# NMAA13034U  Introduction to K-theory (K-Theory)

Volume 2018/2019
Education

MSc Programme in Mathematics

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
• The Grothendieck construction af K-theory
• 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.

Functional Analysis (FunkAn) and Introduction to Operator Algebra (IntroOpAlg)
4 hours of lectures and 3 hours of exercises per week for 8 weeks.
Credit
7,5 ECTS
Type of assessment
Continuous assessment
Evaluation during the course of 6 written assignments. Each assignment counts equally towards the grade.
Aid
All aids allowed
Marking scale
Censorship form
No external censorship
One internal examiner.
Re-exam

Oral, 30 minutes. Several internal examiners. 30 minutes preparation time with all aids.

##### Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

• Category
• Hours
• Lectures
• 32
• Theory exercises
• 24
• Preparation
• 150
• Total
• 206