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
Type of course
Mode
Prerequisites (description)
Course coordinators
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
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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