Sets where a typical Lipschitz function is differentiable

This talk is devoted to the study of properties of Lipschitz functions, mainly
between finite-dimensional spaces,
and especially their differentiability properties.

The classical Rademacher Theorem guarantees that every Lipschitz function
between finite-dimensional spaces is
differentiable almost everywhere. This means that for every set E of positive
Lebesgue measure and for every
Lipschitz function f defined on the whole space the set of points from E where f
is differentiable is non-empty
and is 'much larger' than the set of points where it is not differentiable.

A major direction in geometric measure theory research of the last two decades
has been to explore to what extent
this is true for Lebesgue null sets. Even for real-valued Lipschitz functions,
there are null subsets S of R^n
(with n>1) such that every Lipschitz function on R^n has points of
differentiability in S; one says that S is a
universal differentiability set (UDS).

Moreover, for some sets T which are not UDS, meaning that they are necessarily
Lebesgue null and there are some bad
Lipschitz function nowhere differentiable in T, it may happen that a typical
Lipschitz function has points of differentiability in T.
In a recent joint work with M. Dymond we give a complete characterisation of
such sets: these are the sets which
cannot be covered by an F-sigma 1-purely unrectifiable set. We also show that
for all remaining sets
a typical 1-Lipschitz function is nowhere differentiable, even directionally, at
each point.
In this talk I will present the latest results on UDS and the dichotomy of
typical differentiability sets.