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K-theories

General data

Course ID:	1000-1S22KT
Erasmus code / ISCED:	11.1 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	K-theories
Name in Polish:	K-teorie
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	Seminars for Mathematics
ECTS credit allocation (and other scores):	6.00 Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Main fields of studies for MISMaP:	mathematics physics
Type of course:	elective seminars
Prerequisites (description):	Basic courses taught at I and II year with extended knowledge of topology and algebra.
Mode:	Blended learning
Short description:	Foundations of important versions of K-theories: topological, algebraic, operator algebras and relations between them.
Full description:	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.
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
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 methods and assessment criteria:	Presented papers and activity during the seminar.

Classes in period "Academic year 2024/25" (future)

Time span:	2024-10-01 - 2025-06-08	Selected timetable range: weekly course term Navigate to timetable
Type of class:	Monographic seminar, 60 hours more information
Coordinators:	Stefan Jackowski
Group instructors:	Stefan Jackowski
Students list:	(inaccessible to you)
Examination:	Grading

Course descriptions are protected by copyright.
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