Seminar Geometric Analysis 2017 (WISM 560)

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Announcements:

  • The first meeting will be on Monday, February 13.
  • This site is still under construction
  • If you plan to participate in this seminar, please send me a mail describing your background in analysis and geometry.

    • Supervisor:

    E.P. van den Ban, HFG office 604, Contact

    Seminar meetings:

  • Weekly, Spring 2017, weeks 6 - 24.
  • Day, time and location: Mondays, 15:00 - 17:00, Hans Freudenthal Gebouw, room 610.
  • Description of topics covered
    • Plan for the seminar:
  • Geometric analysis concerns the interplay of analysis and differential geometry. In the seminar, we will discuss the theory of pseudo-differential operators on compact manifolds, as well as several applications.
  • At the initial stage of the seminar we will focus on aspects of the theory of distributions and Sobolev spaces, Fourier transform, the local definition of pseudo-differential operators, the method of stationary phase and symbol calculus.
  • We will then consider the definition of pseudo-differential operators on smooth compact manifolds, and discuss applications to elliptic differential operators such as the construction of inverse operators modulo smoothing operators, the elliptic regularity theorem and the existence of harmonic forms representing de Rham cohomology.
  • Other possible subjects are:
    - the analytic index of an elliptic operator
    - the spectrum of the Laplace operator on a compact Riemannian manifold and its relation to the geometry of the manifold (Weyl s law, Can you hear the shape of a drum?).
    • At the start of the seminar we will inventarise the knowledge of the participants, and agree on a proper strategy for addressing inadequate background knowledge. The precise choice of subjects and literature references will depend on this.

    Prerequisites for the seminar:

  • Basic functional analysis
  • Basic differential geometry
    • In case you consider registering, but are in doubt whether your background knowledge is up to it, do not hesitate to send me a mail to discuss this. If you decide to register, I would appreciate it if you could inform me by email as well.

    Literature:

  • We will use the Lecture Notes Analysis on Manifolds, by myself and Crainic.
    • Some background material:
  • Prerequisite material on manifolds.
  • Prerequisite material on Lie derivatives and tensors.
  • Prerequisites on Fredholm operators.


    • Here is some extra literature that may be useful to be consulted throughout the semester:

    Material on Functional Analysis, Distribution theory, Analysis on Manifolds, Pseudodifferential operators:

  • J.J. Duistermaat, J.A.C. Kolk, Distributions, Birkhauser, 2010.
  • R.O. Wells, "Differential analysis on complex manifolds", Graduate Texts in Mathematics, 65. Springer-Verlag, New York-Berlin, 1980. x+260 pp.
  • J. Chazarain and A. Piriou, "Introduction a la theorie des equations aux derivees partielles lineaires" [Introduction to the theory of linear partial differential equations] Gauthier-Villars, Paris, 1981. vii+466 pp. (there is also an english version)
  • J.J. Duistermaat, "Fourier integral operators", Progress in Mathematics, 130. Birkhäuser Boston, Inc., Boston, MA,/ 1996. x+142 pp.
  • L. Hörmander. The analysis of linear partial differential operators. III. Pseudodifferential operators. Grundlehren der Mathematischen Wissenschaften 274. Springer-Verlag, Berlin, 1985.
  • L. Hörmander. The analysis of linear partial differential operators. IV. Fourier integral operators. Grundlehren der Mathematischen Wissenschaften, 275. Springer-Verlag, Berlin, 1985.
  • F. Treves, "Topological vector spaces, distributions and kernels", Academic Press, New York-London 1967 xvi+624 pp.
  • L. Schwartz, "Functional Analysis",
  • P. Gilkey, "Invariance theory, the heat equation, and the Atiyah Singer index theorem", Mathematics Lecture Series, 11. Publish or Perish, Inc., Wilmington, DE, 1984. viii+349 pp. Courant Institute of Mathematical Sciences, 1964
  • M.W. Wong. An introduction to pseudo-differential operators, World Scientific, 2nd ed., 1999.
  • M.E. Taylor. Pseudo-differential operators, Princeton University Press, 1981.

    • Literature on index theorems:
    Material on the Atiyah-Singer theorem:

  • M. Atiyah and I.M. Singer, "The index of elliptic operators" I, II and III, Ann. of Math. 87 (1968), pp. 484-604.
  • P. Shanahan, "The Atiyah-Singer index theorem. An introduction", Lecture Notes in Mathematics, 638, Springer, Berlin, 1978. v+224 pp.
  • N. Berline, E. Getzler and M. Vergne, "Heat kernels and Dirac operators", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 298. Springer-Verlag, Berlin, 1992. viii+369 pp.

    • Material on vector bundles and characteristic classes (besides the material already mentioned):
  • M. Atiyah, "K-theory", W. A. Benjamin, Inc., New York-Amsterdam 1967 v+166+xlix
  • R. Bott and L. Tu, "Differential forms in algebraic topology", Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982. xiv+331 pp.
  • D. Husemoller, "Fibre bundles", Third edition. Graduate Texts in Mathematics, 20. Springer-Verlag, New York,/ 1994. xx+353 pp.
  • J. Milnor and J. Stasheff, "Characteristic Classes", Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. vii+331 pp



    • Last change: 23/11-2016