Convexity in Hadamard spaces

Hadamard spaces are metric spaces of nonpositive curvature, where curvature is defined as a very simple feature of the metric, and unlike in Riemannian geometry, no differentiable structure is needed. Even though Hadamard space theory has become classical part of geometry, it is still a very active area of mathematical research.

The aim of this course is to systematically develop the theory of Hadamard spaces with a special emphasis on convexity. It turns out that Hadamard spaces are a natural nonlinear setting for convex sets and convex functions. We will see that many of the convexity results in linear spaces can be extended to Hadamard spaces. Although we will be primarily concerned with analytical aspects of convexity, there will be a strong interplay with geometry and probability as well. After building the rudiments of Hadamard space theory, we will get to more advanced topics such as resolvents and gradient flow semigroups, convex optimization, martingale theory, Markov processes, etc. A nice application to biology will be presented in the end of the course.

The course is intended to be rather elementary. No preliminary knowledge is expected.

Date and time info
Tuesday, 11:00 - 12:30 h