Analys- och sannolikhetsseminarium

Abstrakt finns enbart på engelska: In this talk I will explain how algebraic invariants can be enriched with additional "complexity-theoretic" information. This approach allows one to "extract" from such invariant a whole sequence of new "phantom" invariants indexed by countable ordinals. These new invariants allow us to obtain finer classification results, which measure how difficult it is to build the objects under considerations starting from some "elementary" building blocks.

I will focus on applications on the theory of operator algebras and C*-dynamics, considering invariants such as the Picard group, the outer automorphism group, (Banach algebra) cohomology, and Kasparov's KK-theory.

I will also explain how the phantom invariants extracted from each of these yields a corresponding notion of "length". This is yet another (ordinal-valued) invariant, which can be seen as a measure of complexity. For instance, the length associated with the automorphism group can be seen as a "measure of noncommutativity" of a C*-algebra, and correspond to the complexity of determining whether a given automorphism is inner. The problem of computing this and other notions of length vastly generalizes several classical problems in analysis, such as the problem of determining when inner automorphisms are closed in the topology of pointwise or uniform convergence.