Abstract:
We show the geometry and heat flow on metric measure spaces. We study the gradient
flows ,diffusion semigroups on metric and Wasserstien spaces under lower curvature bounds
and over compact Alexandrov spaces. We also study the heat equation on manifolds as a
gradient flow in the Wasserstien space. Metric characterizations of Riemannian submersions
are considered. We also consider the open map theorem and Trottere product formula
for gradient flows in metric Spaces. Acharacterizationof transportation distance between
the non-negative measures and applications on gradients flows with Dirichlet boundary
conditions are shown. We discuss the measure contraction property and non-contraction
of heat flow on metric measure and Minkowski spaces.
