Vorträge SS 2025

Speaker: Tessa Kammermeier, Universität Hamburg – Invited by Catherine Meusburger

Abstract: The notion of idempotent morphisms admitting splittings is important in algebraic theories such as K-theory, topological quantum field theories and many more. This is because the existence of idempotent splittings corresponds to the existence of a class of colimits that is, in some sense, minimal. These colimits, which are preserved by all functors, are called absolute. In categories where not all idempotents split, we would like take a completion to add only these absolute colimits. This is called the Karoubi or Cauchy completion of a category.

In this talk, I will sketch the proof of why idempotent splittings correspond to absolute colimits and outline the construction of the Karoubi and the Cauchy completion of a category, which will turn out to be equivalent. Furthermore, I will talk about a 2-categorical analogue of the Karoubi completion and, if time permits, talk about how this corresponds to a higher form of absolute colimits.

Speaker: Kang Li, Friedrich-Alexander-Universität Erlangen-Nürnberg – Invited by Karl-Hermann Neeb

Abstract: A C*-algebra is often considered as a non-commutative space, which is justified by the natural duality between the category of commutative C*-algebras and the category of locally compact, Hausdorff spaces. Via this natural duality, we transfer Lebesgue covering dimension on locally compact, Hausdorff spaces to nuclear dimension on commutative C*-algebras. The notion of nuclear dimension for C*-algebras was first introduced by Winter and Zacharias, and it has come to play a central role in the structure and classification for simple nuclear C*-algebras. Indeed, after several decades of work, one of the major achievements in C*-algebra theory was completed: the classification via the Elliott invariant for simple separable C*-algebras with finite nuclear dimension that satisfy the universal coefficient theorem. Unfortunately, simple C*-algebras suffer from a phenomenon of dimension reduction: every simple C*-algebra with finite nuclear dimension must have nuclear dimension at most one. In order to overcome this phenomenon, we (together with Liao and Winter) have introduced the notion of diagonal dimension for an inclusion of C*-algebras, where D is a commutative sub-C*-algebra of A so that this new dimension theory generalizes Lebesgue covering dimension of D and nuclear dimension of A simultaneously. In this talk, I will explain its future impact on the classification of simple nuclear C*-algebras and its connection to dynamic asymptotic dimension introduced by Guentner, Willett and Yu.

Speaker: Ivan Penkov, Constructor University Bremen – Invited by Karl-Hermann Neeb

Abstract: The Lie algebra sl(infty) is a direct limit of the Lie algebras sl(n) for all n. An sl(infty)-module M is integrable if it is locally finite over U(sl(n)) for every n, and M has semilarge annihilator if the annihilator in sl(infty) of every vector from M has finite corank after intersection with a certain fixed reductive subalgebra of sl(infty).

The aim of this talk is to present a conjectural picture for the structure of the category of sl(infty)-modules with the above two properties. In particular we will discuss indecomposable injectives, Ext’s between simple modules, Koszulity etc.

Speaker: Mikael Rørdam, University of Copenhagen – Invited by Kang Li