NCG Day 2022

On December 2nd, 2022 we will have a small workshop at Leiden University on Noncommutative Geometry and its applications.

The event will take place in the new Gorlaeus Building room DM021PC, from 9:30-17:00.

Programme

9:30 - 10:15: Prof. dr. Walter van Suijlekom (Nijmegen), Noncommutative spaces at finite resolution

10:45 - 11:30: Malte Leimbach (Nijmegen), On spectral truncations of the torus

11:45 - 12:30: Dr. Dimitris Gerontogiannis (Leiden), A geometric Fredholm module representation of the fundamental class for Smale spaces

14:00 -14:45: Dr. Bram Mesland (Leiden), Existence of Levi-Civita connections in noncommutative geometry

15:00-15:45: Dr. Adam Rennie (Wollongong), Uniqueness and curvature of Levi-Civita connections in noncommutative geometry

16:15-17:00: Dr. Peter Hochs (Nijmegen), C*-algebras of symmetric spaces

Abstracts

Walter van Suijlekom, Noncommutative geometry at finite resolution

We extend the traditional framework of noncommutative geometry in order to deal with two types of approximation of metric spaces. On the one hand, we consider spectral truncations of geometric spaces, while on the other hand, we consider metric spaces up to a finite resolution. In our approach the traditional role played by $\text{C}^\ast$-algebras is taken over by so-called operator systems. We consider $\text{C}^\ast$-envelopes and introduce a propagation number for operator systems, which we show to be an invariant under stable equivalence and use it to compare approximations of the same space. We illustrate our methods for concrete examples obtained by spectral truncations of the circle, and of metric spaces up to finite resolution.

Malte Leimbach, On spectral truncations of the torus

It was shown by van Suijlekom that the metric spaces obtained from spectral truncations of the circle converge in Gromov-Hausdorff distance. This was done by establishing a so-called $\mathrm{C}^1$-approximate order isomorphism between the algebra of smooth functions on the circle and its compression by the spectral projection - the Toeplitz operator system. We investigate whether similar methods can still be applied in the case of the torus by discussing a candidate for a $\mathrm{C}^1$-approximate order isomorphism. On the way to establishing its properties, we explore connections with the theory of Fourier and Schur multipliers as well as with the lattice point counting problem. We will give a preliminary answer to the above mentioned question for low-dimensions. This is joint work with Walter van Suijlekom.

Dimitris Gerontogiannis, A geometric Fredholm module representation of the fundamental class for Smale spaces

Ruelle algebras are purely infinite $\text{C}^\ast$-algebras from Smale spaces and analogues of higher dimensional Cuntz-Krieger algebras. More than a decade ago, Jerry Kaminker, Ian Putnam and Mike Whittaker proved that Ruelle algebras exhibit Poincaré duality in KK-theory. The fundamental class was represented by an extension by the compact operators. This talk presents a geometric Fredholm module representation of the fundamental class. This work opens the window to Lefschetz fixed point theorems for Ruelle algebras. The construction’s backbone is Markov partitions. This is joint work with Mike Whittaker and Joachim Zacharias.

Bram Mesland, Existence of Levi-Civita connections in noncommutative geometry

In this talk I will discuss the construction of Hermitian and torsion-free connections associated to a spectral triple, under mild “geometric” assumptions. In particular I will explain an unexpected link with the “two-projection-problem” in Hilbert $\text{C}^\ast$-modules. Joint work with Adam Rennie.

Adam Rennie, Uniqueness and curvature of Levi-Civita connections in noncommutative geometry

After giving sufficient conditions for uniqueness of Hermitian and torsion-free connections, I will give some examples, along with their curvatures. Joint work with Bram Mesland

Peter Hochs, C*-algebras of symmetric spaces

Let $G$ be a unimodular locally compact group, and $\widehat G$ the set of equivalence classes of unitary, irreducible representations of $G$. In the abstract Plancherel theorem, the representation $L^2(G)$ of $G \times G$ is decomposed as a direct integral over $\widehat G$ with respect to a measure. In this way, $\widehat G$ is viewed as a measure space. It can also be viewed as a topological space. Its topological properties are encoded by the reduced group $\text{C}^\ast$-algebra of $G$. Harish-Chandra gave an explicit form of the Plancherel measure for semisimple Lie groups, and this and related work by others was used to describe the reduced group $\text{C}^\ast$-algebra by Wassermann, Higson, Clare, Crisp, Song and Tang. If $G$ is a semisimple Lie group, and $H < G$ (a union of connected components of) the fixed-point set of an involution on $G$, then Harish–Chandra’s Plancherel theorem was extended to a decomposition of $L^2(G/H)$ as a representation of $G$, by van den Ban, Schlichtkrull and Delorme. With Afgoustidis, Higson and Nishikawa, we are investigating a generalisation of the reduced group $\text{C}^\ast$-algebra, which should reflect the topology of the set of representations that occur in this decomposition of $L^2(G/H)$. I will aim to keep this talk relatively non-technical, and focus on examples.

Sponsors

This workshop is partially supported by the Netherlands Organisation of Scientific Research (NWO) under the VENI grant 016.192.237