Minicourse 'Introduction to stable groups'
Room: Frank Adams 2 (Alan Turing Building)
Lecture notes
Schedule:
Tuesday 27 September, 10am-11am: lecture
Wednesday 28 September, 10am-12pm: lecture
Tuesday 4 October, 10am-11am: problem session Problem List 1
Wednesday 5 October, 10am-12pm: lecture
Tuesday 11 October, 10am-11am: problem session Problem List 2
Wednesday 12 October, 10am-12pm: lecture
Tuesday 18 October, 10am-11am: problem session Problem List 3
Wednesday 19 October, 10am-12pm: lecture
Tuesday 25 October, 10am-11am: problem session Problem List 4
Wednesday 26 October, 10am-12pm: lecture
Course Description:
The aim of the course is to survey the main results on groups satisfying the model-theoretic property of stability and its strengthenings such as ω-stability and finiteness of Morley rank. The main question about groups of finite Morley rank, so-called Cherlin-Zilber conjecture, asks if every simple group of finite Morley rank is an algebraic group over an algebraically closed field. Despite this main question being still open in rank higher than 3, the study of groups of finite Morley rank has brought about a number of deep structural results, e.g. a striking theorem of Zilber saying that a connected solvable non-nilpotent group of finite Morley rank interprets an infinite field. We will discuss some of these fundamental results on groups of finite Morley rank along with some results on stable groups in broader generality. In particular, we will discuss the concepts of a connected component and a generic element in a stable group, which may be viewed as generalisations of the familiar concepts from the theory of algebraic groups to much wider contexts.
Prerequisites: Basic knowledge of model theory; basic knowledge of group theory (recommended).
Syllabus:
1. Preliminaries: Stable theories, ω-stable theories, Morley rank, central series of a group, solvability, nilpotence, group actions. Basic properties of stable groups: chain conditions, connected components. Examples of stable groups.
2. Zilber’s indecomposability theorem. Binding groups.
3. ω-stable fields.
4. Groups of finite Morley rank.
5. Generics, connected components, and local ranks in stable groups.
References:
B. Poizat, Stable groups,
American Mathematical Society, 2001.
F. Wagner, Stable groups, Cambridge University Press, 2013.
Related MAGIC courses:
MAGIC004 - Applications of model theory to algebra and geometry