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.
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