Author: Kowalski

The complex number obtained by taking the product of the integer two, the imaginary unit i, and the area π of a disc of radius 1 can be written — in principle — in six different ways:

$latex 2i\pi,\ 2\pi i,\ i2\pi,\ i\pi 2,\ \pi 2 i,\ \pi i 2.$

However, as far as I know, only the first two are commonly used. Or, to be more precise, I use exclusively the first form (which is much more euphonious when pronounced in French), and most everyone else uses the second form, with laughable justification.

Every time I meet Peter Sarnak, he mentions his deep chagrin at being unable to blog, a sad state of affair which he attributes to his inability to type (straight quote). However, as he points out, he does the next best thing: he writes letters to various correspondents to explain a variety of topics, and then puts them on the web (together with some other papers, notes for lectures, and a few papers). And all this is well-worth looking at for inspiration and information. And since the address doesn’t seem to be as well-known as it could, here is the link.

Highlights of these are (to my mind at least):

* The Schur lectures on Arithmetic Quantum Chaos (1993); they have been published, but are not so easy to find. There has been a lot of recent progress (which I discussed briefly here; Sarnak also has a short letter on this topic, and is readying a more complete version as he prepares a lecture next week at IAS).

* The letter to Lagarias on integral Appolonian packings, a very beautiful topic where spectral theory of infinite covolume discrete subgroups meets number theory and geometry.

* The Notes on the Generalized Ramanujan conjectures, a great educational read for people who (like me) have been mostly comfortable with classical (GL(2)) modular forms, but want to see and learn what happens in higher rank.

* The letter to Morawetz on the supremum norms of eigenfunctions; again, it’s a topic where new phenomena occur when one moves beyond the classical case of hyperbolic surfaces (which still contains many mysteries, of course).

Until I borrowed it recently from a friend, I hadn’t looked at the autobiography of L. Schwartz. As it turns out, I found that his memories of the invention of the theory of distributions were quite fascinating (some other parts, like learning that he did not immediately dismiss the possibility of becoming head of some trotskyist party after the war, were amusing; as Wikipedia delicately puts it, he was “leaning towards socialism”).

He explains that he was looking for a definition of “generalized function” in order, roughly, to put on a better footing the notion of weak solutions of certain partial differential equations (such notions already existed in the work of Leray, among others): the problem was that there was a way to define what it means for a non-smooth function f to satisfy a PDE

$latex \sum_{p}a_p D^p f=0$

but there was no definition or proper meaning of the various terms in such a sum!

Then he says that, during one night in November 44 (la plus belle nuit de ma vie), he had a flash of insight; he worked feverishly (fiévreusement) until the end of 44, having described the solution to Henri Cartan and Bourbaki and received their enthusiastic endorsement.

Then — and this is the fascinating part I had never heard about — he realized his definition was not the right one! It was more complicated than the definition which he found in Grenoble in late 1944, and from almost any rational analysis (after the facts), it should never have come before.

Basically, his former definition was the following: an 11/44-distribution was supposed to generalize (and encompass) the notion of a smooth function φ (defined on the real line) by generalizing the convolution operator T_φ associated to φ:

$latex T_{\phi}\,:\, f(x)\mapsto f\star \phi (x)=\int_{\mathbf{R}}{f(y)\phi(x-y)dx}$

which he saw (already a new step) as defined on the space of test functions (i.e., smooth functions with compact support), and with image in the space of smooth functions. So he considered all such operators, say

$latex T\,:\, f\mapsto T(f)$

satisfying some additional properties (T must commute with convolution operators associated to smooth functions with compact support, and T must be continuous in some suitable sense). Because of the well-known property

$latex (f\star \phi^{\prime})(x)=(f\star \phi)^{\prime}(x)=(f^{\prime}\star \phi)(x)$

linking convolution and differentiation, he could easily define a derivative for all such operators by defining

$latex T^{\prime}(f)=T(f^{\prime}),$

and thus give a meaning to all the terms at least in a constant coefficient PDE (such as the Laplace equation) applied to such an operator.

What was wrong? As Schwartz tells it, the first “bad taste” (goût amer) was the fact that defining the product of an 11/44-distribution with a smooth function u required a rather painful definition (emberlificotée). The second was that he didn’t succeed at all in defining a Fourier transform.

And then, he said, he suddently realized that he could have used a much simpler point of view; and this is the one which is currently universally used: instead of generalizing convolution operators, a distribution generalizes the linear functional

$latex F_{\phi}\,:\, f\mapsto \int_{\mathbf{R}}{f(x)\phi(x)dx}$

(there is a link however, since one has

$latex F_{\phi}(f)=f\star \tilde{\phi}(0),\quad\quad\text{where}\quad\quad \tilde{\phi}(x)=\phi(-x)$

for all f). The derivative is now obtained by generalizing the classical integration by parts:

$latex F_{\phi^{\prime}}(f)=\int_{\mathbf{R}}{\phi^{\prime}(x)f(x)dx}=-\int_{\mathbf{R}}{\phi(x)f^{\prime}(x)dx=-F_{\phi}(f^{\prime})$

(because f is smooth, we can differentiate it; because it has compact support, the boundary terms vanish), and the product of a distribution with a smooth function u poses no problem:

$latex (uF)(f)=F(uf)$

which is well-defined since uf is again smooth with compact support. So linear partial differential operators can immediately by applied to distributions. The Fourier transform requires some care: the right space of test functions (the “Schwartz space”…) is needed to make sense of the definition suggested by the Plancherel formula:

$latex \hat{F}(f)=F(\hat{f}),$

and the Schwartz space is just what is needed so that this definition work perfectly.

As Schwartz points out, this point of view of linear functionals should probably have been the “obvious” one: it is also a generalization of the approach to measure theory and integration by duality with continuous functions, which was popular in France at the time (after A. Weil’s book on integration in topological groups): a measure μ is defined as a (suitably positive and continuous) linear map,

$latex f\mapsto \mu(f)$

where f is assumed to be continuous and compactly supported. The parallel is clear… at least a posteriori.

I have mentioned the Barnes function already once recently, as it arose in a marvelous identity of Widom. Today, here is a look at the duplication formula…

Recall that the Barnes function is defined by

$latex G(z+1)=(2\pi)^{z/2}e^{-(z(z+1)+\gamma z^2)/2}\prod_{n\geq 1}{(1+z/n)^n\exp(-z+z^2/(2n)}}$

for any complex number z (it is indeed an entire function; here γ is the Euler constant). Besides the ratio

$latex \frac{G(1+z)^2}{G(1+2z)}$

which appeared in the previous post, the following expression

$latex f_{Sp}(z)=2^{z^2/2}\frac{G(1+z)\sqrt{\Gamma(1+z)}}{\sqrt{G(1+2z)\Gamma(1+2z)}}$

also occurs naturally in the asymptotic formula of Keating-Snaith for characteristic polynomials of unitary symplectic matrices. For some reasons, this didn’t look nice enough to me (e.g., the computation of

$latex f_{Sp}(n)=\frac{1}{(2n-1)(2n-3)^2\cdots 1^n}$

for n a positive integer is somewhat laborious). But because of the shape of the term in the square root of the denominator, one can try to simplify this using the basic relation

$latex G(1+2z)\Gamma(1+2z)=G(2+2z)=G(2(1+z)),$

and the duplication formula for G(2w), which “must exist”, since the Barnes function generalizes the Gamma function, for which it is well-known that

$latex \Gamma(2z)=\pi^{-1/2}2^{2z-1}\Gamma(z)\Gamma(z+1/2)$

(usually attributed to Legendre).

So there is indeed a duplication formula; I picked it up from this paper of Adamchik (where it is cleaner than in the paper of Vardi referenced by Wikipedia), who states it as follows:

$latex G(2w)=e^{-3\zeta^{\prime}(-1)}2^{2w^2-2w+5/12}(2\pi)^{(1-2w)/2}G(w)G(w+1/2)^2G(w+1).$

This is promising since in the case considered, the terms involving G, after application of the formula to w=1+z, become

$latex G(z+3/2)^2G(z+1)G(z+2)=\Gamma(z+1)G(z+1)^2G(z+3/2)^2,$

leading to a cancellation with both the gamma and G-factors in the numerator!

And although the term ζ'(-1) is maybe a bit surprising, a moment’s thought shows it can be eliminated by simply plugging in any specific value of z and using the resulting formula to express it in terms of a specific value of G(z). Indeed, taking z=1/2, and watching the dust settling lazily across the page, we get the very nice expression

$latex f_{Sp}(z)=2^{-z^2/2-z-1/2}(2\pi)^{(z+1)/2}\frac{G(1/2)}{G(z+3/2)}.$

In fact, it’s clear then that (at least for such purposes), it is best to write the duplication formula in a way which avoids ζ'(-1) altogether (losing a bit of information, since it’s somewhat interesting to know that this quantity is linked to G(1/2)):

$latex G(1/2)^2G(2w)=(2\pi)^{-w}2^{2w^2-2w+1}\Gamma(w)((G(w)G(w+1/2))^2.$

(This is still not tautological for z=1/2: it contains the value Γ(1/2)=π^1/2.) From the shape of this, number theorists, at least, would probably be curious to see what happens if one replaces the Gamma function with

$latex \Gamma_{\mathbf{C}}(w)=(2\pi)^{-w}\Gamma(w)$

which is the factor at infinity for the local field C. In that case, it seems natural to introduce

$latex G_{\mathbf{C}}(w)=(2\pi)^{-z(z-1)/2}G(w),$

for which we retain the induction relation of G with respect to Γ:

$latex G_{\mathbf{C}}(w+1)=\Gamma_{\mathbf{C}}(w)G_{\mathbf{C}}(w).$

In terms of these functions, the duplication formula is even nicer:

$latex G_{\mathbf{C}}(1/2)^2G_{\mathbf{C}}(2w)=2^{2w^2-2w+1}\Gamma_{\mathbf{C}}(w)((G_{\mathbf{C}}(w)G_{\mathbf{C}}(w+1/2))^2.$

Since I’m not sure these links are well-known (i.e., I didn’t know them myself until yesterday in one case, and until a few weeks in the other), it may be useful to point them out:

* The work-in-progress new edition of SGA 3 (the part of Grothendieck’s algebraic geometry seminar dedicated to algebraic groups), which is edited by Philippe Gille and Patrick Polo; aficionados of the functorial point of view will enjoy it for remarks like:

Pour obtenir un langage qui “colle” sans effort à l’intuition géométrique, et éviter des circonlocutions insupportables à la longue, nous identifions toujours un préschéma X sur un autre S au foncteur
$latex (Sch)_S \rightarrow (Ens)$
qu’il représente, et il est nécessaire de donner de nombreuses définitions de telle façon qu’elles s’appliquent à des foncteurs quelconques, représentables ou non.

which I translate as follows:

To obtain a language which “sticks” effortlessly to geometric intuition, and avoid circumlocutions which quickly become unbearable, we always identify a prescheme X over another S with the functor
$latex (Sch)_S \rightarrow (Ens)$
it represents, and it is necessary to give many definitions in such a way that they apply to arbitrary functors, whether representable or not.

(A footnote says this point of view was first really considered by P. Cartier in the theory of formal groups).

* The Algebraic Stacks Project, which is a book-in-constant-progress about algebraic stacks. It is (as far as I know) the mathematical writing project which is closest in style to Open Source software. Indeed, although one can download a single PDF file with the current version, it also has distributed version control access (using Git), a maintainer (in the sense that Linus Torvalds is maintainer of the Linux kernel), who is A. J. de Jong, a free license (GNU Free Documentation License), complete history, and so on. Looking at the log entries shows that the project is fast moving: the currently 1298 pages long PDF version is just a convenience local in time (the project page indicates that no published version is projected to appear).

New contributors are encouraged to participate, and one can see that it would be quite easy to start by correcting a few typos, then looking at a part where one has particular expertise, and so on.

Thanks to the availability of source files, one can even (say) decide to “fork” the project, and create a new/derivative version where the set-theoretic foundations are done differently; or one where wimpy 2-categories are throughout upgraded to n-categories. More realistically, the license allows (I think: I haven’t read it in full detail) someone to produce a printed edition of a given — possibly edited — version (just like Linux distributions used to be available as physical boxes representing, in effect, a particular state of many independent software projects, before such distribution became almost non-existent — except maybe as DVDs included in books).

I haven’t yet looked much at the actual content of the book, but I must say I find quite appealing the idea of writing such a text this way. Maybe I’ll try to read/browse through a sufficient part to write some kind of review of its style.