See also: exercises
- Fall semester 2009
- The first weeks you are welcome to
attend the intensive reminder on the basic theory of manifolds.
announcements will follow
- (01) Week 39, sep 23
- Lecture notes, Lecture 1:
differential operators, scalar and vectorbundle valued.
Principal symbol. Elliptic operators, differential complex, elliptic complex.
- (02) Week 40, sep 30
Lecture Notes, Lecture 2.
- (03) Week 41, oct 7
Lecure Notes, Lecture 3.
- (04) Week 42, oct 14
Schwartz space, Fourier transform, tempered distributions, Sobolev spaces.
See notes Lecture 4 and Notes on Rellich's lemma.
- (05) Week 43, oct 21
Pseudo-differential operator, symbol space, localization,
full symbol, principal symbol.
- See notes Lecture 5.
- (06) Week 44, oct 28
- See notes Lecture 6 and Appendix to Lecture 6
- Kernel of pseudo-differential operator, adjoint, asymptotics for adjoint symbol, pseudo-locality.
- Week 45, nov 4
- No lecture.
- (07) Week 46, nov 11
- See notes Lecture 7.
Principal symbol of composition of properly supported pseudo-differential operators,
transformation of pseudo-differential operators under diffeomorphisms,
pseudo-differential operators on a manfifold, restriction and gluing, principal symbol on
a manifold, composition and adjoint.
- (08) Week 47, nov 18
See Notes for Lecture 8.
Pseudo-differential operators on vector bundles, construction of parametrices, regularity
theorem for elliptic operators.
- (09) Week 48, nov 25
- One hour lecture, see Notes for Lecture 9.
Definition of local Sobolev space by using elliptic pseudo-differential operator;
invariance of local Sobolev space; local Sobolev space with values in bundle,
continuity of pseudo-differential operator. Fredholm property of elliptic
pseudo-differential operator on compact manifold. Index independent of Sobolev order.
- Two times 45 minutes: Exercises done in front of class.
- (10) Week 49, dec 2
- See Notes for Lectures 10 - 11.
Section 10.1: connections, Section 10.2: curvature,
Section 10.3: characteristic classes
(by applying invariant polynomials on local curvature matrices and glueing),
Start of 10.4.
- (11) Week 50, dec 9
- See Lecture 10 from 10.4 in Notes for Lectures 10-11.
- (13) Week 51, dec 16
- See Lecture 11 in Notes for Lectures 10-11.