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.

Organisers

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

Notes of the talks are available here.

Schedule

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

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