Lectures Analysis on Manifolds (MRI Master Class, Mastermath)

Spring semester 2013

 

(04) Week 10, March 7

Lecture notes, Chapter 4:

Schwartz space, Fourier transform, tempered distributions, Sobolev spaces.

Exercises: 4.1.15, 4.2.4, 4.3.11.

Hand in at the beginning of exercise class March 21: 4.3.13 and 4.3.17.

(05) Week 11, March 14

This lecture will be given by David Martinez Torres

Material: Sobolev embedding theorem, Arzela-Ascoli theorem (statement without proof), Rellich's lemma.

Exercises: 4.6.2, 4.6.4.

Exercise 4.6.5: hand in at the beginning of exercise class of March 28.

(06) Week 12, March 21

This lecture will be given by Erik van den Ban again.

Material: 5.1 -- 5.4: space of symbols, pseudo-differential operators, full symbol

Exercises: 5.1.1, 5.1.3, 5.1.4, 5.1.5

Exercise 5.1.6: hand in at the beginning of exercise class of April 4.

(07) Week 13, March 29

Material: 5.4: symbol maps, 5.5: expansions in symbol space

6.1: distribution kernel of a pseudo-differential operator; smoothness outside diagonal.

localization of the kernel, properly supported kernel.

6.2: the adjoint of a pseudo-differential operator

Exercises: 6.1.5, 6.1.9, 6.1.10.

Homework: 6.1.11, hand in at the beginning of the exercise class on April 11.

(08) Week 14, April 4

6.2 the adjoint of a pseudo-differential operator, continued

6.3: pseudo-locality; Appendix A: will not be discussed, contains proof of a result needed in Prop. 6.2.2.

7.1: composition and adjoint of pseudo-differential operators

Study adjoint yourselves.

Exercise 6.2.4 (page 93 of old lecture notes); Exercises 6.2.5.

This week no hand in exercise

(09) Week 15, April 11

7.2: invariance under diffeomorphisms

7.3: pseudo-differential operator on a manifold

7.4 The principal symbol on a manifold

7.5 Symbol calculus on a manifold

Study Lemma 7.3.8 on your own.

Exercises 7.3.4, 7.3.5, 7.3.7.

Hand in: Exercises 7.3.9, 7.3.10 (hand in April 25) See the revised version for these exercises the bundle of extra exercises. in

(10) Week 16, April 18

8.1: pseudo-differential operators between vector bundles on manifolds

8.2: principal symbols for Pseudo diff operators between vector bundles

8.3: Composition and adjoint

8.4: Elliptic operators and the construction of parametrices.

Exercise: 7.5.6. (No new hand in exercise; however, see the revisions of 7.3.9, 7.3.10.

(11) Week 17, April 24 (Last lecture by Erik).

9.1: Pseudo-differential operators and Sobolev spaces

9.2: Application: invariance of Sobolev spaces.

9.2: Sobolev spaces on manifolds.

9.3: The index of an elliptic operator.

Hand in exercise: Extra exercise 9.3.3. Hand in by May 16.

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