## Topics Geometric Analysis (WISM 560)

• Exercise numbers refer to the following Exercise collection
• Here is a version of the lecture notes with corrected typos indicated
• Here are some new notes on Weyl's law
• : Introduction to Pseudo Differential Operators

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