This semester’s RTG lecture will be split between courses by Karin Melnick on “Introduction to parabolic geometries”, by Beatrice Pozzetti on “Incidence structures on flag varieties and rigidity”, who will cover the morning session, and by Petra Schwer on “CAT(0) groups and geometry”, who will lecture in the afternoon.

**Introduction to parabolic geometries
(Karin Melnick)**

Abstract: As with semisimple Lie groups, the subject of parabolic geometries matches a beautiful general theory with an equally important variety of interesting examples. The mini-course will start with a discussion of parabolic homogeneous model spaces G/P and the G-invariant geometric structures they carry. Then we will define parabolic geometries, which are "infinitesimally modeled" on such homogeneous spaces. We will describe the examples of conformal, projective, and CR structures, including certain of their distinguished curves --- conformal and projective geodesics, or CR chains, respectively.

The second half of the course will focus on automorphisms of parabolic geometries. Via the Cartan connection and distinguished curves, it is possible to obtain a useful local picture of automorphisms from infinitesimal data. I plan to conclude with an illustration of these methods by proving a rigidity theorem, a local version of the Schoen-Webster theorem, about local automorphisms of strictly pseudoconvex CR structures. An aim of the course is to introduce some of the geometry on parabolic homogeneous spaces of SU(m,n), in order to prepare for B. Pozetti's subsequent mini-course.

**Incidence structures on flag varieties and rigidity
(Beatrice Pozzetti)**

Abstract: As a partial continuation of K. Melnick's minicourse we will see how rigidity properties of flat parabolic geometries can be used to deduce rigidity for homomorphisms from lattices to Lie groups. In the three lectures we will focus on the case when the lattice is the fundamental group of a manifold covered by the complex hyperbolic space, and the homomorphism takes values in SU(p,q). We will define a notion of maximality for such homomorphisms, discuss how we can relate it to properties of maps between suitable flat parabolic geometries (or flag varieties), and show the following rigidity result: all maximal representations are restriction of representations from SU(1,n).

**CAT(0) groups and geometry
(Petra Schwer)**

Abstract: The CAT(k) property is a way to measure curvature of a metric space by comparing its triangles to triangles appearing in a model space of constant sectional curvature k.

Intuitively triangles in a CAT(0) space are thinner than their comparison triangles in the Euclidean plane.

As we will see the class of CAT(0) spaces is rather rich and includes smooth objects like Riemannian manifolds as well as more discrete objects like graphs or certain simplicial complexes. More concrete examples are symmetric spaces of non-compact type or buildings.

During this course we will discuss boundary constructions for CAT(0) spaces, criteria for a space to be CAT(0) as well as consequences for groups acting nicely on such a space. In addition will study properties of individual isometries of CAT(0) spaces. If time permits we will see more recent applications and open questions related to the topic.

*Here are the scans of my handwritten notes: *

First session on Basics and examples.

Second session The Bruhat-Tits fixedpoint theorem.

Third session on Gromov's link condition and other ways to test for the CAT(0) property.

Fourth session on CAT(0) groups and Isometries of CAT(0) spaces.

Fifth session on The visual boundary and its relatives.