Heat kernel and analysis on manifolds
Summary
This course develops the theory of the heat equation on Riemannian manifolds. We introduce the Laplace-Beltrami operator, the heat semigroup, and the heat kernel, study its fundamental properties, and explore how it "sees" the geometry of the underlying space.
Content
- Brief recapitulation of Euclidean Sobolev spaces, the Laplace operator, the Euclidean heat equation, and basic Riemannian geometry.
- Setup of the heat flow on Riemannian manifolds: Laplace-Beltrami operator, Riemannian Sobolev spaces, heat semigroup, heat kernel, etc.
- Fundamental properties of the heat flow: maximum principles, regularity, positivity, etc.
- Applications of the heat flow to geometry: spectral properties, parabolicity, links to Brownian motion and Ricci curvature, etc.
Keywords
Heat flow; Heat kernel; Laplace-Beltrami operator; Riemannian geometry; Spectral theory; Ricci curvature; Parabolic PDEs.
Learning Prerequisites
Required courses
- Analysis I-IV.
- Linear algebra I-II.
Recommended courses
- Measure theory.
- Functional analysis I-II.
- Introduction to partial differential equations.
The lecture notes will summarize the necessary background material from these courses in a preliminary section. In addition, elementary background in Riemannian geometry (as covered e.g. in "Differential geometry III - Riemannian geometry") will be useful but is not required, as we will recall adjacent basic notions during the lecture.
Important concepts to start the course
Sobolev spaces, Laplace operator, heat equation, and heat kernel in Euclidean space.
Learning Outcomes
By the end of the course, the student must be able to:
- Formulate the definitions and results of the lectures
- Apply the concepts learned in class to concrete problems
- Analyze problems related to the topics treated in the course
- Choose an appropriate method to solve a given problem
- Prove some elementary statements about the topics of the course
- Solve exercises on the topics
Teaching methods
Weekly lectures with instructor and weekly exercise sessions with assistants.
Expected student activities
Attending the lectures and solving the exercises.
Assessment methods
Oral exam.
Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.
Supervision
| Office hours | No |
| Assistant.e.s | Yes |
| Forum | Yes |
Resources
Bibliography
- Alexander Grigor'yan, Heat kernel and analysis on manifolds, International Press, 2009.
- Dominique Bakry, Ivan Gentil, and Michel Ledoux, Analysis and geometry of Markov diffusion operators, Springer, 2014.
Notes/Handbook
Lecture notes will be available on Moodle.
Moodle Link
In the programs
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Heat kernel and analysis on manifolds
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Heat kernel and analysis on manifolds
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Heat kernel and analysis on manifolds
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Oral (winter session)
- Subject examined: Heat kernel and analysis on manifolds
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional