Large scale geometry (a.k.a coarse geometry) is the paradigm of studying metric spaces from "far away'', neglecting all the local information and just focusing on features of the space at infinity. The aim of this school is to introduce terminology and techniques from large scale geometry and to show how they can be used in a variety of contexts. These include (but are not limited to) geometric group theory, index theory, operator algebras, (geometric) topology.
The school will feature four minicourses by leading experts in the field, each of them consisting of four 50-minutes lectures plus one exercise/discussion session. The minicourses will be aimed at a broad audience.
Young participants are also encouraged to present their research with a short talk in a "gong-talk" session.
One way of studying infinite groups is by analysing their actions on classes of interesting spaces. One such class is that of Hilbert spaces, and important properties, such as Kazhdan's property (T) and a-T-menability, can be defined via actions on Hilbert spaces. In recent years, these properties have been reformulated in terms of actions on Banach spaces, with very interesting results. This mini-course will overview some of these reformulations and their applications. In particular, it will describe various notions of spectrum providing an optimal way to measure ``the strength'' of the property (T) that an infinite group may have, and what can be said about this spectrum, in particular for hyperbolic groups. It will also describe weak versions of a-T-menability, some of which hold for (acylindrically) hyperbolic groups and for mapping class groups.
Gromov-hyperbolic spaces were defined by Gromov in the 80s abstracting a simple, but as it turns out crucial property of geodesic triangles in Riemannian manifolds of negative sectional curvature. Nowadays hyperbolic spaces and groups are very prominent in geometric group theory, with hyperbolic groups including quite a few groups of interest such as free groups, surface groups, and fundamental groups of compact hyperbolic manifolds. I will introduce hyperbolicity, and discuss a few of the properties of hyperbolic spaces that make the theory so rich. I will also talk about recent research directions regarding generalisations of hyperbolic spaces and groups.
The Roe algebra is a C*-algebra encoding the large scale geometry of a metric space. The K-theory of this algebra is a coarse geometric homology invariant of the space and the coarse Baum-Connes conjecture postulates an answer for this invariant. This mini-course will introduce the Roe algebra and its geometric properties, along with the coarse K-homology for a space, and we will discuss how this connects with other versions of the Baum-Connes conjecture. We will also discuss positive results about the conjecture and as well as known obstructions and counterexamples.
I will give an introduction to recent developments in large scale geometry and its applications to manifold topology and geometric analysis. In particular, I will discuss applications of large scale geometry of infinite dimensional spaces to the Novikov conjecture for diffeomorphism groups and spectral analysis of differential operators. I will pose open questions and make every effort to make these lectures accessible to beginners.
Closest airports: Frankfurt Main International, Hannover
Göttingen is well connected via fast trains to most major cities in Germany. The travel time from Frankfurt is about 2h and from Hannover less than 1h.
Walking from the train station to the Mathematical Institut takes no more than 20 minutes passing through the town centre. Most hotels are in walking distance from the main building, however there is also a chance of using the buses.