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math-et-physique

Outsiders and the advancement of science

1 Octobre 2014, 17:05pm

Publié par Fabien Besnard

L'éléphant n°8 will be out tomorrow (more about this later), and it reminds me of a nice discussion with philosopher Karine Prévot about whether outsiders in a field are more likely to make unexpected breakthroughs. This seems to be a widely held belief, but it could easily be a myth kept alive by some spectacular but in fact very uncommon examples.

There are indeed arguments going in either direction. On one hand it seems reasonable that outsiders bring with them some fresh air and a new way of looking at old unsolved problems. On the other hand, why someone lacking a thorough education in a given field would ever come up with interesting ideas ? Spending years working on a subject would be pointless: just wait for an outsider to come and solve the problem. This seems ridiculous.

Still, there are some significant examples. The question is how much significant they are. Surprisingly, I did not find any research paper on this question.

Anyway, below is a little list I have compiled (with the help of many people at mathoverflow, may they be thanked) of famous scientists who made some important contribution in a field other than their own. Feel free to expand it.

From physics to biology:

Erwin Schrödinger: the famous physicist wrote "What is life ?", a book which has been very influential in the biology community, in particular on James Watson.

Francis Crick : speaking about Watson, Crick was primarily educated as a physicist, and only began studying biology in 1947, according to his wikipedia entry.

Max Delbrück was a theoretical physicist who came to be more and more interested in biology. He earned the Nobel prize in physiology and medicine in 1969.

From Chemistry to biology:

Louis Pasteur is best remembered for his invaluable contributions to microbiology and the prevention of diseases, but, as is well-known, he began his scientific career as a chemist.

Linus Pauling could also be put in this section, with his discovery of a new kind of disease.

From mathematics to biology:

Alan Turing is justly known for his contributions to mathematics and computer sciences. Maybe less widely known is his seminal work on morphogenesis.

Also in this section (maybe) is Norbert Wiener for his foundation of cybernetics.

From mathematics to physics, froms physics to mathematics: this is trickiest than one can think. Those disciplines are so close together that a mathematician could be argued not to be an outsider in physics, even less so the other way round. However, it might still come as a surprise that the logician Kurt Gödel discovered one of the very few known exact solutions to the equations of general relativity. And guess what ? It has paradoxical properties...

Also I can't resist mentioning Alain Connes work on the spectral standard model. Aftel all, it was not so obvious that a specialist in functional analysis could come up with entirely new ideas on the unification of the fundamental forces.

In the other direction, it is impossible not to mention Edward Witten. Not only was he the first physicist ever to win the Fields medal, but he did so because he could apply physical intuition from quantum field theory to topological questions. And this exactly the kind of examples I am looking for.

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Frontiers of Fundamental Physics 14

28 Août 2014, 17:46pm

Publié par Fabien Besnard

It's been a while since I haven't posted anything on this blog ! So let me give you some news before I get overwhelmed again by my teaching duties (which will be very dense from next month and until january).

Today I start with a great conference which was held in Marseille this July. First, I have to thank the organizers, in particular Thierry Masson, for their very warm welcome. Thierry, who fell in love with Marseille (the photographs randomly appearing on the conference website are from his personal collection), shared his enthousiasm with all of us. He gave me a 1 meter long panoramic view of the Vieux Port which he assembled himself out of photographs taken in 1869. This gorgeous piece of work now stands in my office and helps me remember the wonderful moments spent drinking some wine while staring at the very same view, on which the sun set. Magical !

Frontiers of Fundamental Physics 14

On the more serious side, this very broad conference is one of those few where you can talk to people well outside your field, and hear from them about the hottest news out there.

For instance Carlo Rovelli told about a conjectural new state of matter : "Planck stars" ! To find what these objects are expected to be, here is a (rather critical) blog review, with a response from the author. The original paper is here.

But of course there was also more specialized sessions, with a very high density of interesting talks (at least in the mathematical physics sessions I attended. By the way the slides of my own talk are here. I let you judge for yourself whether it increased or decreased the density).

One thing that struck me was Fedele Lizzi's and collaborators result about bosons stopping their propagation at high energy in the spectral standard model. With Nadir Bizi, we have a result which says that an almost commutative manifold cannot have a noncommutative causal structure (which we define), unless the causal structure on the commutative part disappears at small scales. The methods are completely different but point in the same direction ! Maybe they simply both point to a breakdown of the theory at high energy scale : after all the smooth manifold structure on the spacetime part of the model cannot be trusted at arbitrarily high energy.

Walter van Suijlekom was there too (you can see his comments about the conference here). He has wonderful pedagogical talents. You can translate "I understood what the hell he was talking about !". I even found his discussion of inner perturbation in NCG very enlightening, which is no small feat since I'm a really slow learner. Given his great clarity of exposition, his new book promises to be one of the best entry point in the field.

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Quadrature

24 Mai 2014, 11:22am

Publié par Fabien Besnard

Le directeur de la revue Quadrature m'a demandé de bien vouloir mettre un extrait gratuit de son magazine sur mon blog. Je le fais bien volontiers, d'autant plus qu'ayant eu l'occasion de lire cette publication, je sais qu'elle est de très bonne qualité, et œuvre depuis 25 ans pour la diffusion de la culture mathématique.


Bonne lecture !

Quadrature

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Documentaire sur Lagrange

7 Avril 2014, 15:27pm

Publié par Fabien Besnard

Un documentaire d'une trentaire de minutes sur le mathématicien et physicien Joseph-Louis Lagrange (1736-1813).

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Two exercises on convex cones in Banach spaces

12 Mars 2014, 21:57pm

Publié par Fabien Besnard

I happened to work in a particular kind of finite dimensional algebras for quite a bit, and it's amazing how quickly you forget the subtleties which can occur in infnite dimensions. Now that I resumed a previous work that I had done, I came across an hypothesis that I put in a theorem, and it took me a while to (re-)understand why this hypothesis was not always satisfied. Here is the thing in the form of two exercises (so that if I forget it again, I can just take a look back here !). The second gives a solution to the first, so if you want to do it the hard way, do the first one first !

 

 

1) Find a closed convex cone C in a (real) Banach space V such that C-C is a strict subspace of V which is dense in V.

 

 

 

 

 

 

 

 

2) Let V be the Banach space of continuous functions on [0;1] with the uniform norm. Let I be the closed convex cone of non-decreasing elements of V.

 

a) Show that I-I is not V.

b) Show that I-I is dense in V.

 

 

You can show 2b by an abstract structural argument... or you can show the density inside a particular subspace of V that you already know to be dense in V.

 

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An exercise on the manifold of derogatory matrices

17 Novembre 2013, 18:21pm

Publié par Fabien Besnard

Here is a nice little exercise. Let us consider the set H_N of all hermitian NxN matrices. In this circumstance, a matrix M is called derogatory if it has at least one multiple eigenvalue (in more general circumstances the definition is more complex, but never mind). In fact we are interested in the set A of hermitian matrices which are block diagonal, with the size (and place) of the blocks fixed. In other words we are looking at the direct sum A of H_{n_1},...,H_{n_k} with n_1+...+n_k=N.

 

We suppose that k>1.

 

Let us call V the manifold of derogatory matrices inside A. It decomposes into different components, according to whether the multiplicities fall inside the same block or not. We are interested in the highest dimensional components, which have dimension N-1 (why ?) and correspond to the case where the multiplicities are across different blocks (say the first two for concreteness).

 

The question is : what is the equation of the tangent hyperplane to V at some matrix a the two first blocks of which share an eigenvalue ?

 

I solved this in a rather heuristic way, guessing the answer from a simple example, and then checking that it holds generally. I wonder if there is a more systematic ( but still simple) way to derive the result.

 

I'll give the answer at the end of the week.

 

 

Update 20/11/2013 : Since nobody gave the answer, here is an indication. You might like to first work out the case where the blocks have all size 1.

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Conférence débat avec Carlo Rovelli et Michel Bitbol

15 Novembre 2013, 13:40pm

Publié par Fabien Besnard

"Mécanique quantique et interprétation relationnelle"

Conférence débat

par Carlo Rovelli (physicien) et Michel Bitbol (philosophe)



Lundi 18 novembre 2013, à 18h
Université Paris Diderot

Amphithéâtre Buffon
Entrée libre
Je ne pourrais malheureusement pas y assister. Si vous y allez, n'hésitez pas à me faire part ici de vos retours.

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L'Éléphant n° 4

13 Octobre 2013, 16:36pm

Publié par Fabien Besnard

Il est sorti cette semaine ! Vous y trouverez l'épisode 3 de "De quoi est fait le monde ?". J'y parle de la révolution chimique, de Fresnel, de Faraday, de Maxwell. En prime, une interview de l'historien des sciences Helge Kragh.

 

Elephant-4-Cahier-COUV--e1381144869331.jpg

 

Je rappelle que L'Éléphant se trouve généralement rangé avec les livres. On le trouve aussi en librairie.

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Ces parasites de l'open access

5 Octobre 2013, 09:44am

Publié par Fabien Besnard

Vous n'avez sûrement pas besoin de moi pour connaître l'info, car elle fait la une du magazine Science, mais au cas où voici de quoi il s'agit : un de leur correspondant, John Bohannon, a envoyé un article bidon, en partie généré par ordinateur,  truffé d'erreurs grotesques, et dont ni l'auteur ni le labo d'origine n'existent à 304 revues en "golden open access" (ça veut dire que l'auteur doit payer pour publier).

 

Résultat :sur les 255 qui avaient terminé le procédé de "peer review" (qui était inexistant dans 60% des cas), 157 l'ont accepté, 98 rejeté.

 

Si vous voulez vous marrer, il y a même la correspondance complète que les journaux ciblés ont entretenu avec les auteurs imaginaires.

 

Certes l'expérience concerne le domaine de la médecine, mais je suis certain que si on la faisait avec toutes les revues au noms ronflants qui envahissent ma boîte mail en m'invitant à soumettre un article dans les domaines les plus divers, le résultat serait le même.

 

Le plus triste dans l'histoire c'est que certains journaux semblent faire leur travail correctement. Or il est clair que quand un système est infecté à ce niveau il n'y a plus aucune confiance possible, ce qui est le cas depuis longtemps avec le golden open access.

 

Si vous n'avez pas accès à l'article original de science, il est bien résumé sur le blog de Sylvestre Huet.

 

Autre lien intéressant dans le même ordre d'idée, le blog retractation watch, qui tient à jour une liste d'auteurs dont les papiers ont été retirés. Une sorte de thermomètre des comportements frauduleux en science. Là aussi cela concerne surtout la médecine et la biologie, les auteurs étant médecins. Mais il y a quand même une section sur les sciences physiques (où, sans surprise, c'est la chimie qui est la mieux représentée).

 

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Terry Tao's blog and the twin prime conjecture

26 Août 2013, 10:27am

Publié par Fabien Besnard

Though my knowledge of number theory is terribly close to zero, the twin prime conjecture is so famous that the recent news of progress towards the result have managed to reach my ears. All of this is well explained here, but in brief let me just remind you what is this conjecture about.

 

A twin prime pair is an (ordered, let's say) pair of prime numbers which differ by 2, like the pair (3,5) for instance. Of course, with the exception of the pair (2,3), this is the closest two prime numbers can get. A little experimentation seems to indicate that there are many such pairs : (3,5), (5,7), (11,13), (17,19), (29,31) are the first ones. This makes the conjecture that there is an infinity of such pairs sound natural : this is the twin prime conjecture. On the other hand, if you know more about the prime numbers repartition, the conjecture, if true, will seem much more striking to you since the prime numbers are on the average farther and farther apart from each other.

 

Now on april this year, Yitang Zhang announced a proof of the following property :

 

(P_N) : there exists infinitely many primes that differ by at most N

 

where N is about 70 000 000. Of course the twin prime conjecture is that (P_2) is true, but anyway Zhang's result (which has now been checked by all the experts in the field) is a major breakthrough. Incidentally, Zhang, who is already in his fifties, was a complete unknown who worked alone and sometimes in difficult conditions.

 

Now enters Terry Tao and his excellent blog. I wish I had the time to talk about this blog before, but I had not, so this is the opportunity. Tao, who always does an incredible job at explaining thoroughly and pedagogically all kinds of subjects, now offers you the opportunity to witness real mathematic research being done, so to speak, right before your eyes. He launched the polymath8 project, which is a collaborative project whose goal is to optimize Zhang methods in order to lower the bound N. The results so far are impressive : they started from 70 000 000 in may, and they reached about 50 000 already by mid-june. Now the bound is down to 4680, and a paper is announced. However they apparently won't be able to get past the barrier of N=16 using the same methods. I wish I understood more about all of this. I encourage the readers who do to go to Tao's blog and keep us informed.

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