Finite group schemes are generalizations of finite groups in an algebro-geometric setting. They occur naturally in algebraic geometry and algebraic groups in positive characteristics (for example, as Frobenius kernels) and they are closely related to restricted Lie algebras. The lectures will first give an introduction to the general theory of finite group schemes over an algebraically closed field, with few prerequisites. Then we will discuss actions on algebraic curves and a partial solution to the inverse Galois problem for infinitesimal group schemes.
Discrete subgroups of SL(2,R) are well understood, and classified by the geometry of the corresponding hyperbolic surfaces. On the other hand, discrete subgroups of higher-rank semisimple Lie groups, such as SL(n,R) for n>2, remain more mysterious. While lattices in this setting are rigid, there also exist more flexible “thinner” discrete subgroups, which may have large and interesting deformation spaces, giving rise in particular to so-called higher Teichmüller theory. We will survey recent progress in constructing and understanding such discrete subgroups from a geometric and dynamical viewpoint.
The goal of this series of lectures is to discuss the interplay between descriptive set theory and the study of geometric and model-theoretic properties of groups. We will begin by covering the necessary background from topology and logic. After that, we will focus on applications of descriptive methods to the study of first-order rigidity, asymptotic invariants, and quasi-isometric diversity of finitely generated groups.
Given the current knowledge of complex representations of finite (quasi)simple groups, obtaining good upper bounds for their characters values still remains a difficult problem, a satisfactory solution of which would have significant implications in a number of applications. We will report on recent results that produce such character bounds, and discuss some such applications, in and outside of group theory.
One Hour Speakers
The space of ends of a groups was defined by Freudenthal in 1930 for finitely generated groups, and the definition was extended by Specker in 1950 to arbitrary groups. Freudenthal and Hopf proved in the 1940s that each finitely generated group the space of ends is either empty, a singleton, two points, or a Cantor set, and furthermore they characterized 2-ended groups as infinite virtually cyclic groups. Specker then proved that every infinitely generated group has either 1 or infinitely many ends. Groups with infinitely many ends were characterized by Stallings in 1968 in terms of amalgams/HNN extensions, and this was extended to arbitrary groups by Dicks and Dunwoody to all non-locally-finite groups; nevertheless, this does not describe the topology of the space when it is infinite.
This work is concerned with the problem of determining the topology of the space of ends of a group with infintely many ends. We check that it has no isolated point. Unlike the case of finitely generated groups, it is not metrizable in general, e.g., for a free group of infinite rank. We present some topological alternatives for these spaces, discussing separability and cellularity (maximal number of pairwise disjoint open subsets).
What do polycyclic groups look like and how can one compute with such groups? The first part of this talk contains a survey of well-known algorithms for this purpose. Then the talk discusses some open problems in this research area and recent advances towards them. In particular, a new algorithm to determine the Frattini subgroup of a polycyclic group is exhibited (joint work with Matthias Neumann-Brosig).
Tensor triangular geometry is the study of tensor triangulated categories via geometric methods. I’ll give a brief (and partial!) introduction to its axiomatics and then focus on the applications to finite tensor categories arising from modular representations. While classifying indecomposable modules in these categories is often an insurmountable task, tensor triangular geometry allows to bring some order and structure into this wild territory, particularly via the geometric classification of tensor ideals.
Bass-Serre Theory is a powerful tool for decomposing groups acting on trees, but its usefulness for constructing non-discrete groups acting on trees is severely limited. Such groups play an important role in the theory of locally compact groups, as they are a rich source of examples of nonlinear simple groups. An alternative `local-to-global’ approach to the study of groups acting on trees has recently emerged based on groups that are `universal’ with respect to some specified `local’ action (i.e., the action of a vertex stabiliser on neighbouring vertices).
In this talk I will discuss some prominent local-to-global constructions for groups acting on trees. I will then introduce some joint work with Colin Reid, in which we aim to advance the local-to-global theory of groups acting on trees by developing a ‘local action’ complement to classical Bass-Serre theory. We call this the theory of local action diagrams. The theory is powerful enough to completely describe all closed groups of automorphisms of trees that enjoy Tits’ Independence Property (P).