# 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 L _{1}*

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 L_{1}, 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.