THEMATIC PROGRAMS

October 20, 2014

Thematic Program on Asymptotic Geometric Analysis
July - December 2010

Graduate Courses held at the Fields Institute
September - December, 2010


All courses will be held at the Fields Institute, Room 230 unless otherwise noted.

Isoperimetric Inequalities and Applications to Asymptotic Geometric Analysis
Instructor : Emanuel Milman
Graduate Course on Isoperimetric Inequalities and Applications to Asymptotic Geometric Analysis

Instructor : Emanuel Milman

Graduate Mini-Course
An introduction to infinity harmonic functions

Instructor :Scott Armstrong (University of Chicago) and Charles Smart (New York University)

Optimal Transportation Minicourse:
Geometry, Regularity and Applications

Instructor :Robert McCann (University of Toronto)

Guest Lecture
Frank Morgan (Williams College)

Graduate Course Isoperimetric Inequalities and Applications to Asymptotic Geometric Analysis
Instructor : Emanuel Milman
Tuesday 2-4 pm & Thursday 10-12 pm

November 16 & 18
Guest lectures by Alexander Litvak (University of Alberta)

1. Classical Isoperimetric Inequalities on Euclidean, Spherical, and Gauss Spaces (2 weeks)

- definition of perimeter and boundary measure.
- proofs by various methods (symmetrization, combinatorial, PDE, Convexity).
- connection to Brunn-Minkowski inequality.

2. Isoperimetric Inequalities on Riemannian Manifolds (2 weeks)

- refresher on Riemannian Geometry and comparison theorems.
- background on Integral Currents.
- the Levy-Gromov isoperimetric inequality.
- if time - the Bakry-Ledoux isoperimetric inequality.

3. Optimal Transport (2 weeks)

- Background on existence, uniqueness and regularity of optimal-transport map.
- McCann's proof of the Euclidean isoperimetric inequality using optimal-transport.
- 1 week advanced topic TBD.

4. Applications in Asymptotic Geometric Analysis (3 weeks)

- Classical Convex Geometry.
- Dvoretsky's theorem.
- Quotient-of-Subspace Theorem.
- Other applications time permitting.


GRADUATE MINICOURSE:
An introduction to infinity harmonic functions


Scott Armstrong (University of Chicago) and Charles Smart (New York University)

Tuesday Oct 26 2:10 - 4:00 Room 230
Thursday Oct 28 10:10 - 12:00 3rd Floor Stewart Library

In this series of lectures, we outline the basics of the theory of infinity harmonic functions, also known as (absolutely) minimal Lipschitz extensions. Armstrong will use the first half of each meeting to discuss the existence and uniqueness of infinity harmonic functions as well as connections to tug-of-war games and the infinity calculus of variations. Smart will talk in the second half of both meetings and present the regularity theory, including the recent result of Evans-Smart on the pointwise differentiability of infinity harmonic functions. The parallel presentations will have contrasting points of view, since it is only the regularity theory that seems to require any PDE methods.


Oct. 19 and 21
Optimal Transportation Minicourse:
Geometry, Regularity and Applications

by Robert McCann (University of Toronto)

In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another,  where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and applications will be discussed, following the notes [47] of McCann and Guillen at http://www.math.toronto.edu/mccann/

Oct 19: The first two hours of the course will establish the basic duality result of optimal transportation:  the existence, uniqueness and characterization of convex gradient maps, as well as optimal maps for more general costs,  and discuss their connection to inequalities and to partial differential equations.

Oct 21: The last two hours of the course will focus on recent developments in the regularity theory for solutions to Monge-Ampere type equations, including a self-contained proof of a Hoelder continuity result for optimal maps. It will also offer a geometric perspective on connections of this problem to questions in geometric measure theory and general relativity such as finding high codimension minimal / maximal surfaces in a suitable (pseudo-Riemannian and symplectic) setting.

Thursday October 7 - Guest Lecture by Frank Morgan (Williams College)

"Soap Bubbles and Mathematics"
Along with the mathematics, there will be a little guessing contest with prizes.

 

Taking the Institute's Courses for Credit
As graduate students at any of the Institute's University Partners, you may discuss the possibility of obtaining a credit for one or more courses in this lecture series with your home university graduate officer and the course instructor. Assigned reading and related projects may be arranged for the benefit of students requiring these courses for credit.