# Geometric Functional Analysis (710, Modern Analysis II), Winter 2009

 Instructor: Roman Vershynin Office: 4844 East Hall E-mail: romanv "at" umich "dot" edu Class meets: T Th 11:40-1:00 in 3866 East Hall. Office Hours: M 3-4:30, W 1:30-3 in 4844 East Hall.

Course description: Geometric functional analysis studies high dimensional linear structures, such as Euclidean and Banach spaces, convex sets and linear operators in high dimensions. A central question is: what do typical convex bodies and typical linear operators look like in Rn when n grows to infinity? One of the main tools of geometric functional analysis is the theory of concentration of measure, which offers a geometric view on the limit theorems of probability theory. Geometric functional analysis thus bridges three areas - functional analysis, convex geometry and probability theory. The course is a systematic introduction to the main techniques and results of geometric functional analysis.

Prerequisites: Analysis II (Math 597) and Probability Theory (Math 525). Real Analysis II (Math 602) will be very helpful. Graduate Probabiilty Theory (Math 625) will also be of help.

Assignments: Homework assignments will be occasionally given in class. Many of them will have a research flavor. The assingments will be divided in two phases. Phase I problems are due Thursday, February 19. Phase II problems are due Tuesday, April 21. You are expected to solve (or make good progress) on approximately 10 problems in each phase. You are welcome to work in groups on the problems, but please write down your work individually.

Course Plan

1. Functional analysis and convex geometry.
Preliminaries in functional analysis: normed spaces, linear operators, duality. Finite dimensional normed spaces: Minkowski functional, polar sets, duality.
2. Banach-Mazur distances.
Definition and properties of Banach-Mazur distance. John's theorem in geometric form [Mat 13.4], [MS 3.3]. Consequence for Banach-Mazur distances. Distance between l_p^n abd l_q^n [see TJ Seciton 38]. Gluskin's theorem without proof. Further possible topics (omitted): extension of John's theorem for contact points (variational proof) and its consequences [GM Euclid 2.3].
3. Concentration of measure and Euclidean sections of convex bodies.
Concentration of measure on the sphere [B], [Mat 14.1]. Johnson-Lindenstrauss Lemma [Mat 15.2]. Epsilon-nets. General Dvoretzky Theorem (by Figiel-Lindenstrauss-Milman) [MS 4]. Euclidean subspaces of l_p^n spaces [MS 5]. Many faces of symmetric polytopes [Mat 14.5]. Dvoretzky-Rogers Lemma [MS 3.4], [Mat 14.6]. Dvoretzky theorem [MS 5]. Volume ratio theorem [Lectures 13, 14 of my course on random matrices].
4. Metric entropy and its applications.
Diameters of projections of convex sets. Metric entropy. Duality problem. Sudakov and inverse Sudakov inequalities [LT 3.3]. Low M* estimate. Ell-position (without proof). Quotient of subspace theorem.
5. Geometric inequalities.
Prekopa-Leindler inequality. Brunn-Minkowski inequality. Applications: Brunn's principle, isoperimetric inequality in R^n, concentration of measure in the ball, sphere and Gauss space, Borell's inequality, Urysohn's inequality, Santalo inequality [MP]. Milman's ellipsoids [GM Euclid], [GM ACG], [M Symm]. Applications [incl. Pisier]: inverse Santalo inequality, inverse Brunn-Minkowski inequality, duality of entropy (on the exponential scale), an alternative proof of quotient of subspace theorem.

Texts: There will be no regular textbook. Please take notes. The following texts cover certain parts of this course:

• [B] K. Ball, An elementary introduction to modern convex geometry. Flavors of geometry, 1--58, Math. Sci. Res. Inst. Publ., 31, Cambridge Univ. Press, Cambridge, 1997. PDF file
• [GM Euclid] A. Giannopoulos, V.Milman, Euclidean structure in finite dimensional normed spaces. Handbook of the geometry of Banach spaces, Vol. I, 707--779, North-Holland, Amsterdam, 2001. PS file
• [GM ACG] A. Giannopoulos, V. Milman, Asymptotic convex geometry: short overview. Different faces of geometry, 87--162, Int. Math. Ser. (N. Y.), 3, Kluwer/Plenum, New York, 2004. PS file
• [L] M. Ledoux, The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001.
• [LT] M. Ledoux, M. Talagrand, Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991.
• [Mat] J. Matousek, Lectures on discrete geometry. Graduate Texts in Mathematics, 212. Springer-Verlag, New York, 2002.
• [MP] M. Meyer, A. Pajor, On the Blaschke-Santalo inequality, Arch. Math. 50 (1990), 82--93.
• [M Symm] V. Milman, Isomorphic symmetrizations and geometric inequalities, Geometric aspects of functional analysis (1986/87), 107--131, Lecture Notes in Math., 1317, Springer, Berlin, 1988.
• [MS] V. Milman, G. Schechtman, Asymptotic theory of finite-dimensional normed spaces. Lecture Notes in Mathematics, 1200. Springer-Verlag, Berlin, 1986. PDF files
• [Pisier] G. Pisier, The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989.
• [T] N. Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, 38. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.

Course webpage: http://www.umich.edu/~romanv/teaching/2008-09/710/710.html