E. Kowalski’s blog

Continuing our popular series of posts on the Barnes function (here and there), here is a useful remark which I picked up in a 1996 paper of Ehrhardt and Silbermann, Toeplitz determinants with one Fisher-Hartwig singularity: instead of the Weierstrass product expansion

where γ is the Euler constant, it may be better to use the alternate product expansion

where the Euler constant is not present anymore. This, in fact, can probably be considered to be the “right” analogue of Euler’s original definition of the Gamma function

(which I’ve also discussed earlier, as it occurred in a computation where it was much more to the point than the Weierstrass product).

Two items of interest concerning ETH and mathematics:

(1) Although being away in Princeton means I won’t be able to attend them, I’d like the mention the two Heinz Hopf Vorlesungen, tomorrow (or today, Tuesday October 20, for those in Europe) and Thursday, in auditorium HG G5 at 7pm; for the first time, these lectures are linked to the new Heinz Hopf Prize, and the prizewinner this year, Robert MacPherson, will speak about How Nature Tiles Spaces?. (There were earlier lectures in 2001, 2003 and 2005, but there was no prize at that time).

(2) Like last year, I would like to encourage all interested young PhD’s to apply to the postdoctoral positions of the mathematics department at ETH (the deadline is November 21 for full consideration). In addition, the Mathematics Institute of the University of Zürich (i.e., of the canton) has also some postdoc positions available, as well as a lectureship (one position is specifically in Arithmetic Geometry, to work with U. Derenthal, who is currently at Freiburg in Germany).

And if you are a senior applied mathematician, note also the full professor position currently open…

Yuri Manin, in a translation of a recent interview, available in the forthcoming issue of the Notices of the AMS:

I’m somewhat apprehensive that its [the Riemann Hypothesis] first solution might be a proof using blunt analytic methods. It will receive every imaginable prize, the solution will be acclaimed in every newspaper in the world, and all of this will be misleading because the “right” solution should be given in a wider context, which we already know. We even know several approaches to a solution.

(Note: I do not speak or read Russian, so I can not check the accuracy of the translation from the original article).

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:

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).