Archive for the ‘mathematics’ Category

The Genestier-V.Lafforgue preprint is out, and V.Lafforgue’s crypto-currency proposal for refereeing

September 6, 2017

[Posted on september 6, 2017.]

Alain Genestier and Vincent Lafforgue have very recently posted a preprint which is the long-awaited follow-up to V.Lafforgue’s 2012 paper on the Langlands Correspondance (whose current version of august 2017 seems to be close to print, since it acknowledges the “extremely thorough” input from several referees).

On V.Lafforgue’s webpage there is a very interesting note from january 2017 titled A proposition to give value to the work of referees which introduces the idea of a crypto-currency to reward referees (!), provides concrete technical specifications to implement and test the idea, and explores potential drawbacks.

I haven’t seen it discussed on the web yet, but I very much hope it will become widely read, and that a suitable version will emerge and be given a try by several large institutions…

Programmer’s laptop by Wall Boat on flickr

Advertisements

Early september 2017 items

September 3, 2017

[Posted on september 3, 2017.]

Various short news :

  • Go Yamashita has posted a few days ago his version of Shinichi Mochizuki’s work on the abc conjecture (and he also has posted an excerpt of an email which, together with a footnote, alludes to some unethical behaviour from a certain I.F. Hopefully, whether or not it is the case, this type of controversy will stay away from the math itself.)
  • the arXiv overlay journal Épiga has finally released its first papers
  • it has been announced today that the french Secretary of State in Digital Affairs, Mounir Mahjoubi, has asked Cédric Villani to write a report on how France should define a strategy for the coming years regarding the rise of AI, due in 3 months (it will start from the report done 6 months ago on the topic)
  • earlier this summer, the Comité de candidature à l’organisation de l’ICM 2022 issued a communiqué saying that although the IMU had stated a preference for St-Petersburg and that it was customary for other candidacies to withdraw before the next Genral Assembly, they would still present the candidacy of Paris next year…
  • a high-quality conference on Differential Geometry will be held at IHÉS next december in memory of the late Marcel Berger
  • US-based mathematicians have the opportunity to apply to the collaboration grants of the Simons Foundation announced this week
  • Norbert Blum has withdrawn his paper on P vs NP
  • today at least Annals of Mathematics is running low on papers in the to appear section (I wonder if an issue ever got delayed due to a lack of accepted papers…)

Logic cookies by Steve Rainwater on flickr

News roundup, and binary Cantor orthogonality

August 15, 2017

[Posted on august 15, 2017.]

Summer news:

***

Completely unrelatedly, here’s probably another useless idea from this blog’s host. Consider a sequence (a_n)_{n\in\mathbb{N}^*} of binary numbers in [0,1] such that the number of digits of a_n is at least equal to n for all n. Then one can apply Cantor’s diagonal argument to extract a number that is uniquely defined (since a digit different from 0 must be 1 and vice versa) and different from all the a_n. Call the resulting number the “binary Cantor orthogonal” of that sequence. Can it have any useful properties?

Let’s look at the following example: a_n:=[1/p_n]_2, the binary version of the inverse of the n-th prime. That is, we do:

\displaystyle \begin{array}{ccl}  \frac{1}{2}& =& 0.1\\  \frac{1}{3}& = & 0.01010101\dots \\  \frac{1}{5}& =& 0.001100110011\dots\\  \frac{1}{7}&=& 0.001001001001\dots\\  \frac{1}{11}&=& 0.0001011101\dots\\  \vdots  \end{array}

 

then extract the diagonal of the decimal parts and invert it modulo 2. The resulting sequence is 0,0,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0… Unfortunately, the OEIS doesn’t know this particular sequence, so there’s probably nothing noticeable here.

Jardin des Roses in Rennes (a small area of this marvel).

August 2017. Public domain

Euclidean geometry, nominalism, and drawings

July 9, 2017

[Posted on july 9, 2017.]

In middle and high school, geometry is a very visual affair, always illustrated by drawings. Indeed, reasoning on a problem is usually made much easier by glancing at a sketch, even one not to scale.

Of course, problems can arise if the sketch is not able to capture the essence of the problem, as in the well-known missing square puzzle, which is a good opportunity to tell pupils about the difference between abstract mathematical thinking, and visual “proofs”.

Picture of the puzzle by wikimedia user Krauss

Please note that I am not saying “difference between ideal mathematical objects and their visual approximation” : that would be a Platonistic view, which is not at all compulsory. For instance, a nominalist take on this issue is completely possible, and preferable as far as I’m concerned. See Jody Azzouni’s take in Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics, in particular the beginning of its section 5:

Now,  in Euclidean geometry we have precise metric statements (a length of 2.5, an angle of \frac{2\pi}{3}), and also some that seem more topological in nature (points belonging to some part of the plane as a consequence of this or that, even though they could also be recast entirely in terms of an angle or length not having some exact value).

So what I’m wondering now is whether there is a framework for approximate geometry, saying that “any set of pseudo-lines [“topological lines”, something not really straight, and of varying thickness, just like on a drawing] and pseudo-points [some “fat dots”], when they are in such and such configuration, imply that another pseudo-line or pseudo-point has a certain property exactly”. In other words, a setting for which every actual drawing on a piece of paper is exactly capturing the essence of the question.

I would be very interested by any relevant comments or references on that topic. Has this been done already?

Some february 2017 newslets

February 18, 2017

[Posted on february 18, 2017.]

Firstly, some recent items:

 

Also to be noted, two upcoming auctions:

  • on march 14 (estimated at €20,000/30,000) in Paris, arare copy of the 1637 first edition of Descartes’ Discours de la Méthode (containing the famous appendix La Géométrie)
  • on february 22 (estimated at €200/300) in Lyon, a 326 pages manuscript c.1810 on dynamics and other topics (author unknown) [edit: won by the floor at €280]

Finally, it appears this blog was started 10 years ago, in what was definitely another era: before Polymaths, MO, arxiv overlay journals…and all the new official youtube channels in math and beyond.

The currency of mathematics: ideas vs proofs

February 12, 2017

[Posted on february 12, 2017.]

Quanta magazine has come up with yet another stellar wide-audience article, this time by Kevin Hartnett on the work of several authors in symplectic geometry.

It contains this great quote by Mohammed Abouzaid:

There are two conceptions of mathematics,” Abouzaid said. “There’s mathematics as: The currency of mathematics is ideas. And there’s mathematics as: The currency of mathematics is proofs. It’s hard for me to say on which side people stand. My personal attitude is: The most important thing in mathematics is ideas, and the proofs are there to make sure the ideas don’t go astray.

It’s probably the most reasonable take on that topic.

Now what are other areas of mathematics that have been impacted by these two conceptions in recent years? Of course, the work of Perelman and the controversy with the Cao-Zhu paper quickly comes to mind, but this was then modified by Cao-Zhu within a few months so that the ideas-conception won in that instance.

Are there others, either form the distant past or the recent few years? Feel free to mention any, that’s be insteresting to study.

The mooring line, by Bernard Spragg NZ on flickr

Mirzakhani, Lindenstrauss, Witten, McMullen, Zelmanov sign petition against Trump’s immigration EO

January 28, 2017

[Posted on january 28, 2017.]

Fields Medalists Maryam Mirzakhani, Elon Lindenstrauss, Curtis T. McMullen, Edward Witten and Efim Zelmanov are, with several other prominent US-based mathematicians, among the earliest signatories of the Academics Against Immigration Executive Order petition, and well done to them ! [Edit: Terence Tao and Vladimir Voevodsky also signed.][Further edit: so have Pierre Deligne, Vladimir Drinfeld and Andrei Okounkov.] [Further edit: the members of the Board of Trustees of the AMS also signed and issued a statement.]

All Cedram journals are now Diamond Open Access

January 17, 2017

[Posted on january 17, 2017.]

As mentionned previously on this blog, starting this month all Cedram journals are now Diamond Open Access, so it adds Annales de la Faculté des Sciences de ToulouseAnnales Mathématiques Blaise Pascal, and  Journal de Théorie des Nombres de Bordeaux to the others.  A fantastic piece of news, and I’ve updated my list of DOA Mathematics Journals to reflect this.

***

In other news:

  •  Olivia Caramello has put online her recent HDR Thesis as well as the (very laudatory) referee report
  • Notices of the AMS has a nice piece by Henry Cohn on the sphere packing breakthrough
  • talks by Emmanuel Lepage and by Wojtek Porowski on Shinichi Mochizuki’s IUT are taking place in Nottingham

Cliffs of Moher by khdc on flickr

(Alternative title: ‘compromise’ is not a swear word)

Further january 2017 items

January 10, 2017

[Posted on january 10, 2017.]

Another quick list:

  • yesterday László Babai announced that he could find a short workaround to restore his quasi-polynomial claim (with modifications to his arXiv paper detailed shortly). If this checks out it is a remarkable story! In any event, Harald Helfgott’s Bourbaki seminar on saturday promises to be highly interesting (live on youtube [link updated to the start of the opera-themed talk] around 4pm Paris time, that’s 10am Eastern) update: paper here.
  • the arXiv submission rates statistics are quite fascinating. I’d be very curious to know how this translates into number of individual authors.
  • several interesting new features in mathscinet, if only it was open access…

Uccello Perspective Study by ArtGallery ErgsArt on flickr

Early january 2017 items

January 4, 2017

[Posted on january 4, 2017.]

Two very recent things:

  • Danylo Radchenko and Maryna Viazovska have released an interesting preprint on Fourier interpolation on the real line, and while I do not understand it in any depth I’ve noticed that a formula at the end of section 7, namely that for odd Schwartz functions f one has f'(0)+\sum_{n=1}^{\infty}\frac{r_3(n)f(\sqrt{n})}{\sqrt{n}}=i\widehat{f'}(0)+ \sum_{n=1}^{\infty}\frac{r_3(n)i\widehat{f}(\sqrt{n})}{\sqrt{n}} (where r_3(n) is the number of representations of n as a sum of 3 squares) is exactly the little-known formula of Guinand that Yves Meyer rediscovered at the end of 2015 (see equation 9 in his now freely available PNAS paper) and which has already been mentionned on this blog and on MO (so I’ve told Radchenko and Viazovska about Meyer’s paper). That’s nice to see such a beautiful formula rediscovered twice and by different means! There must be deep and interesting connections between those two areas of math then…
  • at the Bourbaki seminar next week Harald Helfgott will lecture on the much heralded work of László Babai on the Graph Isomorphism Problem, and Helfgott has just announced that he could check most of the proof but had found an error in the time analysis, partly corrected by Babai, who has just issued a notice about the issue.

Stern Clara und Hilli by Kerstin (aka Ella T.) on flickr