Introduction to Microlocal Analysis: 18.157 (Spring 1998)

• General information

  I will be following my lecture notes (here is the current version of chapter 2, which is undergoing revision chapter2.ps ) at least at the beginning. I hope to get far enough to cover
  1. Global pseudodifferential operators on Euclidean space
  2. Pseudodifferential operators on compact manifolds
  3. An example of pseudodifferential operators on a(nother) class of non-compact manifolds
  and in each case to do some serious application -- determinants, homology, K-theory as time permits.
• Homework:
  1. Due February 19, 1998. Homework1 (Postscript file).
• Lectures (in 2-255 2:30 - 4:00 unless otherwise noted)
  1. Tuesday February 3: Schwartz test functions, tempered distributions.
  2. Thursday February 5: Symbols and approximation.
  3. Tuesday February 10: Pseudodifferential operators defined.
  4. Wednesday February 11: (2-3 in 2-102): Asymptotic summation, kernels of residual operators, reduction theorem (composition theorem?).
  5. Thursday February 19: [Problem set 1 due] Composition theorem. Principal symbol. Ellipticity. Parametrices.
  6. Tuesday February 24: Elliptic regularity, Laplacian. Schur's Lemma. Square root of a symbol.
  7. Wednesday February 25: Approximate square root. Boundedness on $L^2.$ Sobolev spaces.
  8. Thursday February 26: Weighted Sobolev spaces. Polyhomogeneity. Isotropic calculus.
  9. Tuesday March 3: Compact supports. Local coordinate invariance.
  10. Wednesday March 4: No lecture.
  11. Thursday March 5: Transformation of symbol. Pseudodifferential operators on a (compact) manifold.
  12. Thursday March 12: Ellipticity.
  13. Tuesday March 17: Fredholm condition.
  14. Wednesday March 18: Vector bundles, deRham complex.
  15. Thursday March 19: Metrics, inner products and Hodge theorem.
  16. April 7: Dirac operators and manifolds with boundary.
  17. April 9: Boundary problems and the Calderon projector.