Graduate lectures in mathematics

February 22 - Mai 31, 2018

Sebastian Baader

Alexander Kolpakov, Prof. Dr., Université de Neuchâtel, Alessandro Sisto, Prof. Dr., ETH Zurich

Thursday 14.15-15.45 Prof. Dr. Alexander Kolpakov: Reflection Groups and Coxeter Groups

Thursday 16.00-17-30 Prof. Dr. Alessandro Sisto: Geometric Group Theory

University of Bern, room B13 ExWi, Sidlerstrasse 5, Bern

Alexander Kolpakov: Reflection groups and Coxeter groups

Summary: This course is devoted to the study of very hands-on and simultaneously complicated and fascinating objects: symmetry groups generated by reflections. Such groups are abundant in nature, and represent symmetries of many mathematical and natural objects, be them Platonic solids or biological and chemical structures, such as cells, microorganisms, or molecules. We shall starts with a geometric exploration of reflection groups, their root systems and diagrams, as well as associated geometric complexes (Coxeter complex and Davis complex). We shall continue on classifying finite groups. Then we shall generalise this notion gradually, and see how our techniques evolve (and we will be able to treat even the most general case very similarly to what we started with: finite reflection groups). Finally, we will discuss abstract Coxeter groups, their presentations and representations, and associated geometric structures, such as geometric spaces (spherical, Euclidean and hyperbolic), and geometry of their Coxeter and Davis complexes (including an excursion into CAT spaces). As well, we shall devoted some time to the classification of Coxeter groups of a very widespread and important kind: hyperbolic Coxeter groups (and see their connection with Gromov's word-hyperbolic groups). This course will partially complement "Geometric Group Theory", as well as "Geometric Group Theory" will help understanding some of its more advanced chapters.

Alessandro Sisto: Geometric Group Theory

Summary: Geometric group theory is the study of groups (usually infinite, finitely generated ones) using geometric methods. I will first of all introduce a key object in geometric group theory, namely the Cayley graph, which is a metric space associated to a group and a finite generating set, and I will explain its relation to nice isometric actions of the group itself (through the notion of quasi-isometry and the Milnor-Svarc Lemma).
Most of the course will be devoted to studying an important class of groups called (Gromov-)hyperbolic groups. These include many groups of interest in geometry and topology like free groups, fundamental groups of closed surface of genus at least 2, and more generally fundamental groups of closed manifolds of negative sectional curvature.

under:
math.cuso.ch

15