Workshop NCG Day 2025

Thursday August 21, 10:00-17:00 in BW0.20

10:00-11:00 — Chris Bourne (Nagoya): A C*-module framework for interfaces of discrete quantum systems

11:00-12:00 — Lars Koekenbier (Erlangen): A transfer matrix analysis of the asymptotic spectra of Toeplitz matrices and their perturbations

12:15-13:15: Lunch

13:15-14:15 — Yuezhao Li (Leiden): On topological phases of aperiodic matter

14:15-15:00 — Aaron Kettner (Prague): Cuntz-Pimsner algebras of homeomorphisms twisted by vector bundles

15.15-16:00 — Akhila Nelliyamkunnath Satheesan (Prague): Weighted quantum projective spaces as graph C*-algebras

16:00-17:00 — Jesse Reimann (Delft): Split exact sequences and KK-equivalences of quantum flag manifolds

17:00 – Drinks at bar “De Fusie” (inside the Gorlaeus buidling).

Abstracts:

Chris Bourne (Nagoya) A C*-module framework for interfaces of discrete quantum systems

Loosely speaking, an interface describes a spatial mixing of several systems described by a C-algebra of observables such that these mixed dynamics are not felt ‘at infinity’. Building from work by Măntoiu, we give a mathematical description of (discrete) interfaces using C-modules and (discrete) crossed products. Several examples will be given as well as spectral and index-theoretic properties.

Lars Koekenbier (Erlangen) A transfer matrix analysis of the asymptotic spectra of Toeplitz matrices and their perturbations

In this talk I will show how transfer matrix techniques can be used to compute the asymptotic spectra of non-Hermitian tridiagonal finite-block Toeplitz matrices. In this way one can recover Widoms results on the asymptotic spectra of such matrices. Special attention will be given to topological eigenvalues arising from matrices with a chiral symmetry and the associated bulk-boundary correspondence. Going beyond Widoms theory, I will also show how the transfer matrix approach can be used to compute the asymptotic spectra of Toeplitz matrices with a perturbation on a finite number of sites. One can then see how the different parts of the spectra depend on the perturbations. Varying the ranks of the perturbations one can now also interpolate between open and closed boundary conditions. The results will be illustrated by numerics.

Yuezhao Li (Leiden), On topological phases of aperiodic matter

It has been known (thanks to the work of Bellissard) that topological materials living on an aperiodic point pattern ought to be modelled by a groupoid C*-algebra. Its structural properties and K-theory encode the topological information of such materials. Precise computation has been carried out in the comprehensive work of Bourne and Mesland. I shall provide some simple observations from the viewpoint of coarse geometry and topological graphs, which say something about the robustness of topological phases of aperiodic matter, and about the special features in dimension one.

Aaron Kettner (Prague), Cuntz-Pimsner algebras of homeomorphisms twisted by vector bundles

I will talk about a construction that takes a vector bundle together with a (partial) homeomorphism on the bundles base space, and produces a C*-algebra. This provides a class of examples that is both tractable, as well as potentially quite large. Under reasonable assumptions, these algebras are classifiable in the sense of the Elliott program. If time permits I will sketch some K-theory calculations, which are work in progress.

Akhila Nelliyamkunnath Satheesan (Prague), Weighted quantum projective spaces as graph C*-algebras

Weighted quantum projective spaces are defined as fixed point algebras of the quantum sphere with respect to a weighted U(1) action. In this talk, I will sketch how to find graph models for a certain class of such weighted quantum projective spaces. I will give a complete list of their irreducible representations, as well as a description of the topology on the primitive ideal space. The graph can be constructed using the results of Eilers-Ruiz-Sorensen on amplified graph C*-algebras.

Jesse Reimann (Delft), Split exact sequences and KK-equivalences of quantum flag manifolds

In this talk, I will discuss how the construction of explicit split exact sequences between quantum projective spaces of Arici and Zegers extends to general quantum flag manifolds, based on the graph description of quantum flag manifolds of Brzeziński, Krähmer, Ó Buachalla, and Strung. This leads to a KK-equivalence between quantum flag manifolds and multiple copies of the complex numbers. As the graphs of general quantum flag manifolds can become quite large, we explore options of stating this explicit construction in terms of operations on the associated Dynkin diagrams.

Based on ongoing work with Enli Chen and Sophie Zegers.

Sponsors

This workshop is partially supported by the Netherlands Organisation of Scientific Research (NWO) under the VIDI grant The noncommutative geometry of bounded symmetric domains