, 13: Santalo formula for the area of a convex region on the plane, Blaschke-Santalo inequality implies p-Beckner inequality on the circle with p=-2, functional form of the Blaschke-Santalo inequality, Fradelizi-Meyer theorem, linearizing Blaschke-Santalo to get the improved Gaussian Poincare inequality for even functions. An excursion into mass transport (optimal transport with respect to quadratic cost, Brenier theorem and some geometric intuition, Cafarelli’s theorem, all without proofs); The improved Gaussian Poincare inequality on symmetric convex sets via the local form of the Blaschke-Santalo inequality and Cafarelli’s contraction; the B-theorem of Cordero-Erasquin, Fradelizi, Maurey, and the B-conjecture.
Lecture 14: B-theorem (proof), some remarks about the B-conjecture, the multiplicative Prekopa-Leindler inequality on the corner and the B-conjecture in the unconditional case, the Log-Brunn-Minkowski conjecture, the reverse Log-Sobolev inequality via Blaschke-Santalo inequality.
Lectures 15, 16: Further applications of the Blaschke-Santalo inequality: recap and a few remarks regarding the Reverse Log-Sobolev inequality of Artstein-Avidan, Klartag, Schutt and Werner; Fathi’s symmetrized version of the Talagrand transport-entropy inequality and its proof via the Blaschke-Santalo inequality. Concentration of measure: basic concepts, isoperimetry, basic examples (the sphere, the Gaussian, the Hamming cube); non-sharp concentration on the sphere via Brunn-Minkowski; non-sharp Gaussian concentration via Prekopa-Leindler.
Lectures 17, 18: Concentration function, sub-Gaussian and sub-exponential concentration, concentration for Lipschitz functions near the median, concentration for Lipschitz functions near the mean on the sphere, Borell’s lemma (about log-concave measures of dilates of convex sets), an application to a weak version of Paouris’s theorem, Paouris’s theorem (statement), proof of the reverse Holder for the moments of semi-norms of log-concave random vectors, applications to moments of seminorms on the sphere, exponential concentration for linear functions with respect to log-concave measures, Thin Shell conjecture and the KLS conjecture, Thin Shell conjecture for unconditional log-concave random vectors.
Lectures 19, 20: Gromov-Milman theorem (from Poincare to exponential concentration); From Fathi’s inequality to a concentration result via Marton’s argument; concentration via Laplace functional; From Log-Sobolev to sub-Gaussian concentration (Herbst argument); Herbst argument; the Klartag-Vershynin small ball estimate for a norm on the sphere.
Lectures 21, 22: Gaussian measures, optimality of the Gaussian measure for the Poincare constant amid the isotropic measures, isoperimetry and isoperimetric profile, Gaussian notation, the Ehrhard inequality (statement), comparison to the Prekopa-Leindler inequality, Gaussian Isoperimetric Inequality via Ehrhard, From the Gaussian Isoperimetric Inequality to the sharp Gaussian concentration; Bounding the isoperimetric profile of the cube via Gaussian Isoperimetry, and Vaaler’s theorem; Gaussian symmetrizations and the proof of Ehrhard’s inequality.
Lectures 23, 24: Latala’s Functional Ehrhard inequality; linearizing functional Ehrhard to get the Generalized Bobkov inequality, some examples, further linearizing to get a Brascamp-Lieb-Ehrhard type inequality; minimizing centered in-radius for convex sets with a fixed Gaussian measure (symmetric and non-symmetric cases); more isoperimetric-type inequalities for the Gaussian measure; Bobkov’s Gaussian barycenter inequality; improved Gaussian Poincare for linear functions; dilates in a direction minimized on half-spaces; dilates of sets of fixed Gaussian measure, and optimizers in various classes.
Lecture 25: Latala-Olieszkiewicz S-inequality modulo the technical 2-dimensional result; the Bochner formula (the traditional and non-traditional proofs, and a generalization); Sobolev spaces; density of the image of the measure-associated Laplace operator; dual norms and examples; the H-1 norm and its relation to mass transport.
Lectures 26, 27: Proof of the relation between the H-1 norm and infinitesimal Wasserstein distance; the Hormander duality lemma; corollary 1 — Klartag’s H-1 inequality; corollary 2 — Brascamp-Lieb inequality; corollary 3 — Gaussian symmetric Poincare inequality; a mention of uniqueness of PDE solutions and symmetry of solution with symmetric data; the local version of the B-conjecture; a more general conjecture; yet another more general conjecture; proof of the B-conjecture for rotation-invariant log-concave measures by Cordero-Erausquin, Rotem, via the Hormander's lemma.
Lecture 28: Klartag’s improvement of the Lichnerowicz inequality and the conjecture about preferred directions of eigenfunctions. Dimensional Brunn-Minkowski conjecture and the statements of the relevant results.
In Spring 2024 I teach a related Course in High-dimensional Probability.
