### Differentiability of Lipschitz functions with respect to singular
measures and related questions

By Rademacher theorem every Lipschitz function on the Euclidean space
is differentiable almost everywhere with respect to the Lebesgue
measure. In these lectures I will explain how this statement should
be modified when the Lebesgue measure is replaced by a singular
(bounded) measure: it turns out that in this case the
differentiability properties of Lipschitz functions are exactly
described by the decompositions of the measure in terms of
one-dimensional rectifiable measures. This leads to the problem of
understanding the structure of measures that admits many of these
decompositions; to this question, however, there are only partial
answers.

The results I will present are directly related to recent work by
many authors, including myself, David Bate, Marianna Csornyei, Peter
Jones, Andrea Marchese, and David Preiss.