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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.
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