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

Volume 2019/2020
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
• 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.

Functional Analysis (FunkAn) and Introduction to Operator Algebra (IntroOpAlg)

Academic qualifications equivalent to a BSc degree is recommended.
4 hours of lectures and 3 hours of exercises per week for 8 weeks.
• Category
• Hours
• Lectures
• 32
• Preparation
• 150
• Theory exercises
• 24
• Total
• 206
Written
Continuous feedback during the course of the semester
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