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.

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

Further developments around the work of Mochizuki

January 15, 2016

[Posted on january 15, 2016]

Two new things in that story :

  • Vesselin Dimitrov has just posted a preprint showing that, assuming the proof of Mochizuki is correct, it in fact is leading to effective results, something which, if I understand well, was not clear to experts (e.g. in Brian Conrad’s notes at least), so this looks like an important development
  • registration for the july IUT Summit at RIMS is now open

Portrait of Johannes Neudörfer, His Son, and Their Rubik’s Cube, after Nicolas Neufchâtel by Mike Licht on flickr

2016 outlook

December 31, 2015

[Posted on december 31, 2015]

Lots of events to look forward to next year :

In terms of ongoing projects :

Jupiter at a glance, by Hubble ESA on flickr

Hibernation

September 14, 2015

This blog goes into a long hibernation of several months.  It might come back to life in 2016.  Any comment will stay into the moderation queue in the meantime.

Château d’Amboise under snow and ice.

Alternative title : the snow triangle.

(February 2012, CC BY-NC-SA 2.0).

Playing with the Matiyasevich-Stechkin visual sieve

September 9, 2015

Back in 1999, Matiyasevich and Stechkin proposed to visualize the sieving process that singles out the primes by using a parabola, which is fairly natural in hindsight. Here is a graph from a geogebra version :

matiyasevich_stechkin_sieve

That puts the sieve in the realm of Euclidean geometry, and it is very tempting to play with this construction a little bit. Here are some half-baked ideas, feel free to mention better ones…

One could try to map the primes so obtained by various transformations associated naturally to a parabola :

  • trying inversions (in particular through the vertex, and through the focus) for various values of the inversion radius would result in the primes being compactified to various line segments, so not something terribly insightful nor of artistic merit
  • the same would occur if looking at the inverse of the projection from a circle
  • in a different direction, one could of course move the parabola around within conic sections, resulting in lots of elliptic (compact) and hyperbolic (non-compact) images of the set of primes [this has very probably been considered before, any reference and keyword would be welcome!]

Another line of thought would be to set up some sort of billiard dynamics (or indeed wave dynamics) inside the parabola, e.g. sending light rays of pulses from each prime and looking at the resulting interference patterns over time, that might lead to some artistically pleasing visual patterns (an idea that might be explored at a later date)

A final thing that comes to mind is to try and construct constants that encapsulate some special features from the complete construction. One possibility I’ve looked at is to add the areas of the quadrilaterals that surround each prime, but if I got this right the sum is 2\sqrt{2} \sum_{p\in\mathbb{P},\ p\geq 7}\frac{p-1}{(p-2)(p-3)}, which unfortunately diverges…

 

Early september newslets

September 2, 2015

A very anecdotal list of math-related newslets :

  • Mochizuki has posted yesterday a short linguistic document concerning the morphological aspects of the terms “anabelioid” and “Frobenioid”
  • A conference in the memory of Nash is due to take place in Princeton at the end of october
  • a paper with the eye-catching title Defining \mathbb{Z} in \mathbb{Q} by Jochen Koenigsmann has been accepted in revised form by Annals of Mathematics (apparently the process took 5 years!)
  • in Sydney the dream job of Mathematician Game Designer is currently being advertised
  • as for well-known mathematicians from the hexagon, a recent event in Vietnam has featured Ngô Bảo Châu and Cédric Villani (who has cut his hair and grown a beard), while Sophie Morel is taking part in the #Add1Challenge by learning Turkish in 90 days, and today is of course René Thom day

swans_in_Metz
Swans in Metz (summer 2015, Public Domain).

A pick of recent items

August 26, 2015

As usual, in no particular order :

greenvariationsGreen variations (Provence, july 2015. Public domain.)

Recherche en mathématiques par des lycéens

August 18, 2015

Aux USA, il existe des structures pour permettre à des lycéens de faire de la recherche en mathématiques, un sujet que j’ai déjà évoqué sur Images des Mathématiques.

La revue Notices de l’American Mathematical Society, dans son numéro de septembre 2015, publie une interview de trois chercheurs qui sont à l’origine du programme PRIMES qui a lieu au célèbre MIT à Boston. Comme ils le disent en introduction :

Every year we receive numerous questions about our program from prospective students and their parents and also from academics who want to organize a similar program. Here we’d like to answer some of these questions, to share our experience, and to tell a wider mathematical community how such a seemingly impossible thing as mathematical research in high school can actually be done.

Traduction rapide de votre serviteur :

Chaque année nous recevons beaucoup de questions à propos de notre programme de la part d’élèves intéressés et de leurs parents et aussi d’universitaires qui veulent organiser un programme similaire. Ici nous voudrions répondre à certaines de ces questions, partager notre expérience, et dire à une plus large communauté mathématique comment une chose qui paraît aussi impossible que de la recherche en mathématique au lycée peut en fait être effectuée.

Je recommande à toutes et à tous de le lire, pour chasser les idées reçues et voir à quel point il est possible pour des élèves doués de 16 ou 17 ans de publier des résultats d’excellent niveau. Et ce, sans qu’ils deviennent ensuite forcément des mathématiciennes ou mathématiciens, si ils préfèrent ensuite faire médecine ou une école d’ingénieur c’est très bien aussi.

Petits ajouts pour lire le texte avec profit:

  • eleventh-grade” correspond chez nous à la classe de Première (élèves de 16/17 ans donc), et “twelfth-grade” correspond à notre Terminale
  • un des élèves cités en exemple dans le texte est Ravi Jagadeesan, auteur notamment d’un article présentant un nouvel invariant utile dans une des théories de Grothendieck qui est téléchargeable ici en pdf, article qui lui a rapporté une bourse de $50000 de l’Institut Davidson pour financer ses études supérieures. Et ce n’est pas son seul article.
  • Pour toute une liste d’autres profils d’élèves avec leurs travaux, voyez cette page du site du programme PRIMES

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