Lectures 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: Philip Schlösser, Radboud Universiteit – Invited by Tobias Simon
Abstract: Conformal field theories (CFTs) are quantum field theories that not only possess a symmetry under the Poincaré group, but also under scaling and inversion. This requirement completely determines the shapes of the 2- and 3-point correlation functions. The 4-point functions are then the smallest correlators that are not entirely constrained in their shape. In fact, they are usually written as functions that depend on two coordinates (regardless of the dimension of the space we consider the CFT in) and expanded in terms of solutions to the so-called Casimir equation. This equation was studied extensively by Dolan and Osborn (in 2001, 2004, 2012, e.g.) and was more recently shown (by Isachenkov and Schomerus in 2016) to be related to the Calogero-Sutherland model via the eigenfunction equation for the Heckman-Opdam modified Laplacian.
I will delve into the representation theory that underlies these 4-point functions, make plausible how they can be interpreted as invariants of a four-fold tensor product of non-unitary principal series representations of the conformal group SO(p+1,q+1) (q=0 or 1), and also as matrix-spherical functions of the symmetric pair (SO(p+1,q+1), SO(p,q)\times SO(1,1)). Time permitting, I will also introduce a radial part decomposition for a symmetric pair (G,K) with non-compact K that allows us to explain the connection discovered by Isachenkov and Schomerus in more detail.
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
Speaker: Diego Fernandez Silvestre, Universität Burgos – Invited by Catherine Meusburger
Abstract: Quantum mechanics is not fully quantum. In standard quantum mechanics, the quantum states arising as solutions to the Schrödinger equation for a free particle are invariant under transformations between classical reference frames. These symmetry transformations form a Lie group: the centrally extended Galilei group. In this context, classical reference frames are described as abstract entities with no dynamical or quantum properties. However, actual reference frames must be associated with physical systems, which are ultimately quantum systems. This is the idea behind the concept of quantum reference frames.
In this seminar, I will introduce the concept of quantum reference frame transformations and discuss the Lie group structure underlying them. We will see that this Lie group structure departs significantly from that of the Galilei group. I will then address the problem from a quantum group standpoint and examine whether this Lie group structure can be interpreted as emerging from a Hopf algebra deformation of the Galilei group. I will argue that the appropriate mathematical object to describe quantum reference frame transformations is the so-called universal T-matrix, or Hopf algebra dual form. Finally, I will comment on the mathematical properties of the universal T-matrix and explore whether the Hopf algebra dual form can be understood more generally in the context of the skew-pairing of Hopf algebras.