2024 spring
- Research seminar
2023 fall
- Foliations
2022 fall
- Groupoid C*-algebras
2022 spring
- K(K)-Theory
2021 fall
- Deformation Quantization and Index Theory


2023 spring
- Geometry and Operator ALgebras 2023
2022 fall
- NCG Day 2022

Noncommutative Geometry Reading Seminar

2022 Spring: K(K)-theory

In 2022 spring semester, we ran a reading seminar on K-theory and KK-theory at Leiden university. In this seminar we studied some basic knowledge of KK-theory for C*-algebras. We began with a quick recap of the properties of K-theory, focusing on its formal properties like the Bott periodicity and Thom isomorphisms, which became more illuminating in the framework of KK-theory. Then we provided the two basic pictures of KK-theory due to Kasparov and Cuntz. After that we studied the formal properties of KK-theory, which are mostly due to the existence of the Kasparov product. Using the Kasparov product we are able to construct a category KK which is equipped with a triangulated structure, so various toolkits from homological algebra can be adapted to it. In the end, we discussed the unbounded picture of KK-theory, whose theory had been developed rapidly in recent years and proved to be useful in the study of noncommutative differential geometry.

Prerequisites. The participants should be familiar with basic theory of C*-algebras. Knowledge on the K-theory of C*-algebras is not assumed but suggested.


Bram Mesland, Jack Ekenstam, Yufan Ge and Yuezhao Li. If you are interested: please contact Yuezhao Li (y.li AT math.leidenuniv.nl).

Date and Location

Tuesdays, 12.15-14.00, in Gorlaeus building.


Notes of the talks are available here.


Date Location Speaker Title
Feb 15 Gorlaeus DM017PC Jack Ekenstam K-theory of C*-algebras 1
Feb 22 Gorlaeus DM017PC Yuezhao Li K-theory of C*-algebras 2
Mar 1 Gorlaeus DM009PC Jack Ekenstam Hilbert C*-modules 1
Mar 8 Gorlaeus DM017PC Jack Ekenstam, Yufan Ge Hilbert C*-modules 2, Kasparov’s picture of KK-theory 1
Mar 15 No talk
Mar 22 Gorlaeus DM017PC Yufan Ge Kasparov’s picture of KK-theory 2
Mar 29 Gorlaeus DM109 Yufan Ge Cuntz’s picture of KK-theory
Apr 5 Gorlaeus DM017PC Yuezhao Li Examples and properties of KK-theory
Apr 12 Gorlaeus DM017PC Bram Mesland The Kasparov product 1
Apr 19 Gorlaeus DM017PC Bram Mesland The Kasparov product 2
Apr 26 No talk
May 3 Gorlaeus DM017PC Georg Huppertz Extension of C*-algebras and KK-theory 1
May 10 Gorlaeus DM017PC Georg Huppertz, Yuezhao Li Extension of C*-algebras and KK-theory 2, Categorical aspect of KK-theory 1
May 17 Gorlaeus DM017PC Yuezhao Li Categorical aspect of KK-theory 2
May 24 Gorlaeus DM017PC Dimitris Gerontogiannis Finite summability in K-homology
May 31 No talk (due to Noncommutative Geometry Along the North Sea 2022)
Jun 7 Gorlaeus DM017PC Mick Gielen E-theory
Jun 14 Gorlaeus DM017PC Yufan Ge K-theory of graph C*-algebras
Jun 21 Gorlaeus DM017PC Francesca Arici K-theory of Cuntz-Pimsner algebras