Lecture notes   (pdf files)

Chapter 1   Diffusion operators

Chapter 2   Diffusion semigroups

Chapter 3   Contractive diffusion semigroups

Chapter 4   The heat semigroup on a Riemannian manifold

Materials related to the course

A proof of the Riesz-Thorin interpolation  (Credits to Pr. B. Schlein)


Topics for the final exam

1) Give the proof of the Riesz-Thorin theorem (See the file above)

The topics 2)-5) are related to this survey paper by Michel Ledoux.

2) Give the proof of the sharp Sobolev inequality (Theorem 3.1)

3) Give the proof of Myers theorem (Theorem 3.3)

4) Give the proof of the hypercontractivity theorem (Corollary 4.3)

5) Give the proof of the sharp Entropy-Energy inequality (Theorem 4.4)

6) Give the proof of Inequality (5) in the following paper.

7) Prove Theorem 1 in the following paper.

8) Prove the Isoperimetric Inequality Corollary 2.3 in the survey by M. Ledoux

9)  Prove the rigidity Theorem 4.6 in the survey by M. Ledoux.