## Archive for the ‘mathematics’ Category

### 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

### Some december 2016 items

December 31, 2016

[Posted on december 31, 2016.]

Some items spotted recently:

• the next séminaire Bourbaki on saturday january 14 will feature an all-star list of speakers, and should be live on youtube
• the list of the top 10 journals ranked by MCQ for 2015 on mathscinet had Cambridge Journal of Mathematics in 4th place, impressive for a journal launched only in 2013
• the famous paper of Maryna S. Viazovska on sphere packings in dimension 8 has been listed in the ‘to appear’ section of Annals of Mathematics
• Alain Connes will give the first lecture of his annual course at Collège de France next week
• Ivan Fesenko will give next month a Colloquium style talk in Cambridge on Shinichi Mochizuki’s IUT Theory
• major blogs have recently advertised the fine AMS Open Math Notes website, and the reader may like to know that another great site is the math section of Cours En Ligne (hosted by CCSD), which contains a fair amount of notes in english beyond the ones in french
• two new programs are starting at MSRI next month: Analytic Number Theory lead by Terence Tao and Harmonic Analysis lead by Michael Christ and Michael Lacey, while there will be a graduate summer school in june/july on Soergel Bimodules lead by Benjamin Elias and Geordie Williamson and another one in july/august on  Automorphic Forms and the Langlands Program lead by Kevin Buzzard
• Jean-Yves Girard has recently released a new paper which is the third part of a series on transcendental syntax (at the boundary between logic and philosopy), and he also published a less technical book on this topic last september
• some New Year wishes from the President of the EMS (containing a shocking piece of information: “numerous colleagues registered for the Berlin congress but did not then pay, causing a financial headache for the organizers“)
• just a few days left to register a candidacy to become CNRS researcher
• a new website for biographies of mathematicians (as announced in Notices of the AMS)

Amphithéâtre, by Patrick Janicek on flickr

### Remarks on ζ(2n+1)

December 15, 2016

[Posted on december 15, 2016.]

Having for a few days stupidly and incorrectly claimed a proof of the irrationality of all $\zeta(2n+1)$ after getting confused with a definition (sic) for which I express my deepest apologies for any confusion this may have caused, a new version of the preprint has withdrawn that claim.

The little that survives then is the formula $\displaystyle \frac{\zeta(2n+1)}{r^{2n}}=(-1)^{n+1} \int\limits_{[0;1]^n} \left (\prod_{i=1}^{n}\frac{\log(x_i)}{x_i}\right ) \log\left (1-\left(\prod_{i=1}^nx_i\right)^r \right ) dx_1\cdots dx_n$ which for $r=1$, due to its product form, I thought might have some uses that the classical $\displaystyle \zeta(k)=\int\limits_{[0;1]^k} \frac{dx_1\cdots dx_k}{1-x_1\cdots x_k}$ might not have.

Related expressions that I then obtained, namely $\displaystyle \int\limits_{[0;1]^2} \frac{\log(x)}{x}\frac{\log(y)}{y}\log(1-(xy)^1) \log(x)\log(y) (xy)^{2k+1} dxdy =$
$\displaystyle \frac{4\zeta(6)}{(2k+1)} + \frac{4\zeta(5)}{(2k+1)^2} + \frac{4\zeta(4)}{(2k+1)^3} + \frac{4\zeta(3)}{(2k+1)^4} + \frac{4\zeta(2)}{(2k+1)^5} -\frac{4}{(2k+1)^6}\sum_{n=1}^{2k+1}\frac{1}{n}-4\sum_{j=1}^{6}\left (\frac{1}{(2k+1)^j} \sum_{i=1}^{2k+1}\frac{1}{i^{7-j}}\right )$

and

$\displaystyle \int\limits_{[0;1]^2} \frac{\log(x)}{x}\frac{\log(y)}{y}\log(1-(xy)^2) \log(x)\log(y) (xy)^{2k+1} dxdy =$
$\displaystyle \frac{63\zeta(6)}{8(2k+1)} + \frac{31\zeta(5)}{4(2k+1)^2} + \frac{15\zeta(4)}{2(2k+1)^3} + \frac{7\zeta(3)}{(2k+1)^4} + \frac{6\zeta(2)}{(2k+1)^5}$
$\displaystyle +\frac{8\log(2)}{(2k+1)^6}-\frac{4}{(2k+1)^6}\sum_{i=1}^{k}\frac{1}{2i+1}-8\sum_{j=1}^{6}\left (\frac{1}{(2k+1)^j}\sum_{i=0}^{k}\frac{1}{(2i+1)^{7-j}}\right )$

do not seem to be of any help towards a proof of irrationality of at least $\zeta(5)$.  A combination of those two expressions, perhaps with other expressions such as those of Vasilyev, might allow to improve matters, but I was not able to find how.

### Some december 2016 short news

December 4, 2016

[Posted on december 4, 2016.]

Seen recently:

• the december issue of the Newletter of the EMS has appeared, with lots of interesting historical informations on the work and era of Shannon, Atiyah and Vinberg in particular
• results of the TIMSS international study have created a noticeable press coverage in France,  since they show that the abilities of french pupils in mathematics from primary school all the way to baccalauréat have dropped tremendously in the past 20 years, and place France virtually at the bottom in Europe ; as a former math teacher I’m not surprised at all…
• a gallery for middle and high school pupils dedicated to math will open next week in London
• a recent paper of Alain Connes and Farzad Fathizadeh is especially daunting “this term involves exceedingly lengthy expressions and at times involves manipulations on a few hundred thousand terms, only the final outputs of the calculations are written in this paper
• next week at CIRM a Small Group event will discuss Deformation Theory, Completed Cohomology, Leopoldt Conjecture and K-Theory
• a movie about the history of Institut Fourier, featuring among others a 95 years-old Jean-Louis Koszul
• ICM 2022 will take place either in Paris or St.Petersburg (both great choices!)
• still a few weeks left to suggest names for Prize Winners at ICM 2018 (looking at eligible candidates, it looks like it will be an especially difficult job for the Fields Medals committee this time around, with definitely more than 4 super-strong candidates)
• the first part of the Christies sale of a scientific library did very well, including £112,900 for a 1482 first edition of Euclid (more than double the lowest estimate) and a solid £266,500 for a copy of Newton’s Principia Mathematica

Orders of magnitude lower, a recent acquisition:

1892 Licence diploma of Marcel Sibuet

(a biography from Roland Brasseur’s informative