Kursussøgning, efter- og videreuddannelse – Københavns Universitet

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Kursussøgning, efter- og videreuddannelse

Introduction to K-theory (K-Theory)

Practical information
Study year 2016/2017
Block 4
Programme level Full Degree Master
Course responsible
  • Ryszard Nest (5-77736a7879457266796d33707a336970)
Office: 04.1.09
Phone: +45 35 32 07 28
  • Department of Mathematical Sciences
Course number: NMAA13034U

Course content

K-theory associates to a C^*-algebra A a couple of abelian groups K_0(A) and K_1(A) that on one hand contain deep information about the algebra A and on the other hand they can be calculated for great many algebras. K-theory is one of the most important constructions in both 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 stress both viewpoints.

The course will contain the following specific elements:


  • Projections in C^*-algebras and vector bundles
  • The Grassmannian and classification of vector bundles
  • The Grothendieck construction af K-theory
  • Exact sequences and calculation of K-groups.
  • K-theory of C_0(X) and Bott isomorphism
  • Products in K-theory.


The course is intended both for student in Non-commutative geometry and students in Topology.

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 in C^*-algebras and vector bundles
3. translate between the C^*-algebra and the vector bundle approach

After completing the course the student will be able to
1. prove theorems within the subject of the course
2. apply the theory to both topology and non-commutative geometry
3. understand the extensive litterature on elementary K-theory and to read the more advanced parts of the subject.

Recommended prerequisites

Functional Analysis (FunkAn) and either Algebraic Topology (AlgTop) or Operator Algebra (IntroOpAlg)

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MSc Programme in Mathematics


Study Board of Mathematics and Computer Science

Course type

Single subject courses (day)


1 block


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Teaching and learning methods

4 hours of lectures and 3 hours of exercises per week for 9 weeks.


No limit




Category Hours
Lectures 36
Theory exercises 27
Preparation 143
English 206


Type of assessment

Continuous assessment
Evaluation during the course of 7 written assignments. Each assignment counts equally towards the grade.

Marking scale

7-point grading scale

Criteria for exam assessment

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

Censorship form

No external censorship
One internal examiner.


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

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