Avraham (Rami) Aizenbud

אברהם (רמי) איזנבוד

My Group




Tuesday 15:00-18:00, Room 261.


Tuesday 14:30-17:30, Room 208 Goldsmith building.

Office hours:

If you would like to come to the office hours or to fix some other time please e-mail me.


Yotam Hendel


1. General information

The course is planned as a year course. However, in the end of the first semester we will discuss in what manner we would like to proceed. Therefore the topics that are mentioned at the end of the list below are subject to change.

I believe that the course can be of interest to pure and applied math students and those studying theoretical physics.

The first semester is aimed at M.Sc. students and first years Ph.D. students. If you are above this level, but still interested in part of the topics, you can join the e-mail list and you will get an e-mail before each lecture with the topics that will be discussed in the lecture, so you can decide whether you want to come.

2. Overview

We will study the theory of generalized functions and distributions (which are almost the same thing) on various geometric objects, operations with distributions (like pushforward, pullback and Fourier transform), and invariants of distributions (like the support and the wave front set)

The topic by its nature is analytic, but my point of view on this topic is oriented towards representation theory and algebraic geometry, so the course will have some algebraic and geometric flavor.

We will discuss both the Archimedean case (i.e. distributions on real geometric objects) and the non-Archimedean case (i.e. distributions on p-adic geometric objects). We will discuss the similarity and difference of both cases.

During the later stages of the course, we will discuss distributions in the presence of a group action, the notion of an invariant distribution, and different methods to prove vanishing of invariant distributions. Those topics are closely related to representation theory.

In addition to the main topic of the course, we will have “digressions” (i.e. some lectures that are related to the main topic but not part of it) on: Functional analysis, p-adic numbers, Harmonic analysis on locally compact abelian groups, Differentiable manifolds, Nuclear spaces, algebraic and semi-algebraic geometry, D-modules, the Weil representation and geometric invariant theory. Those “digressions” will be done on a very basic level, with the aim of making the students familiar with the basic notions in this topics. In case some of these topics will turn out to be too complicated, we will exclude them together with the related parts of the main topic.

I’ll try to include in the course discussion some open (or semi-open) questions, which might fit M.Sc. or Ph.D. students.

3. Prerequisite

I expect the students to be familiar on a basic level with at least 80% of the following notions:

Linear algebra.

Vector space, linear map, subspace, quotient space, dual space, Tensor product.


Topological space, Locally compact space, metric space, Complete metric space, completion of a metric space.


Differentiable manifold, tangent space, tangent bundle.

Group theory.

Group, group action, abelian group,

Functional analysis.

Hilbert space, Fourier series, measure, Fourier transform

4. Chronological list of topics

Each topic should correspond to approximately one week.

4.1. Introduction.

Intuitive description, generalized functions on the real line (2 definitions), derivative of generalized functions, convolution, Green’s function (time independent case), the distribution xλ. Overview. References: [GS, Volume 1, Chapter 1].

4.2. Digression – Functional analysis.

Topological vector space, locally convex vector space, Frechet space, completion of topological vector space, measure, review of generalized functions. References: [Tre, sections 3-10 and 18,19,21], [GS, Volume 2, Chapter 1, sections 1-5 and Chapter 2, section 1].

4.3. Distributions on real vector spaces and its open subsets.

Continuous dual, measure, Distributions, sheaf property, partition of unity, support, Distributions supported on subspace (intro.) – References: [GS, Volume 1, Chapter 1, Chapter 3, section 1], [Hör, Chapters 1-2].

4.4. Generalized functions on open subsets of real vector spaces (algebraic aspects).

distributions vs. generalized functions, top differential forms, generalized functions supported on subspace. References: [Hör, Chapters 1-2], [GS, Volume 1, Chapter 3, section 1].

4.5. Digression – p-adic numbers and l-spaces.

Definitions, Haar mesure. – References: [Kob, chapter 1].

4.6. Generalized functions on p-adic vector spaces and l-spaces.

Definitions. generalized functions on a product (the non-Archimedean case). Comparison with the Archimedean case. References: [BZ76, Chapter 1, section 1]

4.7. Digression – Differentiable manifolds and non-Archimedean Geometry.

vector bundle, (co)tangent bundle, (co)normal bundle, differential forms, densities, sheaves, analytic p-adic manifolds (smooth p-adic manifolds). References: [GP74, Chapters 1.1-1.4, 3.2, 4.1-4.5], [BZ76, Chapter 1, section 1], [Ser64, Part II, Chapter III].

4.8. Generalized functions on manifolds.

Generalized functions on smooth real manifolds, Distributions on l-spaces, vector bundles and sheaves, generalized sections of vector bundels and shifs. – References: [Hör, Chapter 6, section 3], [BZ76, Chapter 1, section 1].

4.9. Operations with distributions.

Generalized functions supported on a submanifold. pushforward, pullback, Analytic property of the pushforward. – References: [Hör, chapter 6, section 1].

4.10. Digresion – Harmonic analysis on locally compact abelian group.

Haar measure, Pontryagin dual, Fourier transform. – References: [Dei, Part 2], [HR].

4.11. Schwartz functions and Fourier transform.

Schwartz functions on a real vector space, Fourier transform of Schwartz functions, Schwartz distributions and Generalized Schwartz functions, the non-Archimedean case, Poisson summation formula. – References: [Hör, Chapter 7, sections 1,2], [GS, Volume 1, Chapter 2], [Dei, Chapter 4].

4.12. The wave front set.

Definition on open sets in vector space, basic properties, relation to pushforward and pullback, definition on general manifold. – References: [Hör, chapter 8, sections 1-2].

4.13. Digression – D-modules.

D-modules on an affine space, D-modules on smooth Affine varieties, singular support (a.k.a. Characteristic variety), Bernstein inequality, integrability theorem, holonomicity. – References: [Cou, chapters 0-2.1, 3, 5-6.1, 7-11].

4.14. Singular suport of distributions.

D-module attached to a distribution, singular support and holonomicity of distributions, relation to the wave front set, applications. [Hör, chapter 8, section 3].

4.15. Digression – Nuclear spaces.

– References: [Tre, part 3].

4.16. Schwartz kernel theorem.

Generalized functions on a product (the Archimedean case). – References: [GS, Chapter 5], [Tre, part 3, section 51].

4.17. Schwartz functions on manifolds – Introduction.

Motivation (short exact sequence of spaces of distributions), problems, comparison with the non-Archimedean case, different approaches. – References: [AG08, section 1].

4.18. Digression – algebraic and semi-algebraic geometry.

Spaces with a sheaf of functions, complex algebraic variety, real algebraic variety, Nash manifolds. Zaidenberg-Tarski theorem. – References: [Kem, Chapter 1], [Shi, chapters 1-4], [AG08, section 2-3], [BCR].

4.19. Schwartz functions on Nash manifolds.

Definitions, basic properties, Schwartz distributions supported on a submanifold, Schwartz functions on a product, pushforward, pullback of Schwartz functions. – References: [AG08].

4.20. Group actions distributions.

Group action on manifolds, invariant distributions, equivariant distributions, group actions on bundles, invariant generalized sections. The wave front set and the singular support of invariant distributions.

4.21. Vanishing of invariant distributions.

Stratification and extension (higher co-homology), product, Schwartz vs. general distributions.

4.22. Localization principle.

The non-Archimedean case, problems with the Archimedean case. – References: [Ber82, section 1.4].

4.23. Frobenius descent (a.k.a. Frobenius reciprocity).

– References: [Ber82, section 1.5], [AGS08, Appendix A].

4.24. Uncertainty principle.

Introduction, Heisenberg uncertainty principle, qualitative uncertainty principle – simple example.

4.25. The wave front set and uncertainty principle.

– References: [AG09b, section 2.3], [Aiz, section 4].

4.26. Homogeneity and uncertainty principle.

Formulation, example of usage, comparison of the Archimedean and non-Archimedean case. – References: [AGS08, Section 4.3].

4.27. Digression – The Weil representation.

The one dimensional case: Weil representation of the Lie algebra sl2, Weil representation of the Lie group SL2, relation with Heisenberg representation. The general case. – References: [LV, Part 1].

4.28. The Weil representation and uncertainty principle.

The non-Archimedean case, the Archimedean case, example of usage. – References: [AGa, section 6], [AG09b, section 2.2.4].

4.29. Digression – Luna’s slice theorem.

– References: [Dre00].

4.30. Specific examples of vanishing of invariant distributions.

– References: [AG09aAGRS10AG09b]

4.31. Invariant distributions vs. co-invariant test functions.

– References: [AGb, section Appendix(es)].

5. Literature List

Most of the material of the first semester is contained in

  • [GS, Volume 1, Chapters 1-3.1, and Volume 2, Chapters 1-3]
  • [Hör, chapters 1-8]
  • [BZ76, Chapter 1, section 1].

However the approach in those sources is slightly different and they contain a lot of additional information. For some of the lectures I’ll need other references as mentioned in the topics list above.

Note that the “digression” lectures are usually on a much more basic level than the corresponding source, so do not be surprised if it looks much longer than you expect from one-lecture material.

In the beginning of each lecture I’ll try to mention the sources that I will be using. I’ll also do it one lecture ahead.

It is also a good idea to use less formal web resources (like the ones that are linked above), especially for the “digression” lectures.

Below you can find the full alphabetical reference list, but everything except [HörGSBZ76] should be considered as supplementary material rather than the main source.

[AG08]    A. Aizenbud, D. Gourevitch, Schwartz functions on Nash Manifolds, International Mathematic Research Notes 2008/5 (2008). See also arXiv:0704.2891 [math.AG].

[AG09a]    Aizenbud, A.; Gourevitch, D.:Generalized Harish-Chandra descent, Gelfand pairs and an Archimedean analog of Jacquet-Rallis’ Theorem. Duke Mathematical Journal, 149/3 (2009). See also arxiv:0812.5063[math.RT].

[AG09b]    A. Aizenbud, D. Gourevitch, Multiplicity one theorem for (GLn+1(R),GLn(R)), Selecta Mathematica, 15/2,(2009). See also arXiv:0808.2729 [math.RT].

[AGa]    A. Aizenbud and D. Gourevitch, A proof of the multiplicity one conjecture for GL(n) in GL(n + 1)., arXiv:0707.2363v2 [math.RT] (2007).

[AGb]    A. Aizenbud, D. Gourevitch, Smooth Transfer of Kloosterman Integrals (the Archimedean case), arxiv:1001.2490, to appear in the American Journal of Mathematics.

[AGRS10]    A. Aizenbud, D. Gourevitch, S. Rallis, G. Schiffmann, Multiplicity One Theorems, Annals of Mathematics 172/2 (2010), see also arXiv:0709.4215 [math.RT].

[AGS08]    A. Aizenbud, D. Gourevitch, E. Sayag : (GLn+1(F),GLn(F)) is a Gelfand pair for any local field F, Compositio Mathematica, 144 (2008), see postprint: arXiv:0709.1273[math.RT]

[Aiz]    A. Aizenbud, A partial analog of integrability theorem for distributions on p-adic spaces and applications, arXiv:0811.2768, To appear in the Israel Journal of Mathematics.

[BCR]    Bochnak, J.; Coste, M.; Roy, M-F.: Real Algebraic Geometry Berlin: Springer, 1998.

[Ber82]    J. Bernstein, P-invariant Distributions on GL(N) and the classification of unitary representations of GL(N) (non-Archimedean case), Lie group representations, II (College Park, Md., 1982/1983), 50–102.

[BZ76]    J. Bernstein, A.V. Zelevinsky, Representations of the group GL(n,F), where F is a local non-Archimedean field, Uspekhi Mat. Nauk.

10/3, (1976).

[Cou]    S. Coutinho, A Primer of Algebraic D-Modules Cambridge University Press.

[Dei]    Anton Deitmar, A First Course in Harmonic Analysis

[Dre00]    J.M.Drezet, Luna’s Slice Theorem And Applications, 23d Autumn School in Algebraic Geometry ”Algebraic group actions and quotients” Wykno (Poland), (2000).

[GS]    I.M. Gelfand, G. Shilov Generalized functions volumes I,II.

[GP74]    V. Guillemin, A. Pollack, Differential Topology Englewood Cliffs, N.J. : Prentice-Hall, 1974.

[Hör]    L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Grundlehren der Mathematischen Wissenschaften 256. Springer-Verlag, Berlin, 1990.

[HR]    E. Hewitt, K. Ross, Abstract harmonic analysis. Volume I

[Kem]    G. Kempf , Algebraic Varieties. Cambridge University Press, 1993.

[Kob]    N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, Volume 58 of Graduate Texts in Mathematics Series Edition 2, Springer-Verlag, 1984

[LV]    G. Lion, M. Vergne, The Weil representation, Maslov index and theta series, Progress in Mathematics, 6, Boston: Birkhauser.

[Tre]    F. Treves, Topological vector spaces, Distributions and Kernels. Academic Press, New York and London, 1967

[Ser64]    J.P. Serre: Lie Algebras and Lie Groups Lecture Notes in Mathematics 1500, Springer-Verlag, New York, (1964).

[Shi]    Shiota, M: Nash Manifolds. Lecture Notes in Mathematics 1269 (1987)

6. E-mail list

To join/un-join the course e-mail list send me an e-mail (from the address you wish to join/un-join) with subject join/un-join me to gen-fun-5776”. To send a message to the course mailing list send me an e-mail with subject “e-mail to gen-fun-5776 – the subject of your message”.

7. How to get credit for the course?

All of the information below is conditional, it might change if it will turn out that my expectations are too high or too low. My aim is that any student who meets the prerequisites will be able to get at least 90 if he or she does a reasonable amount of work (3-4 hours a week in addition to the lectures and practice sections).

The course grade will consist mainly of the following 3 equal parts.

Writing detailed lecture notes of 1 lecture. It should be well-written (preferably in LaTex) and should include details even if they were omitted in class (except those which were declared as homework or out of scope).
Writing a detailed solution of 1 homework assignment. The homework will be given during the lecture and possibly some additional assignment will be given in the end of the lecture or by e-mail. Again the solution should be well written (preferably in LaTex).
Presenting a solution to 1 homework assignment in the class.

Some remarks:

  • It might be that instead of doing 3 different assignments, you will be asked to do 2 (or maybe 3) assignments of the first type and 1 (or 0) of one of the other types.
  • You should not do 2 of the above assignments in the same week (and preferably not on adjacent weeks).
  • Each of the above assignments should be done in a week.
  • If you wish to make one of the assignments in the first week, you should approach me in the end of the lecture. If there will be more than one student who wants to do the same assignment, the priority will be given to M.Sc. students, and then it will be decided randomly.
  • The other weeks will be decided in the end of the second lecture. Again, with priority to M.Sc. students.
  • You may discuss the your assignment with me or with anyone else :-), but you should write/present it by yourself.
  • If the written assignment is done on unsatisfactory level (less then 80% credit), you will be asked to make corrections, this can add you up to 80% of the remaining credit.

Bonus assignments.

In addition, throughout the course, there will be given bonus assignments. Each assignment can give one student up to 15 points bonus. If 2 (or 3) students want to do one bonus assignment together, the points will be split. If more than one student wishes to do one bonus assignment (and they do not want to do it together) the priority will be given first to those who did not do any bonus assignments (or did a smaller part) and then to M.Sc. students.


Although the regular homework will not be graded, you are strongly encouraged to do it and write it down. It is crucial for the understanding of the material.

It is a good idea to do it in groups of 2-3 students. Each group can meet once a week (preferably not on the day of the lecture or an adjacent day, so your study will have no large gaps). Please let me know about the membership of your group and when do you meet, so I can try to be around, in case you need me.

8. Lecture notes1

Lectures 1-9 (by Itay Glazer; based on previous lecture notes by Noam Kahalon)

Exercises sessions (by Yotam Hendel)

9. Homework and solutions1

Homework 1-10 (by Yotam Hendel)

HW 1 solutions (by Yotam Alexander)

HW 2 solutions (by Dan Mikulincer)

HW 3 solutions (by Dan Mikulincer)

HW 4 solutions (by Yotam Alexander)

HW 5 solutions (by Shai Keidar)

HW 6 solutions (by Shai Keidar)

HW 7 solutions (by Yotam Alexander)

HW 8 solutions (by Dan Mikulincer)

HW 9 solutions (by Shai Keidar)

10. See also

5774 coures

Coures dropbox folder

1. The notes and the solutions are writen by the students, they contain some mistakes. Students that wish to upload corrected versions are wellcome to send them to me.