Aristotelis Panagiotopoulos: Dynamical obstructions to classification II
Donnerstag, 22.10.2020 10:30 im Raum SRZ 216/217
One of the leading questions in many mathematical research
programs is whether a certain classification problem admits a
'satisfactory' solution. What constitutes a satisfactory solution depends,
of course, on the context and it is often subject to change when the
original goals are deemed hopeless. Indeed, in recent years several
negative anti-classification results have been attained. For example: by
the work of Hjorth and Foreman-Weiss we know that one cannot classify all
ergodic measure-preserving transformations using isomorphism types of
countable structures as invariants; and by the work of Thomas we know that
higher rank torsion-free abelian groups do not admit a simple
classification using Baer-invariants such as in the rank-1 case.
In this talk I will provide a gentle introduction to *Invariant Descriptive
Set Theory*: a formal framework for measuring the complexity of such
classification problems and for showing which types of invariants are
inadequate for a complete classification. In the process, I will present
several anti-classification criteria which come from topological dynamics.
In particular, I will discuss my recent joint work with Shaun Allison, in
which we provide a new obstruction to classification by (co)homological
invariants and use it to attain anti-classification results for Morita
equivalence of continuous-trace C*-algebras and for the isomorphism problem
of Hermitian line bundles.