K-theory assigns to each C*-algebra A two abelian groups K_0(A) and K_1(A). The K-theory of a C*-algebra contains deep information about the algebra A and there are strong tools that allow you to compute the K-theory. K-theory is probably the most important invariants in operator algebras, non-commutative geometry and in topology with a host of applications in mathematics and in physics. For commutative unital C*-algebras, aka 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 equivalence of) projections and in the other description it classifies (stable equivalence of) 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.
MSc Programme in Mathematics
MSc Programme in Mathematics with a minor subject
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
4. Understand the significance of Bott periodicity
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.
4 hours of lectures and 3 hours of exercises per week for 8 weeks.
Functional Analysis (FunkAn) and Operator Algebra
Academic qualifications equivalent to a BSc degree is recommended.
- 7,5 ECTS
- Type of assessment
- Type of assessment details
- Evaluation during the course of 4 written assignments. Each assignment counts equally towards the grade.
- All aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- No external censorship
One internal examiner.
Criteria for exam assessment
The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended learning outcome.
Single subject courses (day)
- Theory exercises
- Course number
- 7,5 ECTS
- Programme level
- Full Degree Master
- Block 4
- No limit
The number of seats may be reduced in the late registration period
- Study Board of Mathematics and Computer Science
- Department of Mathematical Sciences
- Faculty of Science
- Mikael Rørdam (6-767376686571447165786c326f7932686f)
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Courseinformation of students