Tag - Riemannian geometry

This is a 22-lecture course, with each lecture being between one and two hours, given by Nikolay Bogachev.

Modern research in the geometry, topology, and group theory often combines geometric, arithmetic and dynamical aspects of discrete groups. This course is mostly devoted to hyperbolic manifolds and orbifolds, but also will deal with the general theory of discrete subgroups of Lie groups and arithmetic groups. Vinberg's theory of hyperbolic reflection groups will also be discussed, as it provides a lot of interesting examples and methods that turn out to be very practical. One of the goals of this course is to sketch the proof of the famous Mostow rigidity theorem via ergodic methods. Another goal is to talk about very recent results, giving a geometric characterization of arithmetic hyperbolic manifolds through their totally geodesic subspaces, and their applications. Throughout the course we will consider many examples from reflection groups and low-dimensional geometry and topology. In conclusion, I am going to provide a list of open problems related to this course.

This is a 24-lecture course, with each lecture being about 80 minutes, given by Spiro Karigiannis.

This is a second course in Riemannian geometry. The emphasis will be on the intimate relationship between curvature and geodesics.

This is a 22-lecture course, with each lecture being about 90 minutes, given by Nicola Gigli. Note that the 14th and 18th lectures were not recorded.

Created by Professor Nicola Gigli, the aim of the course is to provide an introduction to the world of synthetic description of lower Ricci curvature bounds, which has seen a tremendous amount of activity in the last decade: by the end of the lectures the student will have a clear idea of the backbone of the subject and will be able to navigate through the relevant literature.

We shall start by studying Sobolev functions on metric measure spaces and the notion of heat flow. Then following, and generalizing, the intuitions of Jordan-Kinderlehrer-Otto we shall see that such heat flow can be equivalently characterized as gradient flow of the Cheeger-Dirichlet energy on L² and as gradient flow of the Boltzmann-Shannon entropy w.r.t. the optimal transportation metric W₂. This provides a crucial link between the Lott-Villani-Sturm (LSV) condition and Sobolev calculus on metric measure spaces and, in particular, it justifies the introduction of 'infinitesimally Hilbertian' spaces as those metric measure structures for which W^1;2(X) is a Hilbert space. By further developing calculus on these spaces we shall see that on infinitesimally Hilbertian spaces satisfying the LSV condition (these are called Riemannian curvature dimension spaces, or RCD for short) the Bochner inequality holds. 

We shall then discuss more sophisticated calculus tools, such as the concept of differential of a Sobolev function, that of vector field on a metric measure spaces and the notion of Regular Lagrangian Flow on RCD spaces.

We shall finally see how these are linked to the lower Ricci curvature bound - most notably we shall prove the Laplacian comparison theorem - and finally how they can be used to prove a geometric rigidity result like the splitting theorem for RCD spaces. It is worth noticing that such statement gives new information - compared to those available through Cheeger-Colding's theory of Ricci-limit spaces - even about the structure of smooth Riemannian manifolds.

Prerequisites: some familiarity with Riemannian geometry and optimal transport theory in the case cost=distance² is preferred, but not required: I shall provide the necessary background when needed.