K-theories 1000-1S22KT
1. Algebraic functor K
2. Vector bundles and itshomotopy classification
3. Projective modules
4. Homotopy groups
5. Bott periodicity theorem
6.Topological K-theory as generalized cohomologoy theory
7. Maximal ideal spectrum of ring of continuous functions. Gelfand theorem.
8. Vector bundles as projective modules. Swan theorem
9. Milnor's algebraic K-theory
10. Classifying spaces of topological groups and small categories
11. Quillen's algebraic K-theory
12. Banach algebras. Operator algebras. C^*-algebras.
13. Periodicity theorem in K-theroy of operators.
Main fields of studies for MISMaP
mathematics
Course coordinators
Type of course
Mode
Prerequisites (description)
Learning outcomes
The student:
1. notices the analogies and differences of theories called K-theories in the context of various branches of mathematics.
2. Can search and analyze scientific mathematical texts and on their basis prepare a lecture / presentation.
3. Can prepare an outline of a paper and a presentation of a paper in the form of slides. .
4. Can present mathematical content in a manner adapted to the audience.
Assessment criteria
Presented papers and activity during the seminar.
Bibliography
Atiyah, M.F., K-theory. W.A. Benjamin, Inc. 1967
Friedlander, E.M. , An Introduction to K-theory. Lecture Notes. Northwestern University, 2007.
Husemoller,D., Fibre Bundles. Graduate Texts in Mathematics (GTM, volume 20), Springer
Grayson, D.R., Quillen’s work in algebraic K-theory. J. K-Theory 11 (2013), 527–547
Hatcher, Allen (2003). Vector Bundles & K-Theory
Karoubi, M., K-theory. An Introduction. Grundlehren der mathematischen Wisseschaften 226. Springer
Karoubi, M., K-theory. An elementary introduction. Conference at the Clay Mathematics Research Academy
Mlnor, J, Introduction to Algebraic K-Theory. Annals of Mathematics Studies. Vol. 72
Rordam, M.; Larsen, Finn; Laustsen, N. (2000), An introduction to K-theory for C∗-algebras, London Mathematical Society Student Texts, vol. 49, Cambridge University Press, ISBN 978-0-521-78334-7
Swan, R. Algebraic K-Theory. Lecture Notes in Mathematics (LNM, volume 76), Springer
Matthes R., Szymański, W., Lecture Notes On The K-Theory Of Operator Algebras
Weibel, Ch. A., K-Book. An Introduction to Algebraic K-Theory (Link do draftu) Graduate Studies in Math. vol. 145, AMS, 2013
Zakharevich, I. Attitudes of K-theory Topological, Algebraic, Combinatorial. Notices of The American Mathematical Society Volume 66, Number 7. p. 1034
Hatcher, Allen (2003). Vector Bundles & K-Theory
Karoubi, M., K-theory. An Introduction. Grundlehren der mathematischen Wisseschaften 226. Springer
Karoubi, M., K-theory. An elementary introduction. Conference at the Clay Mathematics Research Academy
Mlnor, J, Introduction to Algebraic K-Theory. Annals of Mathematics Studies. Vol. 72
Rordam, M.; Larsen, Finn; Laustsen, N. (2000), An introduction to K-theory for C∗-algebras, London Mathematical Society Student Texts, vol. 49, Cambridge University Press, ISBN 978-0-521-78334-7
Swan, R. Algebraic K-Theory. Lecture Notes in Mathematics (LNM, volume 76), Springer
Matthes R., Szymański, W., Lecture Notes On The K-Theory Of Operator Algebras
Weibel, Ch. A., K-Book. An Introduction to Algebraic K-Theory (Link do draftu) Graduate Studies in Math. vol. 145, AMS, 2013
Zakharevich, I. Attitudes of K-theory Topological, Algebraic, Combinatorial. Notices of The American Mathematical Society Volume 66, Number 7. p. 1034