Recherche en mathématiques par des lycéens

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

Another short set of summer items

Recently spotted over the web :

• lots of new videos recently added on the CIRM’s youtube channel, including Boxer’s construction of torsion Galois representations
• there’s still a postdoc position available as part of the 4-year project (see blurb here) on the Langlands Program in the wake of Scholze’s ideas (it is not listed by either of the otherwise informative EMS jobs page and Mathjobs page for France)
• some of the slides from the Ghys conference are now available
• also available are videos of the last summer school at Institut Fourier on Geometric Measure Theory and Calculus of Variations
• the Ramanujan Biopic is due to be screened at the Toronto International Film Festival next month

diamond-shaped synchronicity by Karen Green on flickr

Mid-summer bits and pieces

1) There exists a handy map of french Masters in mathematics (note that it’s obviously too late to apply for this year in most cases).  I wonder if such a map exists in other countries.

2) Some people are really great speakers, here are two examples with 15 minutes proofs in videos at the upper undergraduate level :

•  Dror Bar-Natan on Arnold’s version of Abel-Galois non-resolubility of the quintic
•  Benson Farb on the Poincaré-Hopf index theorem

3) as an outsider, I’ve never understood the incredibly one-sided voting results of SMF elections : here they are for 2015, then 2014, and 2013. Anyhow, there’s about 500 to 600 people involed in pure maths in France these days then (yes, some people won’t vote or are not members, but still it’s probably the right order of magnitude).

 

Primes on space-filling curves

This is a variation on a theme by Ulam.  It is well-known that Ulam’s spiral is related to the high density of primes along some quadratic polynomials, as per Hardy & Littlewood’s conjecture F.

What about other curves, in particular space-filling curves like that of Peano or that of Hilbert ? Would one expect to see particular patterns ?

Well, not knowing what to expect, I’ve tried to look at Hilbert’s. Below are thus primes marked on iterations 7 and 8 of Hilbert’s curve, with also primes shown without the curve. (I’ve chosen the convention that has the first vertex at the top left corner, and I prefer not to show the quick and ugly code.)

End result : unfortunately I can’t quite spot anything too noticeable (that the primes avoid some diagonals is an easy consequence of the curve being built from units of 4 vertices, but beyond that…). Also, for some reason worpress.com wouldn’t allow my svg files, so these are uglier png versions…

Edit (3rd august 2015): since primes do rarefy, it was tempting to look at a few more iterations and see if at least there are more around the top left corner on a larger scale. This is indeed seen in the plots of iteration 10 and 11 that I’ve added after the original ones. Iteration 11 contains 4,194,304 integers, and thus includes a fair amount of primes. But apart from the top left corner denser region the plots in fact look really uniform, despite lots of not too small prime gaps in these ranges, so there’s  really no pattern there.

And now iteration 10 and 11 (making the points on the 10th somewhat larger for better viewing).

Short news

In no particular order:

• Canadian Zhuo Qun (Alex) Song has just become the most renowned IMO contestant ever, with 6 medals, including 5 gold and a perfect score a few days ago
• Terence Tao turns 40 tomorrow, july 17th, (which happens to be a good day for space exploration, but a bad day for a french polymath)
• the MSRI Summer Graduate School on Gaps between Primes and Analytic Number Theory has started this week
• the name is probably not copyrighted, so there’s a company called Nicolas Bourbaki, and it is currently hiring
• and now back to…

…Pluto!

æstas

L’été dans le Luberon by decar66 on flickr

A subjective pick of recent papers

In no particular order, some interesting-looking fairly recent papers :

• Special values of the Riemann Zeta function capture all real numbers, by Emre Alkan in Proc. AMS.  Shows that any real number can be approximated by a linear combination of Zeta at odd integers, with the coefficients explicitely known. Ideally, there might be something to extract that is possible to explain to undergraduates.
• Syntactic categories for Nori motives, by Luca Barbieri-Viale, Olivia Caramello and Laurent Lafforgue. (Obviously this is way over my head ; it is a collaboration that makes sense in light of the conference previously mentionned ; and this appears to be L.Lafforgue’s first collaboration, and first paper in english too.)
• Chromatic numbers of hyperbolic surfaces, by Hugo Parlier and Camille Petit to appear in Michigan Math. J.  Lots of interesting results there, including a hyperbolic surface with infinite chromatic number!
• An Interpolating Distance between Optimal Transport and Fischer-Rao, by Lenaic Chizat, Bernhard Schmitzer, Gabriel Peyré and François-Xavier Vialard, in the context of the Σ-Vision project.

In other news, the Epi-Science website is progressing apparently, with the documentation dated may and june 2015 for instance.

Quick observations concerning the Séminaire Bourbaki

Since the next Séminaire Bourbaki is due to take place next saturday, it is a good opportunity to make some quick observations.

Started in 1948, its goal as an expository seminar was clear from the start: the very first exposé was by H. Cartan on the work of Koszul rather than on his own results, and many recent advances were presented at the seminar (e.g. the same year Pisot reviewed the elementary proof of the PNT by Selberg and by Erdős).

The first issues regroup several years, and in 1968 things settled on the current format : 4 sessions during the academic year, held in november, january, march, and june. The year 2014-2015 is its 67th year, and will end with the 1103th exposé. One can notice the following :

• first exposé by someone not holding an academic position in France : Eilenberg in 1950
• first exposé in a language other than french : Tate in 1957  quickly followed by Harish-Chandra
• most cited exposé : according to Google Scholar (if I got it right) it is Grothendieck’s 1955 Produits tensoriel topologiques et espaces nucléaires
• first exposé by a woman : apparently in June 1968 took place an exposé by Claire Delaroche jointly with Kirillov (no idea whether both spoke), and for a woman alone it occured in november 1969 with Michèle Vergne
• were the work of all the Fields Medalists the topic of a seminar prior to the ICM announcements ? Nearly so, up to 1994 the only exceptions seem to have been (unless mistaken) Roth-F58 (but reviewed in the Dubreil seminar in 1956), Cohen-F66 (reviewed in november after the ICM), and Zelmanov-F94 (never reviewed!?).

 

Most of Grothendieck’s manuscripts to be digitized

According to an article by Stéphane Foucart in Le Monde, it has been announced at the Grothendieck conference in Montpellier that :

• about 15,000 pages of manuscripts held by the university were now in the process of being digitized (that’s the archive that was given to Jean Malgoire in the 1990’s, which in turn he donated to the university)
• about 50,000 pages found in Grothendieck’s Lasserre home will be digited by the Bibliothèque Nationale de France (of Gallica fame), once legal hurdles are cleared, and which contain a mix of mathematical and non-mathematical texts

Let’s see : assuming 50% of mathematical content, that’s (15,000+50,000)/2=32,500 pages.  Chopping that into bunches of 200 pages, that’s about 162 books. In comparison, Euler’s Collected Works are said to fill 60 to 80 books…

Recent talks on youtube: Manolescu, Ghys, Starr

Here are some interesting videos of talks that have been put on youtube in the past month or so :

– Ciprian Manolescu in Edinburgh

– Etienne Ghys, who is the first laureate of the Clay Award for Dissemination of Mathematics (richly deserved!), on his favorite groups (a series of 8 lectures in Portuguese at IMPA)

– Jason Starr, a series of 3 lectures in Moscow