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 :


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 (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

Another short set of summer items

August 15, 2015

Recently spotted over the web :

diamond-shaped synchronicity by Karen Green on flickr

Mid-summer bits and pieces

August 8, 2015

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

July 26, 2015

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.

primesonhilbert7 primesonhilbert7nocurve primesonhilbert8 primesonhilbert8nocurveAnd now iteration 10 and 11 (making the points on the 10th somewhat larger for better viewing).


Short news

July 16, 2015

In no particular order:



June 26, 2015

L’été dans le Luberon by decar66 on flickr


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