MAT 1502 University of Toronto Fall 2017

MAT 1502 Topics in Geometric Analysis: Convexity

Fall 2017
Almut Burchard, Instructor

How to reach me: Almut Burchard, BA 6234, 8-3318.
almut @ ,
Lectures W 3:10-4, R 3:10-5pm in BA 6180
(Lectures missed when I travel will be made up on Wednesdays 4:10-5pm).
Office hours: Monday late-afternoon
About the course: We will explore a variety of topics in convex and geometry and analysis, moving from classical theorems and techniques to recent results and open problems. Possible topics include: Brunn-Minkowski and isoperimetric inequalities; symmetrization methods; concentration phenomena; geometric versions of Hahn-Banach; Legendre transform; extreme points and the Krein-Milman theorem; monotone and convex matrix functions.
  • "Asymptotic Geometric Analysis, Part I", by S. Artstein-Avidan, A. Giannopoulos, and V. Milman.
    Mathematical Surveyys and Monographs, Vol. 202 (2015)
  • "Convexity: An Analytic Viewpoint", by Barry Simon.
    Cambridge Tracts in Mathematics, Vol. 187 (2011)
  • "Convex Analysis", by R. Tyrell Rockafellar.
    Princeton University Press (1970)
  • "Functional Analysis, Sobolev Spaces, and PDE", by H. Brézis (Chapter 1).
    Springer Universitext (2011).
Format: Students enrolled in the course should work out assignments, and give a one-hour presentation on a topic of their choice (typically, Wednesdays).
Guests are welcome to participate as they wish.


Week 1 (September 11-15)
W: Overview
R: Convex bodies. The Brunn-Minkowski inequality
Week 2 (September 18-22)
W: Steiner symmetrization
R: Proof of Brunn-Minkowski (by "competing symmetries")
Week 3 (September 25-29)
W: The space of convex bodies: Compactness, Hausdorff vs. Nikodym distance
R: Two-point symmetrization and isoperimetric inequality on spheres
Week 4 (October 2-6)
W: Justin Ko: Isoperimetry and concentration on Gauss space
R: Support function and polar bodies
Week 5 (October 9-13)
W: Applications of Brunn-Minkowski
R: Christian Despres: John's ellipsoid
Classical positions of convex bodies

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