Malte Leimbach: Spectral truncations and operator systems in noncommutative geometry

Spectral truncations are compressions of spectral triples by spectral projections for the Dirac operator. This formalism was introduced by Connes--van Suijlekom to reflect constraints on the availability of spectral data, and they advocate for considering operator systems rather than C*-algebras in noncommutative geometry. Connecting to the setting of Rieffel's compact quantum metric spaces and Kerr--Li's operator Gromov--Hausdorff distance, it makes sense to ask about convergence of spectral truncations. After giving a gentle introduction to some of the relevant concepts from noncommutative geometry and quantum metric spaces, I will report on recent convergence results about truncations for tori and compact quantum groups.