The Lecture Notes of the courses and further information
will also be posted.
The course will focus on some recent
concerning in particular
the optimal transport theory, the variational approach to
nonlinear evolution equations, functional
and differential geometry.
Deep results have been obtained during the last decade:
a brief (and largely incomplete) account of the main topics,
considered in the present course,
Distances between probability measures
induced by optimal transportation problems,
according to the formulations of
Kantorovich-Rubinstein-Wasserstein, and their link
with Hamilton-Jacobi equations.
The construction of an
``infinite dimensional (almost) Riemannian''
structure on measures and the interpretation
of many (nonlinear) diffusion PDE's as gradient flows.
The link between optimal transport,
kinetic formulations, and statistical mechanics.
Entropy/Entropy dissipation methods for studying nonlinear
evolution equations and functional inequalities.
The interplay between geometric properties of the underlying
domains (usually ``nice'' Riemannian manifolds) and
the probability spaces constructed on them.
Geometric inequalities in the framework of
non-smooth Metric-Measure Spaces.
The metric/energetic theory
for gradient flows and rate-invariant evolution problems.