tcc6.html

The Banach-Tarski Paradox, the von Neumann-Day conjecture

Taught Course Centre (May-June 2019)

Time: Friday 10-12 (except on Friday the 17th of May, when it is 2 pm-4pm). The TCC timetable can be found here.
NB: Due to unforeseen circumstances, the course starts in week 2 (May 10).
Place: Videoconference Room VC1, Mathematical Institute.

Synopsis: I plan to cover the following topics, listed in the order of their appearance in lectures.

The Banach-Tarski Paradox.

Amenable graphs, Amenable groups: equivalent definitions, stability under group operations.

The von Neumann-Day conjecture, Jacques Tits' Alternative Theorem for linear groups.

The course is based on Chapters 15, 17 and 18 of the book "Lectures on Geometric Group Theory", written jointly with Misha Kapovich. A slightly older version of the book is available here.

Recommended reference for algebraic groups (unfortunately out of print): the book of Onishchik and Vinberg.

Lecture notes:

Lecture 1 (slides).

Lecture 1 (whiteboard notes).

Lecture 2 (slides).

Lecture 2 (whiteboard notes).

Lecture 3 (slides).

Lecture 3 (whiteboard notes).

Exercise Sheet 1. For any type of assessment (broadening courses or others) answers must be sent in by email before June 12 (details on the sheet).

Lecture 4 (slides).

Lecture 4 (whiteboard notes).

Lecture 5 (slides).