Avraham (Rami) Aizenbud
אברהם (רמי) איזנבוד
Tuesday 15:00-18:00, Room 261.
Tuesday 14:30-17:30, Room 208 Goldsmith building.
If you would like to come to the oﬃce hours or to ﬁx some other time please e-mail me.
1. General information
4. Chronological list of topics
5. Literature List
6. E-mail list
7. How to get credit for the course?
8. Lecture notes
10. See also
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.
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)
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.
I expect the students to be familiar on a basic level with at least 80% of the following notions:
Each topic should correspond to approximately one week.
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].
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].
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].
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].
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].
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.
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].
Deﬁnitions, basic properties, Schwartz distributions supported on a submanifold, Schwartz functions on a product, pushforward, pullback of Schwartz functions. – References: [AG08].
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.
4.22. 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).
Introduction, Heisenberg uncertainty principle, qualitative uncertainty principle – simple example.
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.
4.29. Digression – Luna’s slice theorem.
– References: [Dre00].
– References: [AGb, section Appendix(es)].
Most of the material of the ﬁrst semester is contained in
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.
[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].
[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.
[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.
[Dre00] J.M.Drezet, Luna’s Slice Theorem And Applications, 23d Autumn School in Algebraic Geometry ”Algebraic group actions and quotients” Wykno (Poland), (2000).
[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.
[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
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”.
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.
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)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.