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:
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Some multilinear algebra including the Kernel Theorem and Volterra composition,
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Some Fourier analysis including the Bochner-Schwartz Theorem,
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An introduction to wavelets with a view to structure theorems for spaces of distributions or function spaces,
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Probability theory on spaces of distributions including the Lévy Continuity Theorem,
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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.
Best Answer
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...
Genuinely distributional proof of Poisson summation: http://www.math.umn.edu/~garrett/m/fun/poisson.pdf
Meromorphic continuation of distributions $|x|^s$ and ${\mathrm sgn}(x)\cdot |x|^s$ http://www.math.umn.edu/~garrett/m/fun/notes_2013-14/mero_contn.pdf
(Fancier version of the previous: mero cont'n of $|\det x|^s$ on $n\times n$ matrices (if this is interesting, I have some notes, or maybe it's a fun exercise). In particular, stimulated by a math-overflow question of A. Braverman some time ago, there are equivariant distributions (e.g., on two-by-two matrices) such that both they and their Fourier transform are supported on singular matrices... Wacky! http://www.math.umn.edu/~garrett/m/v/det_power_distn.pdf
Decomposition of $L^2(A)$ for compact abelian topological groups $A$ (by Hilbert-Schmidt, hence compact, operators). http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06c_cpt_ab_gps.pdf
Reconsideration of Sturm-Liouville problems (with reasonable hypotheses), to really prove things that are ... ahem... "suggested" in usual more-naive discussions.
Quadratic reciprocity over number fields (and function fields not of char=2) ... cf. http://www.math.umn.edu/~garrett/m/v/quad_rec_02.pdf (This presumes Poisson summation for $\mathbb A/k$...)
Explanation that Schwartz' kernel theorem is a corollary of a Cartan-Eilenberg adjunction (between $\otimes$ and $\mathrm{Hom(,-)}$), when we know that there are genuine (i.e., categorically correct) tensor products for "nuclear Frechet spaces", ... which leads to the issue of suitable notions of the latter. http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06d_nuclear_spaces_I.pdf
The idea that termwise differentiation of Fourier series is "always ok" (with coefs that grow at most polynomially), if/when the outcome is interpreted as lying in a suitable Sobolev space on the circle. And that polynomial-growth-coefficiented Fourier series _always_converge_... if only in a suitable Sobolev space. http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/03b_intro_blevi.pdf
** 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.