THEMATIC PROGRAMS

October 20, 2014

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th ANNIVERSARY YEAR

Distinguished Lecture Series
August 28,29,30 at 3:30 p.m.
Assaf Naor
Courant Institute of Mathematical Sciences
New York University

The Lipschitz Extension Problem


The Lipschitz extension problem asks for geometric conditions on a pair of metric spaces X and Y implying that there exists a positive constant K such that for every subset A of X, every L-Lipschitz function f from A to Y can be extended to a (KL)-Lipschitz function defined on all of X. When Y is the real line then this is always possible with K=1 (the nonlinear Hahn-Banach theorem), in which case one asks for an extension of f with additional desirable properties. For general metric spaces X,Y it is usually the case that no such K exists. However, many deep investigations over the past century have revealed that in important special cases the Lipschitz extension problem does have a positive answer. Proofs of such theorems involve methods from a variety of mathematical disciplines, and when available, a positive solution to the Lipschitz extension problem often has powerful applications. The first talk will be an introduction intended for non-experts, giving an overview of the known Lipschitz extension theorems, and an example or two of the varied methods with which such theorems are proved. The following two lectures will deal with more specialized topics, including the use of probabilistic methods, some illuminating counterexamples, examples of applications, and basic problems that remain
open.



Speakers in the Distinguished Lecture Series (DLS) have made outstanding contributions to their field of mathematics. The DLS consists of a series of three one-hour lectures.

Index of Fields Distinguished and Coxeter Lectures