Geometric Analysis, 2nd semester

1. Week 07 (Feb 13):
Motivational overview:

Differential operators, distributions, Sobolev space, symbols,

pseudo-differential operators, ellipticity, inversion modulo smoothing operator, elliptic regularity, spectrum.

2. Week 08 (Feb 20):
First hour (BW):

Basic facts about pdo’s on open subsets of R^n

Def 1.1.6: definition of pdo on a manifold (scalar case)

Definition 1.1.10 of full and principal symbol of an operator on R^n

The example of the Laplace operator on R^n

Exercise 1.1.11, Lemma 1.1.12

The interpretation of the principal symbol as a function on the cotangent bundle of U

Exercise 1.1.13

Definition of principal symbol as a function on the cotangent bundle of a manifold, which is possible in view of 1.1.12.

Second hour (IS):

The calculation of the principal symbol of the Laplace Beltrami operator on a Riemannian manifold

Definition of differential operators between vector bundles, see Def 1.2.2.

grad and div as examples of those.

Lemma 1.2.7.

The symbols of grad and div, and the principal symbol of the Laplace Beltrami operator again.

Exercise 1.2.9: Exterior differentiation as an example, and the associated principal symbol.

Lemma 1.2.10.

To do for all: homework exercise 1 (hand in by Feb 27)

3. Week 09 (Feb 27 ):
First hour (KR)

2.1: reminder, locally convex vector spaces; topology definable by seminorms

Example 2.1.8:, Example 2.1.9: topological dual with strong topology

discussion of inductive limit and mention of Thm 2.1

discuss the important examples of E(Omega) and D(Omega)

Second hour (GB):

definition of D’(Omega), support of distribution, Definition 2.2.4.

Skip 2.2, change of coordinates

Lemma 2.2.7, Definition 2.2.8, Example 2.2.9, Def. 2.2.10

To do for all: homework exercise 2 (hand in by March 13). Please use LaTeX font size 12pt.

4. Week 10 (March 6):
First hour (MM):

Section 2.3: the global theory

to prepare: change of coordinates, 2.2.

topology on E(M, E), Exercise 2.3.2, D(M,E), D’(M,E): generalized sections.

needs preparation: reminder of densities and their integrals after lemma 1.2.10

the subspace $E’(M,E).$

invariance of the spaces under isomorphisms.

Second hour (LS):

Section 4.1: the Schwartz functions, Lemma 4.1.8,

Fourier transform, Lemmas 4.1.9, 4.1.10

Theorem 4.1.14 and a sketch of the proof.

5. Week 11 (March 13):
First hour (BK):

convolution of Schwartz functions (section 4.2)

relation to Fourier transform, Lemma 4.2.2

Prop 4.2.5.

(from section 4.3: ) definition of tempered distributions, def 4.3.1

the embedding E’ < S’ < D’, Exercise 4.3.2, Lemma 4.3.3.

Second hour (LZ):

definition L_s^2

Exercise 4.3.4

Proposition 4.3.5: Fourier transform on tempered distributions

Lemma 4.3.7, skip Lemma 4.3.8

Lemma 4.3.9: characterization of Sobolev space.

Definition 4.3.12 generalization of def Sobolev space, Lemma 4.3.14.

6. Week 12 (March 20):
First hour (PC):

Sect. 4.4, (4.4.5)

Lemma 4.4.1,

Lemma 4.4.2, Corollary 4.4.3

Lemma 4.4.4

Second hour (DA):

Section 4.5, Lemma 4.5.1

Lemma 4.5.2

Lemma 4.5.3

Proposition 4.5.6

To do for all: Exercise 4.3.13. Hand in: Monday March 27.

7. Week 13 (March 27):
First hour (AW):

Par. 5.1 Exercise 5.1.1 and Def 5.1.2 of the symbol space

Observation of Exercise 5.1.3 and the subsequent text

The invariance result on symbol space (text following exercise 5.1.6),

the definition of symbol space on a smooth manifold, and transformation under diffeo’s.

Par 5.2, defi of pseudodiff op, Definition 5.2.2

Lemma 5.2.3, Def 5.2.4, Lemma 5.2.5, in statement (a) of the latter, the unique $K$ need not be compactly supported (misprint)

Exercise 5.2.6, Exercise 5.2.8.

Second hour (BW):

par 5.3, Localization of pseudo-diff ops; Prop. 5.3.1

par 5.2, Def 5.4.1, Lemma 5.4.2, Prop. 5.4.4

Thm 5.4.5, Defs 5.4.6, 5.4.7 and Cor 5.4.10.

8.Week 14 (April 3):
First hour (IS):

par 5.5: expansions in sympbol space: Lemmas 5.5.1,2,3.

par 2.3: general operators and kernels: Schwartz kernel theorem.

Second hour (KR):

Schwartz kernel theorerm, continuation

par 6.1: the distribution kernel of a pseudo-differential operator

Exercise 5.1.6, to be handed in Monday, April 10.

9. Week 15 (April 10):
First hour (LZ):

the adjoint of a pseudo-differential operator

Second hour (MM):

the composition of pseudo-differential operators

Here is a pdf with a precise description of the material

x. Week 16 (April 17): No meeting, Easter/Pasen.

10. Week 17 (April 24):
First hour (GB):

Ch 7, Cor 7.2.3 (page 132)

Lemma 7.2.4

Prop 7.2.5

Thm. &.2.6

Def 7.3.1

Second hour (BK):

Lemma 7.3.3

Lemma 7.3.6

Lemma 7.3.8

Def 7.4.1

Lemma 7.4.3

11. Week 18 (May 1):
Take home exercise 5. Deadline for handing in: Monday, May 8.
First hour (LS):

Thm 7.4.4

Lemma 7.5.1

Lemma 7.5.2

Lemma 7.5.3

Second hour (DA):

ch 8 up to Def 8.1.1

Remark 8.1.2

8.2 up to Def 8.2.1.

12. Week 19 (May 8):
First hour (BW): The principal symbol for operators between vector bundles

text following Def 8.2.1

Lemmas 8.2.2, 8.2.3, 8.2.4

Lemma 8.2.5: localizing symbol

Thm 8.2.6: characterization symbol for ops between vector bundles

Corollary 8.3.2: natural introduction of densities

Second hour (AW): Symbol of adjoint and composition: vector bundle case

Lemma 8.3.3 (symbol of adjoint)

Lemma 8.3.4: pseudo locality

Lemmas 8.3.5, 8.3.6

Theorem 8.3.7: symbol of composition

13. Week 20 (May 15):
First hour (LZ): Parametrices for elliptic operators

Def. 8.4.1, Lemma 8.4.2, Cor. 8.4.3

Def 8.4.4, Thm 8.4.5

Thm 8.4.6 (proof should be given!)

Second hour (KR): Psdo’s and Sobolev space

Corollary 8.4.8 (Elliptic regularity theorem, with proof!!)

Lemma 9.1.1, Lemma 9.1.4 (proof!)

Prop. 9.1.5, Prop. 9.2.1.

14.Week 21 (May 22):
First hour (GB): Sobolev spaces on a manifold

Lemma 9.2.2, Thm 9.2.3

Lemmas 9.2.4, 9.2.5, Cor 9.2.6

Indicate how the definition of Sobolev space extends to a manifold, by using the ideas of Chapter 3. Lemma 9.2.7.

Second hour (MM): Action of Psdo on Sobolev spaces with coefficients in a vector bundle, basics of Fredholm operataors

Theorem 9.2.8, Lemma 9.2.9

Def 1.4.1 of Fredholm operator between Banach spaces

Theorem 1.4.2

Def. 1.4.5, Thm 1.4.9, Thm 1.4.10 (most of these results should be well known)

section 1.4.4.

15. Week 22(May 29):
First hour (BK):

Section 9.3 entirely

Elliptic self-adjoint operators: New notes on Weyl's law, chapter 10, section 10.2 up to (and including) Thm 6.

Second hour (LS):

Spectral decomposition for a self-adjoint elliptic operator: section 10.2 Lemma 7 - Theorem 9.

New notes, section 11: reformulation of Weyl's law. See also Notes Bouclet.

xx. Week 23 (June 5): No meeting: Pentecost/Pinksteren

16. Week 24 (June 12).

First hour: AW: Hilbert-Schmidt and trace class operators

12.1: Hilbert-Schmidt operators, Lemma 10, def 11, Lemma 13, Cor 14, Lemma 15

12.2: Polar decomposition, Thm 21, just result, proof is in lecture notes; Lemma 24 with proof

12.3: Ops of trace class: Def 25, Lemma 27, Lemma 28, Thm 30, Cor. 31 without continuity statement

Second hour: IS: Trace of kernel operators in geometry

12.3: Lemma 33 and defi of norm; Lemma 34, Thm 31 (b) just mention

12.4: Lemma 37; Lemma 39, statement and rough idea of proof

Lemma 40 with proof

Cor 41, just mention

Thm 42 just mention.

Thm 43: sketch of proof, at least in the scalar case.

17. Week 25 (June 19), last meeting.

EvdB: proof of Weyl’s law for the Laplacian on a compact manifold.

---

---

Last change: 16/2-2016