Late april items

April 30, 2016

[Posted on april 30, 2016]

In no particular order :

  • the latest issue of Gazette de Mathematiciens has just appeared, it  includes an introduction to the Berkovich line by Jérôme Poineau, and also part of an exchange between Alice Jacquet and Claire Mathieu on the work of the latter and her collaborators on the emergence of a glass ceiling in social networks (here is the paper of Mathieu et al itself, and note also a talk about this at BNF next month) among lots of other things
  • Claire Voisin has very recently been elected to the newly created Chaire de Géométrie Algébrique du Collège de France
  • an article from the Harvard Crimson about tenure choices in the math departement
  • a curious case of nearly full Open Access : Elsevier-owned Comptes Rendus Mathématiques will henceforth provide some of its articles for free, for that the corresponding author must have a french affiliation (technically, an email at an insitution based in France) otherwise the article will stay behind a paywall for readers but with the strange and fortunate freedom to post the final pdf on the institutional server of the authors (but not on a preprint server for 3 years).
  • the Spiegel has recently featured Peter Scholze (behind a paywall)

Low clouds over the lake, by claudiadea131 on flickr

That arXiv questionnaire, and other news

April 16, 2016

[Posted on april 16, 2016]

Filling that arXiv questionnaire (now offline) was interesting. No idea how many people answered, and more importantly how biased that sample will turn out to be. I hope, for the sake of transparency, that they’ll quickly make the numbers freely available (the free-comment sections are of course private).

Also, it’s informative to see how varied opinions can be. I do agree with some points made by Izabella Laba in her blog post : no comments please (think low quality MO questions that quickly and deservedly get many downvotes, or the sometimes very irrelevant comments made by amateurs on blogs). That would surely drive lots of serious folks away. Some people agree, others don’t. On the other hand,  flagging “substantial overlap” could be useful if properly defined, IMHO.

As for citations stats and tools, taking a well-known preprint that never got formally published, by just clicking on the NASA ADS link one easily gets useful citation tools, while the blog trackbacks are suitably moderated. Not sure what could be added on top of that.

In other news :

  • topically, a math.GM paper on Navier-Stokes made it to a local story
  • a wonderfully clear and interesting talk by Mireille Bousquet-Mélou at CIRM (in french, but with slides in english) on plane lattice paths avoiding a quadrant, a topic related to a series of works done in the past 15 years by lots of people (and where many nice things occur, like the issue of finiteness or not of a certain group naturally associated to the path counting method)

  • an interview of Manjul Bhargava in CNRS News made after the conference mentioned in the previous post
  • a job ad for a mathematician in the gaming industry in Dublin

Early april news

April 5, 2016

[Posted on april 5, 2016]

In no particular order :

  • Laurent Lafforgue recently gave a colloquium (direct link to the mp4 file) on Grothendieck Toposes in Nantes “based on his conversations with Olivia Caramello” (he is announcing a long common text in the process of being written, to appear on his website at some point)
  • registration opened a few days ago for the summer school at Institut Fourier on Geometric Analysis, Metric Geometry and Topology
  • a workshop on Geometric Langlands and a conjecture of Fargues is taking place this week in Oberwolfach, while a workshop on the work of Vincent Lafforgue will take place at AIM next december
  • a recent and highly interesting AI paper that takes into account what makes things look natural to humans (inner sense of what is physically possible or not, and first-person view of the world) by Brenden M. Lake, Tomer D. Ullman, Joshua B. Tenenbaum and Samuel J. Gershman ; the fact that these gentlemen work at high-profile institutions will hopefully turn this paper into a kind of benchmark along which other folks will want to test their systems
  • Manjul Bhargava gave several talks in Paris yesterday, including one on Ramanujan’s work (the movie is announced for september in France, but will be released in just a few weeks in several countries), as part of the UNESCO’s International Conference on the Zero

Blue in the shades, by coniferconifer on flickr

Poisson, Guinand, Meyer

March 27, 2016

[Posted on march 27, 2016]

A recent highly interesting paper by Yves Meyer (PNAS paywalled, local version at ENS Cachan, and seminar notes) constructs explicitly new Poisson-type summation formulas (building on previous little known work of Andrew-Paul Guinand  and an existence result of Nir Lev and Aleksander Olevskii) : the big difference with Poisson summation is that the new formulas do not have support on a lattice but only on a locally finite set (and then provide new examples of crystalline measures).

Since these new results involve some arithmetic (see below) I’ve asked over at MO whether this was known to number theorists, but there hasn’t been any immediate answer, so perhaps not and there’s probably room for interesting further work on the topic.

To state things very explicitely (for my own benefit, but also just for the beauty of it), here are the formulas taken directly from Meyer’s paper :

Poisson (Dirac comb case): on a lattice \Gamma\subset\mathbb{R}^n and its dual \Gamma^* we have for any function f in the Schwartz class \mathcal{S}(\mathbb{R}^n) that

\displaystyle \mbox{vol}(\Gamma) \sum_{\gamma\in\Gamma}f(\gamma) = \sum_{\eta \in\Gamma^*}\widehat{f}(\eta)

Poisson (corollary of Dirac comb case) : for every \alpha,\beta \in\mathbb{R}^n we have (in terms of distributions to make the comb more explicit still)

\displaystyle \mbox{vol}(\Gamma) \sum_{\gamma\in\Gamma +\alpha} e^{2i\pi\beta .\gamma}\delta_{\gamma} = e^{2i\pi\alpha .\beta} \sum_{\eta \in\Gamma^* +\beta} e^{2i\pi\alpha .\eta}\delta_{\eta}

Guinand : define for any n\in\mathbb{N} the number of sums of three squares that equal to n by r_3(n) (by Legendre’s theorem this is possible only for those n not of the form 4^j(8k+7)). Then introducing Guinand’s distribution (acting on functions of the variable t)

\displaystyle \sigma := -2\frac{d}{dt}\delta_0 + \sum_{n=1}^{+\infty} \frac{r_3(n)}{\sqrt{n}} (\delta_{\sqrt{n}}-\delta_{-\sqrt{n}})

then we have \langle \sigma ,f\rangle = \langle -i\sigma ,\widehat{f}\rangle .

Meyer (first example) : introducing the function \chi on \mathbb{N} (this is a clash of notation with Dirichlet characters) by \chi(n)=-\frac{1}{2} when n\not\equiv 0\pmod{4}, \chi (n)=4 when n\equiv 4\pmod{16} and \chi (n)= 0 when n\equiv 0\pmod{16} then with the distribution

\displaystyle \tau := \sum_{n=1}^{+\infty} \frac{\chi(n)r_3(n)}{\sqrt{n}} (\delta_{\frac{\sqrt{n}}{2}}-\delta_{-\frac{\sqrt{n}}{2}})

we have \langle \tau ,f\rangle = \langle -i\tau ,\widehat{f}\rangle .

The support of \sigma and \tau are thus defined as subsets of \{\pm\sqrt{n}|n\in\mathbb{N}\} and \{\pm\frac{\sqrt{n}}{2}|n\in\mathbb{N}\} by their respective arithmetic conditions, and thus are definitely not equally spaced lattice points.

Meyer (second example) : with the distribution

\displaystyle \rho := 2\pi\delta_{\frac{1}{2}} +2\pi\delta_{-\frac{1}{2}} + \sum_{n=1}^{+\infty} \frac{\sin(\pi\sqrt{n})r_3(n)}{\sqrt{n}} (\delta_{\frac{\sqrt{n}}{2}+\frac{1}{2}}+\delta_{\frac{\sqrt{n}}{2}-\frac{1}{2}}+\delta_{-\frac{\sqrt{n}}{2}+\frac{1}{2}}+\delta_{-\frac{\sqrt{n}}{2}-\frac{1}{2}} )

we have \langle \rho ,f\rangle = \langle \rho ,\widehat{f}\rangle (very nice!).

There are several other examples in Meyer’s paper, as well as higher-dimensional constructions (that I haven’t absorbed yet, so I’ll stop here).

Update (march 27): two relevant papers I’ve just found

  • On the Number of Primitive Representations of Integers as Sums of Squares by Shaun Cooper and Michael Hirschhorn published in Ramanujan J (2007) 13:7–25, which in particular provides the explicit formula r_3(n)=\sum_{d^2|n}r_3^p\big ( \frac{n}{d^2}\big ) where the function r_3^p is in turn explicited (p is a label standing for ‘primitive’)
  • Irregular Poisson Type Summation by Yu. Lyabarskii and W.R. Madych, published in SAMPLING THEORY IN SIGNAL AND IMAGE PROCESSING Vol. 7, No. 2, May 2008, pp. 173-186, which does prove a Poisson-type formula with irregularly spaced sampling points (but if I understand well the examples they mention at the end show it is still different from the results of Guinand and Meyer, to be confirmed)

Sphere packings and other news

March 23, 2016

[Posted on 23 march, 2016]

Apparently there have been recently two great advances on sphere packings in dimension higher than 3 :

  • a paper by Maryna Viasovska establishing the densest packing in dimension 8
  • another paper by Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Maryna Viazovska solving the 24-dimensional case (with the Leech lattice)

In other news :

Oranges, by dncnH on flickr

Early march items

March 5, 2016

[Posted on march 5, 2016]

For immediate consumption :

  • the latest issue of the Newsletter of the EMS is out, with several outstanding introductory pieces, a great read.   There’s also an article by Fabian Müller and Olaf Teschke who have measured the amount of published papers that are on the arXiv thanks to new tools provided by zbMATH. Among other things, they mention that  “for the publication year 2014, about 55% of research in algebraic geometry, algebraic topology and K-Theory is available through the arXiv but only about 10% in numerical mathematics and 1% in mathematics history or mathematics education“. I’m slightly surprised that the algebraic geometry figure isn’t into the 70-80% territory.
  • as most people know by now, Discrete Analysis, the arXiv orverlay journal managed by Sir Timothy Gowers aimed at challenging for-profit publishers, is now online (and has lots of nice features)
  • CNRS medals in mathematics have been awarded : a Silver Medal to Isabelle Gallagher (PDEs, in particular Navier-Stokes), and a Bronze Medal to Simon Riche (Geometric Representation Theory)
  • the 2015-2016 Cours Peccot will be given by Nicolas Curien (next may at Collège de France) and by Marco Robalo (TBA)
  • the 2016 Abel Prize is due to be announced soon, on march 15
  • Olivia Caramello has chosen to update her public controversy with category theorists
  • a 177 pages paper by Tye Lidman and Ciprian Manolescu identifies two Seiberg-Witten Floer homology theories. To the outsider, this looks very much like the length and depth of papers that cement the status of a future Fields Medalist. (As an anecdote, one can notice a math.SE question in the bibliography — yes, math.SE not MO).
  • a word of caution taken from the website of the IHP conference in honor of Margulis’ 70th birthday next june, which applies to all other conferences in France in that period : “Due to the European Football Championship (EURO 2016) to be held in France, June 10th-July 10th 2016, participants are encouraged to book their hotel room a long time in advance“.

 

064

Centre Pompidou Metz and the moon, winter 2016. Public domain.

February newslets

February 21, 2016

[Posted on february 21, 2016]

Spotted recently:

  • a fascinating discussion is taking place between Sir Timothy Gowers and Nikolai V. Ivanov on the future of mathematics with respect to the development of AI (the background being the two cultures paper of Sir Gowers, several posts on Ivanov’s blog starting here, and the 2013 paper of Ganesalingam-Gowers –which has yet to appear, apparently–, which Sir Gowers discussed again last year in Cambridge and in London)
  • the theoretical aspects of the recent discovery of gravitationnal waves by LIGO are being presented at IHÉS :  there is a dedicated website and also videos of a very clear four-part physics crash course by Thibault Damour, a central contributor in analytical relativity. Asking about these LIGO results on MO is/was apparently contentious… As for the mathematics involved in the signal detection, the CNRS website mentions Wilson wavelets.
  • Cédric Villani, in a very well written and documented article, discussed the current position of english as the universal langage for research. Trying to guess what would happen in the next few years or decades when automatic translation of scientific papers reaches maturity he sees 3 possibilities : Babel Tower, Universalist, Altruistic (for more see the article). Of course we are not there yet, so he concludes “Learning English is still a good idea.”  In comments, some other opinions are expressed, to which he replies.
  • the initial list of candidates for one of the 11 junior permanent positions at CNRS in pure & applied maths has appeared, it has 261 names so the ratio this year is 23.7 candidates per position…

Gonepteryx rhammi by xulescu_g on flickr

Elliptic curves over Q, and other news

February 10, 2016

[Posted on february 10, 2016]

Last week, a paper on the arxiv by Jennifer Park, Bjorn Poonen, John Voight and Melanie Matchett Wood presented heuristics which “suggests that there are only finitely many elliptic curves of rank greater than 21“, something which divides experts as I once read on MO. And just a few days ago, a new type of database of elliptic curves over \mathbb{Q} was announced by Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, William Stein and James Weigandt, which seems to go in the same direction as the aforementioned heuristic.

One thing leading to another, I wondered what related topic could be made into a good polymath project (i.e. with a full proof not too distant a prospect). One candidate may be the characterization of non-torsion rational points on elliptic curves over \mathbb{Q} of rank \geq 2 (the torsion part is obviously well understood, I see there is an undergraduate book by Joseph H. Silverman and John Tate which should be a nice read, and lots of stuff on the web exists, e.g. see this short undergraduate presentation).

It seems to have all the ingredients required (computational aspect, wide-audience topic, …). Five years ago Harald Helfgott asked questions on that topic on MO, and while it initially attracted interest, there has been no news since, unfortunately. That area of math is extremely active obviously, and has been for a long time, e.g. there’s a conference for database pioneer John Cremona’s 60th birthday that will take place next april in Warwick. So there would surely be enough people interested, while lots of software exists obviously which was used to compile the recent database, and also here’s a standalone plotter by Stefan Kebekus under the GPL just to mention one more item.

Update (february 15): well it wouldn’t have been wise to start a polymath on that topic just now, as today appeared a preprint by Sara Checcoli, Francesco Veneziano and Evelinda Viada which appears to solve this (if I understand correctly…), and much more! This paper looks highly important indeed.

In other news :

  • a day of festivities around the issue of a stamp celebrating Sophie Germain will be take place at IHP on march 18
  • the next day a Séminaire Bourbaki is scheduled, including a talk by Geordie Williamson. It is noticeable that the list of authors following the customary “[d’après…]” tend to be longer and longer these days
  • mathblogging.org had a change of URL
  • Cédric Villani has found yet another idea to make mathematians look like they’re not necessarily elitist ivory tower types, reviewing a concert of a symphonic metal band

Libros Libres, by Alan Levine on flickr

Errata in the age of Open Access journals

January 31, 2016

[Posted on january 31, 2016]

Nearly nine years ago (time flies!) on this blog I wondered aloud about errors in mathematical papers (errors that are made even by the best authors, in the best journals with very competent referees), and the problem of the subsequent errata that are possibly being missed by readers (at least for a good few months maybe). So I proposed at the time as a solution a centralised system listing all errata in a single place.

In the meantime, MO was born, and I asked there a question about when to take published results for granted at all, the consensus seemingly being: never.

So in the world (indeed, the era) of paper journals, it was hoped that the readers would find the erratum early enough that it doesn’t affect too nastily any work building on the problematic proofs. But in the age of Open Access online journals, one might wish things to change a bit.

To wit, the Open Access Electronic-only journal Algebraic Geometry has issued its first erratum at the end of last year. But, as of today, if one downloads the original paper, nothing warns the reader about the erratum.

Wouldn’t it be wise, and very feasible technically, to add above the title in the pdf of the original article a note like “an erratum for this paper has been issued” with a link to it ?

Some january newslets

January 30, 2016

[Posted on january 30, 2015]

In no particular order :

  • the january issue of Gazette des Mathématiciens has just appeared, including among many other things a very nice overview of the proof of the Willmore Conjecture by Fernando C. Marques and André Neves, and also a dossier on Open Access publications in general and the example of the recent conversion of Annales de l’Institut Fourier in particular
  • for another example of a topic that is being well digested, there’s the work of Vincent Lafforgue on the Langlands program, which beyond the course by Michael Harris at Columbia, will also be the subject of a course by Sophie Morel at IPM in Tehran
  • two recent interviews of mathematicians in video, both at CIRM : Sylvia Serfaty and David Ruelle
  • the ceremony awarding the 2015 Fermat Prize to Laure Saint-Raymond and Peter Scholze will take place in Toulouse at the end of march. Note that the wording for Scholze’s citation starts with “pour l’invention des espaces perfectoïdes…“, a welcome non-Platonistic choice
  • another useless experiment of this blog’s host : turn left by \pi/2 at n iff n\pm1 are twin primes, else make a fixed step (say of +50) on a straight line. Plot. Here are the results up to n=1500 ; n=100,000 ; n=500,000 ; n=1,500,000. Resulting insight : nil…

turnattwin1500_gscale

 

turnattwin100000_gscale

 

turnattwin500000_gscale

 

turnattwin1500000_gscale


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