NSCPHD1097  Non-commutative geometry (NCG)

Volume 2015/2016
Content

 

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Algebra of compact operators, Fredholm operators and index, the Toeplitz algebra, reminder on K-theory and the proof of Bott periodicity, introduction to K-homology.
Elements of periodic cyclic homology, characteristic classes of manifolds and Chern character, Index theorem.

Learning Outcome

Knowledge:
The student will obtain detailed understanding of K-theory and learn basic facts about K-homology, cyclic cohomology and characteristic classes. The student will have a basic understanding of the applications of homological methods for both topological spaces and non-commutative C*-algebras

Skills:
At the end of the course the student will be able to prove basic properties of topological K-theory and K-homology, demonstrate the ability to compute it in some examples. He/she will have a basic knowledge of yclic theory

Competences:
The student will be able to use K-theory, K-homology and cyclic homology in both topological and C*-algebraic problems.

Notes,
N. Higson, J. Roe Analytic K-homology, Oxford Mathematical monographs, Oxford University Press, 2000.

Elementary knowledge of K-theory, operator algebras and de Rham cohomology.
5 lectures and 3 exercise classes per week for 9 weeks,
obligatory assignments.
Credit
7,5 ECTS
Type of assessment
Continuous assessment
7 written assignments during the course of which 5 must be approved
Marking scale
passed/not passed
Censorship form
No external censorship
One internal examiner.
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
  • 35
  • Class Exercises
  • 22
  • Course Preparation
  • 149
  • Total
  • 206