A Universal Differentiability Set, UDS for short, is a set such that
every Lipschitz function is differentiable in at least one point of the
set. The existence of measure zero UDS, which is strictly related to
Rademacher's theorem, has already been established in the Euclidean
setting and for certain classes of Carnot groups. The key technique
underlying the construction of UDS is a result that relates maximality
of a directional derivative for a Lipschitz function with
differentiability. In this seminar we will investigate this connection
and we will try to establish how things change when we work in a
nonlinear setting. To do so we will focus our attention on the Laakso
space, one of the best known and most interesting instances of a metric
measure space which admits a differentiable structure but not a linear
structure. An introduction to Laakso spaces and their structure will be
provided during the seminar.