Analysis on Manifolds (MRI Master Class, MasterMath)

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  • The oral exams will take place on Thursday and Friday, February 18 and 19, in my office: 613. The schedule for the exams has been posted on the door.
  • Dates for the concluding oral examination have been fixed during the last exercise class. If you have not got a date yet and want to fix one, send an email to E.P. van den Ban.
  • The lecture notes for Lectures 10-11 are now available.
  • The last take home exercise, 10.6.7, should be handed in before Christmas.
  • See Exercises for the take home exercise.
  • The take home exercise for lecture 9 has been revised. Some extra suggestions are given.
  • The take home exercise for Lecture 9 is available. Deadline for handing in: December 7.
  • The notes for Lecture 9 are now available.
  • A detailed solution to Exercise 3.6.3, written by Ionut, is now available.
  • The notes for Lecture 8 are now available.
  • Lemma 7.3.8 in the lecture notes is correct as stated, but was intended to be presented in a stronger form. The stronger form is obtained by slightly changing the conditions in (a) and (b). The correct formulation of the lemma is contained in this file.
    The proof remains correct. Note that the stronger form of the lemma is needed in the take home exercise.
  • The deadline for handing in the take home exercise for lecture 6 has been postponed to November 18.
  • We will change the formula of the exercise class. See information on exercise class.
  • The take home exercise for lecture 7 was formulated too sketchily for those not familiar with the definition of sheaf. I have therefore made a revised version.
    It this take home exercise is too abstract to your taste, you are perfectly welcome to instead hand in the following alternative take home exercise, which is more concrete.
  • The notes for Lecture 7 are now available. These will be used on Wednesday, November 11.

  • Location and time:

  • Wednesdays, 10:00 - 13:00; Location:  Buys Ballot Laboratory (BBL) 276, Utrecht University
  • First lecture: September 23  , 2009.
  • Last lecture: December 16, 2009.
  • Teachers:
  • Erik van den Ban (UU)
  • Marius Crainic (UU)
  • Organization:
  • Lectures: 2 times 45 minutes
  • Assisted exercise session: 12:00 - 13:00. Assistants:
  • Gijs Heuts
  • Ivan Struchiner
  • Ionut Marcut

  • Note: Before the start of the course, we will give an "intensive reminder" consisting of 5 afternoons of lectures.
    In these sessions we will review some of the basics of differential geometry needed for the rest of the course. For schedule and other details, see the programme.

    The programme of the intensive reminder course
    Material discussed in lectures

    The aim of this course is to develop the mathematical language needed to understand the Atiyah-Singer index formula.

    In the 1960's M. Atiyah and I. Singer proved their index formula, which expresses the analytic index of an elliptic differential operator on a compact manifold in topological terms constructed out of the operator. This formula is one of the main bridges between analysis and topology- a bridge which stimulated a lot of further research and interplay between geometry, analysis and mathematical physics. In 2004 both mathematicians were awarded the Abel prize for their mathematical work. The goal of this course is to develop the mathematical
    language needed to understand the Atiyah-Singer index formula.

    In the first part of the course we will discuss the language of vector bundles on a manifold, and of differential operators between the spaces of smooth sections of these bundles. Such operators have a principal symbol. An operator with invertible principal symbol is called elliptic. An elliptic operator $D$ between vector bundles on a compact manifold is a Fredholm operator on the level of Sobolev spaces. We will discuss the proof of this result, which makes use of the construction of parametrices (inverses modulo smoothing operators) via pseudo-differential operators. The theory of pseudo-differential operators will be developed from the start, a quick review of distributions
    and Sobolev spaces will be given.

    The Fredholm property implies that the kernel of $D$ has finite dimension, and its image finite codimension. The difference of these natural numbers is called the analytic index of the operator.

    The elliptic operator $D$ also has a topological index. The second half of the course will be devoted to the description of this index. The description makes use of the Chern classes of a complex vector bundle. These are cohomology classes on the base manifold, which can be described in terms of the curvature of a connection on the given bundle. The principal symbol $\gs(D)$ of the operator $D$ gives
    rise to a particular vector bundle. The topological index of $D$ can be defined in terms of the Chern classes of this bundle.

    The Atiyah-Singer index formula states that analytic and topological index of $D$ are equal. During the course we will also discuss special examples of the formula, such as the Hirzebruch-Riemann-Roch formula.

    Lecture Notes.

  • Notes Lectures 10 - 11.
  • Notes Lecture 7, Notes Lecture 8, Notes Lecture 9.
  • Notes Lecture 5. Notes Lecture 6, together with an appendix.
  • Notes Lecture 4 with an additional Proof of Rellich's lemma. (the first version needed quite a number of corrections).
  • Notes Lecture Lecture 1, Lecture 2, Lecture 3.
  • Additional notes on Fredholm operators.

  • Literature:
    In the first half of the course, on elliptic operators, we will use lecture notes.
    Here is some extra-literature which may be useful to consult throughout the semester (some of which will also be used for the course):
    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.
  • 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.
  • 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

  • Material on Functional Analysis, Analysis on Manifolds, Pseudodifferential operators (besides the papers of Atiyah-Singer and the book of Gilkey mentioned above):
  • 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.
  • F. Treves, "Topological vector spaces, distributions and kernels", Academic Press, New York-London 1967 xvi+624 pp.
  • L. Schwarz, "Functional Analysis", Courant Institute of Mathematical Sciences, 1964

  • Exam:
    There will be  take home exercises. They will count for 50 percent of your grade. At the end of the course there will be a final examination. Each week we will list several exercises. We advise you to do them in the order suggested. You should be able to do at least three of them each week. You are not allowed to hand these exercises in for correction. However, three assistants will be present in class to help you with their expertise. In addition, each week there will be a take home exercise to be handed in during next week's exercise session. It will be graded by one of the teaching assistents with a grade from 1 to 10. At the end of the course, the average grade will contribute to your final grade with a weight of 50 percent. count for 50 percent

    Analysis of several variables, basic theory of manifolds, in particular differential forms. Basics of functional analysis.

    Last update: 4/9-2009