Time and Location:

• The lectures take place on Tuesdays from 10:40am-12:10pm in Room 122 of the Math. Sc. building.

Course Outline:

• The goal of this course is to present the theory of pseudodifferential operators and some of their geometric applications.

• The following topics should be covered:
  • Pseudodifferential operators (symbolic calculus, kernel functions, parametrices and ellipticity).
  • Spectral asymptotics (heat kernel asymptotics, Weyl asymptotics, Riemannian invariants, zeta functions)
  • Dirac operators and the Atiyah-Singer index theorem.
  • Fefferman's program in conformal geometry (Fefferman-Graham's ambient metric, conformal invariants, conformally invariant operators, singularities of kernel functions of invariant operators).

Contents:

• Chapter 1: Pseudodifferential Operators (6 lectures).

• Chapter 2: Spectral Asymptotics (3 lectures).

• Chapter 3: The Atiyah-Singer Index Theorem (3 lectures)

• Chapter 4: Fefferman's Program in Conformal Geometry (3 lectures).

References:

[BEG] Berline, N.; Getzler, E.; Vergne, M.: Heat kernels and Dirac operators.
Grundlehren der Mathematischen Wissenschaften, 298. Springer-Verlag, Berlin, 1992.
[BGS] Beals, R.; Greiner, P.C.; Stanton, N.K.: The heat equation on a CR manifold.
Journal of Differential Geometry 20 (1984), no. 2, 343-387.
[FG1] Fefferman, C.; Graham, C.R.: Conformal invariants.
Elie Cartan et les Mathématiques d'Aujourd'hui, Astérisque, hors série, 1985, pp. 95-116.
[FG2] Fefferman, C.; Graham, C.R.: The ambient metric.
E-print, arXiv, October 2007.
[Fo] Folland, G.: Real analysis. Modern techniques and aplications.
2nd edition. John Wiley & Sons, Inc., New York, 1999.
[Fr] Friedlander, G.; Joshi, M.: Introduction to the Theory of Distributions
2nd edition. Cambridge University Press, Cambridge, UK, 1998.
[Gi] Gilkey, P.B.: Invariance theory, the heat equation and the Atiyah-Singer index theorem.
2nd edition, CRC Press, Boca Raton, 1994.
[GJMS] Graham, C.R.; Jenne, R.; Mason, L.J.; Sparling, G.A.: Conformally invariant powers of the Laplacian. I. Existence.
Journal of the London Mathematical Society (2) 46 (1992), no. 3, 557-565.
[Hö] Hörmander, L.: The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis.
2nd edition (or newer, but not older). Grundlehren der Mathematischen Wissenschaften, 256. Springer-Verlag, Berlin, 1990.
[Po1] Ponge, R.: A new short proof of the local index formula and some of its applications.
Communications in Mathematical Physics 241 (2003) 215-234.
[Po2] Ponge, R.: The logarithmic singularities of kernels of conformally invariant operators.
To appear.
[Ta1] Taylor, M.E.: Pseudodifferential operators
Princeton University Press, Princeton, NJ, 1981.
[Ta2] Taylor, M.E.: Partial differential equations. I. Basic theory.
Applied Mathematical Sciences, 115. Springer-Verlag, New York, 1996.
[Ta3] Taylor, M.E.: Partial differential equations. II. Qualitative studies of linear equations.
Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996.
[Yo] Yosida, K.: Functional analysis.
6th edition (or newer, but not older). Grundlehren der Mathematischen Wissenschaften, 123. Springer-Verlag, Berlin, 1980.

• Most of the metrial from Chapter 1 will be extracted from [Ta1], except for the description of the kernels of pseudodifferential operators in terms of homogeneous distributions which is alluded to in [Ta3]. Background on locally convex topological spaces and dsitributions (aka generalized functions) can be found in [Fo] and [Yo] (see also [Fr]). References for Schwartz kernel theorems are [Fr] and [Hö]. A good reference for Sobolev spaces on R^n is [Fo]. Material on homogeneous distributions can be found in [Hö] and [Ta2].

• The main references for the material of Chapter 2 are [BGS], [Gi] and [Po1].

• The main references for the material of Chapter 3 are [BGV] and [Po1].

• The main references for the material of Chapter 4 are [FG1], [FG2], [GJMS] and [Po2].