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I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are:

  1. Some multilinear algebra including the Kernel Theorem and Volterra composition,

  2. Some Fourier analysis including the Bochner-Schwartz Theorem,

  3. An introduction to wavelets with a view to structure theorems for spaces of distributions or function spaces,

  4. Probability theory on spaces of distributions including the Lévy Continuity Theorem,

  5. A study of homogeneous distributions and elementary solutions to linear PDEs.

My question is: What "cool topics/applications" would it be nice to include in such a course? I am particularly interested in examples with a high return on investment, i.e., that would not take too long to cover yet would provide the students with valuable tools for eventually a future research career in analysis. Please provide references where I can learn more about your suggestions. I would like some variety if possible. I got suggestions pertaining to probability, PDEs and mathematical physics, but it would be nice to get apps related to other areas of math.

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    $\begingroup$ I would say QFT in physics, but that's my favourite application; not sure if it's suitable for your course though. $\endgroup$ – Alan Jan 17 '17 at 18:23
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    $\begingroup$ Indeed. In fact one of my goals is to provide students with basic tools for going into mathematical QFT if they so wish. $\endgroup$ – Abdelmalek Abdesselam Jan 17 '17 at 18:26
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    $\begingroup$ I should add that prep for qft is not my only goal. I would like the course to provide a broad base that would allow the student to continue in a variety of directions. $\endgroup$ – Abdelmalek Abdesselam Jan 17 '17 at 22:16
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    $\begingroup$ There's always the starting point of "so what the heck is this Dirac delta thing I keep hearing about, really?" And for QFT people: "And why is it so hard to square it?" $\endgroup$ – Nate Eldredge Jan 17 '17 at 22:21
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    $\begingroup$ An interesting topic : some sufficient conditions for $f(z)$ an analytic function being the Laplace transform of a distribution. $\endgroup$ – reuns Jan 18 '17 at 0:20
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Unsurprisingly, the topics that occur to me have various connections to number theory (and related harmonic analysis) (and unclear to me what might have already been done in your course...):

EDIT: inserted some links... EDIT-EDIT: one more...

** Possibility of ranting about the limitations of pointwise convergence... especially when placed in contrast to convergence in Sobolev spaces...

  • Use of Snake Lemma to talk about mero cont'n of the Gamma function, via mero cont'n of $|x|^s$. :) http://www.math.umn.edu/~garrett/m/v/snake_lemma_gamma.pdf

  • Peetre's theorem that any (not necessarily continuous!) linear functional on test functions that does not increase support is a differential operator. (I have a note on this, which may be more palatable to beginners than Peetre's paper.) http://www.math.umn.edu/~garrett/m/v/peetre.pdf

  • Uniqueness of invariant functionals... As the easiest case (which is easy, but cognitive-dissonance-provoking, in my experience), proving that $u'=0$ for a distribution $u$ implies that $u$ is (integration-against) a constant. (Maybe you'd do this along the way...) http://www.math.umn.edu/~garrett/m/v/uniq_of_distns.pdf

  • ... this is not a stand-alone topic, but: the usual discussions of pseudo-differential operators (e.g., "symbols") somehow shrink from talking about quotients of TVS's in a grown-up way... If that hadn't been done earlier, and/or people had a (reasonable!) feeling of discomfort about the usual style of chatter in the psi-DO world, perhaps this could be happy-making.

  • Meromorphic/holomorphic vector-valued functions (cf. Grothendieck c. 1953-4, and also various of my notes) e.g., meromorphic families of distributions... E.g., the $|x|^s$ family on $\mathbb R$ has residues which are the even-order derivatives of $\delta$, and ${\mathrm {sgn}}(x)\cdot |x|^s$ has as residues the odd-order derivatives of $\delta$. http://www.math.umn.edu/~garrett/m/fun/Notes/09_vv_holo.pdf Depending on context, there are somewhat-fancier things that I do find entertaining and also useful. Comments/correspondence are welcome.

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    $\begingroup$ Wow! Thank you for your list of topics. I was indeed hoping for things related to number theory and group representations. When you say "I have a note" I suppose this is not just some scribbles in a drawer somewhere but a reference to a vignette or other file on your webpage. If so, I would appreciate it if you could, at your leisure, put URL links to them in your answer. $\endgroup$ – Abdelmalek Abdesselam Jan 27 '17 at 14:57
  • $\begingroup$ @AbdelmalekAbdesselam, yes, I have PDFs which I'll link-to in the near future. $\endgroup$ – paul garrett Jan 27 '17 at 17:15
  • $\begingroup$ Great! That would be very helpful. $\endgroup$ – Abdelmalek Abdesselam Jan 27 '17 at 18:06
  • $\begingroup$ Your notes are always great! But some of the links are dead. I specifically wanted to look at the determinant one (your third bullet point). Thanks in advance! $\endgroup$ – liuyao Dec 22 '17 at 8:46
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    $\begingroup$ Do you have any convenient references for the stuff about psi-DOs and quotients of TVS? $\endgroup$ – Willie Wong Dec 23 '17 at 3:22
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Two (edit: now four) not-so-usual examples come to my mind:

  • There is the proof of the central limit theorem using Fourier analysis, as done in Chapter 7 of Hörmander's book. It's a cornerstone of probability theory and the Fourier-analytic proof presented by Hörmander is a nice showcase of the power of distribution theory: one gets a conceptually deep and useful theorem in full generality with relatively little effort;
  • Something I have been meaning to try is to work out some "mathematical toy examples" of continuum limits of distributions supported on a lattice, as done e.g. in Chapter 15 of the book by R. Fernández, J. Fröhlich and A.D. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory (Springer, 1992). I do not know if it is a feasible task, but since you are a specialist is constructive QFT, you should be able to do it... Edit 3: a classical and rather charming example of this has just came to my attention: the so-called Bernstein polynomials, used by their namesake Sergei N. Bernstein (the same mathematician who found the family of inequalities - also named after him - for smooth functions with compactly supported Fourier transform which underlie the Littlewood-Paley theory discussed in my Edit 1 below) to provide a probabilistic proof of the Weierstrass approximation theorem through the weak law of large numbers. Taking formal adjoints of the sequence of Bernstein polynomial operators provides a general lattice approximation for distributions on compact rectangular domains. A rather extensive discussion can be found in the short and beautiful book of the same title by G. G. Lorentz (the mathematician who invented the so-called Lorentz interpolation spaces, not the Dutch physicist).
  • (Edit 1) A third one, which I am particularly fond of, is the dyadic Fourier analysis of Littlewood and Paley, which is strongly connected to point (3). The connection to renormalization group ideas is quite obvious, and it has become a standard tool for obtaining important inequalities in nonlinear analysis (Gagliardo-Nirenberg, Moser, Schauder), all the way to the monumental Nash-Moser implicit function theorem (as shown by Hörmander - him again - in the 70's). A nice pedagogical exposition of these ideas can be found the book by S. Alinhac and P. Gérard, Pseudo-Differential Operators and the Nash-Moser Theorem (AMS, 2007) - just watch out for the numerous typographic errors!
  • (Edit 2) A fourth one, which is not actually unusual in itself but rather in its standpoint, is that many interesting examples of distributions (principal values, partes finies, extensions of homogeneous distributions to the origin, etc.) are obtained as extensions of distributions initially defined in (often finite-codimensional) subspaces of test functions, collectively called renormalizations. What would be interesting, also regarding applications to physics, is to treat such examples from this unified viewpoint. Indeed, as it can be seen e.g. by (Bogoliubov-Parasiuk-)Hepp's and Epstein-Glaser's treatments of perturbative renormalization in QFT, the latter really amounts to renormalizing distributions in the above sense. As a famous dictum of K. Hepp goes (Proof of the Bogoliubov-Parasiuk Theorem on Renormalization, Commun. Math. Phys. 2 (1966) 301-326):

    "The renormalization theory (of Dyson) is in this framework a constructive form of the Hahn-Banach theorem." (pp. 302 - parentheses mine)

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    $\begingroup$ two good ideas. For the first one I should definitely include it because the cost in time would be minimal. For the second one, in fact I did just that in a previous mathematical physics topics course. I proved a very special case of Newman's CLT, namely the scaling limit of 1d Ising to white noise... $\endgroup$ – Abdelmalek Abdesselam Jan 18 '17 at 14:17
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    $\begingroup$ ...using cluster expansion techniques as in link.springer.com/article/10.1007%2Fs00023-011-0103-6 The point of this rather academic exercise was no so much the result itself but introducing the cluster expansion method and seeing the Levy Continuity Thm at work. However, this took about a week and a half. An easier thing to do is the Donsker invariance principle in S', i.e. the scaling limit of the simple random walk to Brownian motion seen as probability measure on S' as in these slides people.virginia.edu/~aa4cr/TalkLyonVfinal.pdf $\endgroup$ – Abdelmalek Abdesselam Jan 18 '17 at 14:21
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One of my favourite applications of 'basic' distribution theory, which actually requires most of the tools in your list to be fully apreciated, is the Malgrange--Ehrenpreis Theorem on the local solvability of arbitrary constant coefficients PDE. There are a few proofs, but the proof in M.Taylor's book on Pseudodifferential Operators (last Section of first Chapter) is especially suited for your course I think. It is a substantial theorem though, so it might or might be not a good idea to include it, depending on the length and size of the course.

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    $\begingroup$ A simpler result is Poisson's summation formula, which however can be a pointer to much deeper theories. I guess it is not necessary to give a reference for this one. $\endgroup$ – Piero D'Ancona Jan 17 '17 at 23:27
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    $\begingroup$ Thanks! I thought about the ME Thm in relation topic 5 that I mentioned but thank you for pointing out Taylor's treatment of this result. I'll have a look. $\endgroup$ – Abdelmalek Abdesselam Jan 18 '17 at 0:23
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    $\begingroup$ BTW which deeper theories do you have in mind in relation to Poisson summation? I can think of functional equations for the zeta functions but if you have other things in that vein I would be interested in that too. $\endgroup$ – Abdelmalek Abdesselam Jan 18 '17 at 0:37
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    $\begingroup$ A nice physical application of the Poisson summation formula is Kramers-Wannier duality in statistical mechanics, as done in Kadanoff's book on the subject. $\endgroup$ – Pedro Lauridsen Ribeiro Jan 18 '17 at 2:01
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    $\begingroup$ I know there is a connection with theta functions and modular forms, but my knowledge is limited to the immediate application to functional identities for the theta functions. This is included e.g. in the Wikipedia article on Poisson summation formula $\endgroup$ – Piero D'Ancona Jan 18 '17 at 10:51
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No so much as applications, but I would love to learn about the basics of hyperfunctions (of the Japanese school) in such a course, if only in one dimension. Specifically I think it gives an easier way to define the wavefront set, which is what's required to explain why we can't square the delta function (referring to a comment above). See also the question Is square of Delta function defined somewhere?

For what it's worth, I found the book

  • Kato & Struppa, Fundamentals of Algebraic Microlocal Analysis

particularly well written. The article in EoM is also great

https://www.encyclopediaofmath.org/index.php/Hyperfunction

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    $\begingroup$ Thanks for these references. I had a contact with this stuff, Fourier-Bros-Iagolnitzer transform etc. through a course by Lebeau a long time ago, but it was a bit hard for me. $\endgroup$ – Abdelmalek Abdesselam Dec 22 '17 at 10:28
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Let me mention two nice applications to Sobolev spaces:

The first result shows that if $f\in \mathcal{D}'$ and $\nabla f\in L^p_{\rm loc}$, then $f\in L^p_{\rm loc}$. For a short proof of this classical result see https://mathoverflow.net/a/296464/121665

For applications of this result to higher order Sobolev spaces see:https://mathoverflow.net/a/297392/121665

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