Faculty of Science

Introduction to K-theory (K-Theory)
Provider: Faculty of Science

Activity no.: 5600-21-07-31
	Enrollment deadline: 26/04/2021

Place

Department of Mathematical Sciences

Date and time

26.04.2021, at: 08:00 - 25.06.2021, at: 16:00

Regular seats

ECTS credits

7.50

Contact person

Nina Weisse E-mail address: weisse@math.ku.dk

Enrolment Handling/Course Organiser

Mikael Rørdam E-mail address: rordam@math.ku.dk

Written language

English

Teaching language

English

Semester/Block

Block 4

Scheme group

A (Tues 8-12 + Thurs 8-17)

Exam form

Continuous assessment

Exam form

Evaluation of the participant's assignments prepared during the course

Exam details

Evaluation during the course of 6 written assignments. Each assignment counts equally towards the grade.

Course workload

Course workload category	Hours
Lectures	32.00
Theoretical exercises	24.00
Preparation	150.00

Sum	206.00

Content

K-theory associates to a C*-algebra A two abelian groups K_0(A) and K_1(A) that on the one hand contain deep information about the algebra A and on the other hand can be calculated for great many algebras. K-theory is one of the most important constructions in operator algebras, non-commutative geometry and in topology with a host of applications in mathematics and in physics. For commutative unital C*-algebras, alias continuous functions on compact spaces, there are two equivalent descriptions of the K-groups, each with its own advantages. In one description K_0 classifies (stable) projections and in the other description it classifies (stable) vector bundles over the compact space(the spectrum) associated to the algebra.

The course will contain the following specific elements:
- Projections and unitaries in C*-algebras
- Definition, standard picture and basic properties of the K-groups: K_0 and K_1.
- Classification of AF-algebras
- Exact sequences and calculation of K-groups.
- Bott periodicity.
- The six term exact sequence in K-theory.

Learning outcome

Knowledge: The student will obtain knowledge of the elements mentioned in the description of the content

Skills: After completing the course the student will be able to
1. calculate K-groups
2. classify projections and unitaries in C*-algebras
3. understand AF-algebras and their classification

Competences:
After completing the course the student will be able to
1. prove theorems within the subject of the course
2. apply the theory to concrete C*-algebras
3. understand the extensive litterature on elementary K-theory and to read the more advanced parts of the subject.

Literature

See Absalon

Target group

Teaching and learning methods

4 hours of lectures and 3 hours of exercises per week for 8 weeks.

Content

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