2013-2014 Bowen Lectures

The 2013-14 Bowen Lectures will be delivered by Jeff Cheeger (Courant Institute of Mathematical Sciences, NYU) on March 4th, 5th, and 6th, 2014. Each lecture begins at 4:10pm and ends at 5:00pm.

Series Title: Quantitative Behavior of Singular Sets

Tuesday March 4th
Lecture 1: Quantitative behavior of singular sets, I
105 North Gate Hall

Wednesday March 5th
Lecture 2: Quantitative behavior of singular sets, II
10 Evans Hall

Thursday March 6th
Lecture 3: Quantitative degeneration of Lipschitz maps from the Heisenberg group to L1
105 North Gate Hall


Series Abstract:

It is well known that solutions to nonlinear elliptic and parabolic pde's such as Einstein metrics, harmonic maps, minimal hypersufaces and mean curvature flows, can have nonempty singular sets. In the first two lectures, we will describe methodology developed jointly with Aaron Naber, which enables one to promote known lower bounds on the Hausdorff codimension of such singular sets, to corresponding upper bounds on the volume of the set of points outside of which the solution has any small but definite degree of regularity. The third lecture concerns joint work with Bruce Kleiner and Assaf Naor, showing that a 1-Lipschitz map from the unit ball in the Heisenberg group with its Carnot-Caratheodory metric, to the Banach space L1, must fail to be bi-Lipschitz in a precise quantitative sense. This leads to a counterexample to a conjecture of Goemans and Linial from theoretical computer science. A common theme in the lectures is the replacement of classical blow-up arguments by a multiscale analysis in which one does not pass to the blow-up limit. A key point is that the special structure that exists in a blow-up limit (a.k.a. a tangent cone) is already present to any given degree of approximation at "most" locations and scales.