Introduction to K-theory (K-Theory)

Course content

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.

 

Education

MSc Programme in Mathematics

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

4. Understand the significance of Bott periodicity

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.

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

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

Academic qualifications equivalent to a BSc degree is recommended.

Written
Continuous feedback during the course of the semester
ECTS
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
7-point grading scale
Censorship form
No external censorship
One internal examiner.
Criteria for exam assessment

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

Single subject courses (day)

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