Julian Kranz

Cantor Minimal Systems Seminar

In the summer term 2022, Rafaela Gesing, Petr Naryshkin, Grigoris Kopsacheilis and I run a seminar on Putnam's book Cantor Minimal Systems. A physical copy of the book is available in the library. We meet every Thursday at 14:00 in SRZ 214, starting on April 7.

Abstract

Cantor minimal systems (i.e. minimal homeomorphisms of the Cantor set) are a particularly simple class of dynamical systems and can be classified up to orbit equivalence (i.e. up to homeomorphisms of the space mapping orbits to orbits) by a simple algebraic invariant. Along the way, we will study fascinating interactions between AF-equivalence relations, Bratteli diagrams, and dimension groups. The motivation for this theory comes from the classification of C*-algebras by K-theory. Although the book does not mention C*-algebras or K-theory, we plan to take a detour and learn all the relevant tools to classify AF-algebras and crossed products of Cantor minimal systems. We will discuss these applications in the last talk.

The required prerequisites are basic knowledge in point-set topology, elementary algebra, and elementary measure theory. The seminar is aimed mainly at Master- and PhD students. However, it should also be accessible to Bachelor students who have attended courses such as Introduction to Algebra and Analysis, Topology and Geometry, or Functional Analysis.

Preliminary meeting

We will have a preliminary meeting for distributing talks on March 21 at 14.00 in the seminar room on the first floor of Orleans-Ring 10 (first room on the right on the floor above the common room of the Cluster). If you are interested in attending the seminar, please write an email to one of the organizers. Please also indicate if you want to get credit points from the seminar for your Bachelor/Master studies.

List of talks

You can find detailed descriptions of the contents of the talks here.

References

  1. I. F. Putnam, Cantor Minimal Systems, University Lecture Series vol. 70, American Mathematical Society, 2018.
  2. T. Giordano, I. F. Putnam, C. F. Skau, Topological orbit equivalence and C*-crossed products, Journal für die reine und angewandte Mathematik 469 (1995), 51-112.
  3. G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), 29-44.