A KK-theoretic perspective on the spectral localiserOberseminar Global Analysis and Operator Algebras

by Dr Koen van den Dungen (Mathematical Institute)

Tuesday Nov 3, 2026, 2:15 PM → 3:15 PM Europe/Berlin

Endenicher Allee 60/1.008 (Mathezentrum) (Mathezentrum)

The index pairing between K-theory and K-homology is usually expressed as a Fredholm index on an infinite-dimensional Hilbert space. The construction of the spectral localiser (due to Loring and Schulz-Baldes) allows to reduce such an index pairing to the computation of a finite matrix, thus making the index pairing accessible to numerical calculations. 
In this talk, I will describe a new perspective on the spectral localiser in the context of KK-theory. The KK-theoretic approach provides a spectral flow expression for the even or odd index pairing in terms of the "infinite volume" spectral localiser, and has the main advantage that the construction of the spectral localiser becomes immediately apparent from the computation of the index pairing via a Kasparov product. 
Next, in a second step of "spectral truncation" (closely following Loring and Schulz-Baldes), I will explain how this spectral flow expression can be computed in terms of the signature of the "finite volume" spectral localiser.