Analysis on Manifolds

Course content

Basic properties of differential operators on manifolds,

Relations between ellipticity and Fredholm properties.

Riemannian geometry, Levi - Civita connection and Laplace operator.

Hodge theorem.

Dirac operators and properties of heat kernels

 

Education

MSc Programme in Mathematics

MSc Programme in Mathematics w. a minor subject 

Learning outcome

Knowledge: 
The student will obtain detailed understanding of the properties of elliptic differential operators on manifolds and their applications to topology and geometry.

Skills: 
At the end of the course the student will be able to prove basic properties of elliptic differential operators and demonstrate the ability to use them in applications. 

Competences: 
The student will be able to use analysis of differenial operators on manifolds to study their topological and geometric properties.

5 lectures and 3 exercise classes per week for 9 weeks

Steve Rosenberg, "The Laplacian on a Riemannian Manifold" or equivalent textbook

Previous contact with the theory of differentiable manifolds and functional analysis.

Academic qualifications equivalent to a BSc degree is recommended.

Written
Continuous feedback during the course of the semester
Peer feedback (Students give each other feedback)
ECTS
7,5 ECTS
Type of assessment
Continuous assessment
7 written assignments during the course of which the 5 best counts equally towards the final 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
  • 45
  • Class Exercises
  • 27
  • Course Preparation
  • 134
  • English
  • 206