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.

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
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