Analysis on Manifolds (Mastermath/Wonder master course, Spring 2013)

Note: The lecture of May 23 is cancelled. It will be given one week later, on May 30 (same time and place).

Announcements: Keep an eye on this web-page for further announcements of any kind related to the course (change in schedule/location, lecture notes, exercises, etc etc):
  • keep in mind the INTENSIVE REMINDER (Wednesday, January 30 and Thursday, January 31; location: 611 (Mathematics building, at UU)).

  • note: we added one file with chapters 1-3 in which various typos were corrected.

  • Teachers:
  • Erik van den Ban (UU)
  • Marius Crainic (UU)

  • Location and time: TBA
  • Time: Thursdays, from 10:00 to 13:00 (the first two ours are for the lectures, the last one is for the exercise classes).
  • Location:
  • Week 6 (7 febr): BBL 205
  • Week 7 t/m 13: MIN 027
  • Week 14 t/m 17: MIN 022
  • Week 18 (2 may): in MIN 019
  • Week 19 t/m 21: in MIN 022
  • Intensive reminder: Wednesday, January 30 and Thursday, January 3.
  • First lecture: week 6 (February 7), 2013.
  • Last lecture: week 21 (May 23), 2013.

  • Lectures: 2 times 45 minutes
  • Assisted exercise session: 1 hour per week (immediately after the lectures). Assistants:
  • Ralph Klaasse (r.l.klaasse/at/
  • Reinier Storm (reinier.storm/at/

  • 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 two weeks later, at the start of the lectures. 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.

    Analysis of several variables, basic theory of manifolds, in particular differential forms. Basics of functional analysis. If you lack some of these, we strongly advise you to follow the intensive reminder (see below).

    Intensive reminder: Before the start of the course, we will have an "intensive reminder" consisting of one day in which we will review some of the basics of differential geometry and one day in which we will review some basics of functional analysis.

    Lecture notes: Lecture notes will be made available during the course.

  • Here is Chapter 1 (Differential operators), Chapter 2 (Distributions on manifolds) and Chapter 3 (Functional spaces) in one file with various typos corrected. The most important correction is the change of coordinates formula for distributions on opens in $R^n$, for which I added the discussion on change of coordinates from page 39/40 (with equiation (2.2) as final conlcusion) and the global interpretation of Example 2.3.9 (page 46/47). Lots of other small typos were fixed (thanks to Ralph). If you find more yourself, please let us know.


  • You can have a look at the old lecture notes (2009/2010). However, we advise you not to print/use the old notes, since revised versions will be provided during the course (and will be made available through this web-page).
  • Notes on Prerequisites from differential geometry (E.P. van den Ban), useful to refresh your memory, or for the intensive reminder.
  • Notes on Lie derivatives, tensors and forms (E.P. van den Ban), for the same purpose as the last ones.
  • Additional notes on Fredholm operators (M. Crainic)(old notes).

  • The schedule week by week (here we will try to add, after each lecture, a description of what was discussed in the lectures + the exercises):

    - Intensive reminder 1 and 2 (Wednesday, January 30): Differential Geometry day.

    - Intensive reminder 3 and 4 (Thursday, January 31): Functional Analysis day.

    - WEEK 6/Lecture 1 (February 7): A short overview of the course and of the main key-words starting from the (preliminary) version of the Atiyah-Singer index theorem (ellipticity, Fredholmness, pseudo-differential operators, characteristic classes). Then we started with chapter 1: differential operator and symbols (first trivial coefficients, then general vector bundles).

    Hand-in exercise (to be handed on February 21, at the start of the lecture) : 1.1.9 from the 2012/2013 lecture notes.

    - WEEK 7/Lecture 2 (February 14): Densities and integration, formal adjoints of differential operators, ellipticity and Fredholmness of operators, differential complexes, elliptic complexes, how to deduce results about elliptic complexes from similar results about elliptic operators (e.g.: any elliptic complex is Fredholm). Then we started chapter 2. However, as it seemed that everyone knew the basics of locally convex vector spaces and distributions on opens in R^n, we aggreed that I will assume all of these known (kindly asking you to refresh your memory by looking at the notes), and I will continue next time assuming them (so, next time, we will go directly to manifolds).

    Exercises for the werkcollege: (1.2.9 and 1.3.7), (1.3.3, 1.3.13 and 1.3.15), (1.3.5 and 1.3.16).

    Hand-in exercise (to be handed on February 28, at the start of the lecture) : 1.3.14 from the 2012/2013 lecture notes.

    - WEEK 8/Lecture 3 (February 21): The rest of Chapter 2 (distributions on manifolds, kernels), then we started the chapter on functional spaces, discussing the main axioms and results, but without proofs.

    Hand-in exercise: 2.4.6.

    - WEEK 9/Lecture 4 (February 28): We went back to the notion of functional spaces, the main axioms and the main results, giving the proofs. Then we showed how from a functional space on R^n (which is local and invarian) one obtains one on any open in R^n, and then one can pass to general n-dimensional manifolds and vector bundles; then we looked at the particular case of Sobolev spaces (hence we finished Chapter 3).

    Exercises for the werkcollege: 3.6.3.

    Hand-in exercise: 3.8.7.

    - WEEK 10/Lecture 5 (March 7):

    - WEEK 11/Lecture 6 (March 14):

    - WEEK 12/Lecture 7 (March 21):

    - WEEK 13/Lecture 8 (March 28):

    - WEEK 14/Lecture 9 (April 4):

    - WEEK 15/Lecture 10 (April 11):

    - WEEK 16/Lecture 11 (April 18):

    - WEEK 17/Lecture 12 (April 25):

    - WEEK 18/Lecture 13 (May 2):

    - WEEK 19/Lecture 14 (May 9):

    - WEEK 20/Lecture 15 (May 16):

    - WEEK 21/Lecture 16 (May 23):

    Aim/content of the course:
    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.

    We will use lecture notes (see above).
    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

  • Last update: 20/11-2012