NMAK13008U Index Theorems

Volume 2013/2014
Education
MSc Programme in Mathematics
Content
A typical index theorem expresses the index of a differential operator in topological terms. The best-known example of such a result, which includes many others as special cases, is the formula of Atiyah and Singer for the index of an elliptic operator on a closed manifold. This course will study the Atiyah-Singer index theorem, along with its various specialisations, generalisations, and counterparts in other contexts. There are many possible approaches to the index theorem; in this course, the emphasis will be on K-theory and cyclic homology.
Learning Outcome
Knowledge: By the end of the course, the student will be familiar with the Atiyah-Singer index theorem and some of its special cases; other related index theorems; and the techniques from K-theory and cyclic homology used to formulate and prove these results.

Skills: The student will be able to formulate a variety of index theorems, and explain their proofs using the tools developed in the lectures.

Competences: The student will apply the general theorems to perform explicit index computations; and will use the K-theoretic framework to understand the connections between the various index theorems.
5 hours of lecture and 3 hours of exercises each week.
  • Category
  • Hours
  • Lectures
  • 35
  • Preparation
  • 150
  • Theory exercises
  • 21
  • Total
  • 206
Credit
7,5 ECTS
Type of assessment
Continuous assessment
The assessment will be based on regular written assignments.
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner.
Re-exam
30 minutes oral examination with preparation time. All aids permitted during the preparation time. During the examination only a one page outline is permitted. Several internal examiners
Criteria for exam assesment

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