Lectures Analysis on Manifolds (MRI Master Class, Mastermath)

See also: exercises

Fall semester 2009
The first weeks you are welcome to attend the intensive reminder on the basic theory of manifolds.
Later 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. Fredholm operators.
(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.

Last update: 21/6-2009