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G+ recovery 1 : Grothendieck

Published February 3, 2019 by lieven

My Google+ account is going away on April 2, 2019, and all attempts to automatically backup my G+ posts seem to fail so far. So i’ll try to rescue here some of them, in chronological order and around one theme. Today, Grothendieck-stuff, part one.

May 30th, 2013

Recordings of a 1972 talk by Grothendieck at Cern “Réflexions sur la science- responsabilité du savant”.
If you don’t have time to listen to all 138 minutes, try to grab from part1 the fragment 29:10 – 30:40 on “the strange ritual of inviting experts to give a talk on some esoteric subject for an audience of 50 to 100 people, one or two of whom will perhaps be able to painfully understand a few bits and pieces, and all others find themselves in a position of humiliation, as they gave in to social pressure to be there, even though the topic itself didn’t interest them at all” (poor translation on my part)
These recordings are illustrative for Grothendieck’s talks in his ‘Survivre’ period, early 70ties. 
(h/t Matilde Marcolli on FB)

June 8th, 2013

Grothendieck’s christmas tree

In the pdf-version of “Recoltes et Semailles” Grothendieck writes on page 463 in the Yin-Yang chapter:

“j’ai fini par aboutir à un diagramme, vaguement en forme d’arbre de Noël”

Here’s the actual diagram, from the original typescript of “Les portes sur l’univers”, the appendix to the ‘Clef du Yin et du Yang’.

Sadly, this appendix (and the many drawings contained in it) didn’t make it into the pdf-release of RecS…

June 9th, 2013

Grothendieck’s yin-yang sunflower

Grothendieck’s ‘Les Portes sur l’Univers’ (Gateways to the Universe(?)) is a truly fascinating text, containing several mysterious drawings (and even a bit of icosahedral-math towards the end).

On PU46, he draws the sunflower of yin and yang, having 12 leafs (he claims, corresponding to 12 yin-terms on the inner circle, 12 yang-terms on the outer circle, as well as to the 12 signs of the zodiac…).

He continues: “On l’appellera, au choix, l’accordeon cosmique, ou l’harmonica cosmique, ou (pour mettre tout le monde d’accord) l’harmonium cosmique”.
(One might call it, as one prefers, the cosmic accordion, or the cosmic harmonica, or (in order to seek general consensus) the *cosmic harmony*).

June 10th, 2013

Grothendieck’s icosahedral theorem

On april 12th 1986, Grothendieck decides to add a mathematical annexe to his esoteric text ‘Les portes sur l’univers’. 

“Par contre, c’est peu pour mon ardeur de mathématicien, laquelle s’est a nouveau réveillée ces jours derniers – et voila repartie ma réflexion sur l’icosaèdre, cet amour mathématique de mon âge mur! Je vais donc peut-être rajouter a ces notes quelques compléments sur la combinatoire de l’icosaèdre et sur la géométrie des ensembles a six éléments…”

He starts with a set S of 6 elements (the vertices), any pair of elements is an edge and any triple a triangle. He then calls a set of triangles F an *icosahedral structure* provided every edge is contained in exactly two triangles in F.

His main result is that all such icosahedral structures are isomorphic (and has exactly 60 isomorphisms), an icosahedral structure consist of exactly 10 triangles and a choice of triangle determines the structure uniquely. Moreover, there are exactly 12 different such octahedral structures and there is an involution on this set coming from ‘complementary’ structures.

At a first glance, Grothendieck’s result appears to be closely related to one of the surprises in finite group theory: the outer automorphism of the symmetric group on 6 letters.

For more on this and related mathematical mysteries of the octahedron, try:

+John Baez  ‘Some Thoughts on the Number 6’  

+Noah Snyder  ‘The Outer Automorphism of S_6’

my own ‘Klein’s dessins d’enfant and the buckyball’

December 18th, 2013

for Grothendieck aficionados

a chance discovery last month en route from Les Vans – Lablachere (in the Ardeche region), a ‘ferronnerie d’art’ (a wrought-iron workshop) called ‘La Clef des Songes’.

All 315 pages of this Grothendieck meditation from 1987 can be found here.

The 691 pages of ‘Notes pour la clef des songes’ are a bit harder to get. Fortunately, the mysterious website ‘l’astree’ offers them as a series of 23 pdfs here. Enjoy the read!

January 3rd, 2014

Why did Grothendieck quit mathematics?

After yesterday’s post on the striking similarities between the lives of Grothendieck and JD Salinger it sure felt weird to stumble upon this footnote in “La Clef des Songes”  

Probably I’m reading way too much into it, but it appears to indicate that Grothendieck stopped doing mathematics to become … a writer!

April 23rd, 2014

Grothendieck documentary available on DVD

+catherine aira and Yves Le Pestipon made a 90 minute long documentary “Alexander Grothendieck, sur les routes d’un genie” which had successful showings in universities, at the Novela science festival, on Toulouse television, and elsewhere. It will be shown in Nantes, Toulouse, Montpellier, and Montreal.

Yves Le Pestipon is one of the people behind the mysterious website lastree.net which has (among many other things) posts on Grothendieck containing hints to his present whereabouts…

Here are some YouTube clips:
clip1

clip2

Here’s the tumblr page of the project:

All of us who cannot attend the viewings can still order the DVD for 25 Euros (20 Euros in France) by sending an email to catherine.aira@gmail.com.

A new release of the DVD, containing English subtitles, will be available soon.

Thanks to +Adeel Khan Yusufzai +David Roberts and +catherine aira 

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RH and the Ishango bone

Published December 8, 2018 by lieven

“She simply walked into the pond in Kensington Gardens Sunday morning and drowned herself in three feet of water.”

This is the opening sentence of The Ishango Bone, a novel by Paul Hastings Wilson. It (re)tells the story of a young mathematician at Cambridge, Amiele, who (dis)proves the Riemann Hypothesis at the age of 26, is denied the Fields medal, and commits suicide.

In his review of the novel on MathFiction, Alex Kasman casts he story in the 1970ties, based on the admission of the first female students to Trinity.

More likely, the correct time frame is in the first decade of this century. On page 121 Amiele meets Alain Connes, said to be a “past winner of the Crafoord Prize”, which Alain obtained in 2001. In fact, noncommutative geometry and its interaction with quantum physics plays a crucial role in her ‘proof’.



The Ishango artefact only appears in the Coda to the book. There are a number of theories on the nature and grouping of the scorings on the bone. In one column some people recognise the numbers 11, 13, 17 and 19 (the primes between 10 and 20).

In the book, Amiele remarks that the total number of lines scored on the bone (168) “happened to be the exact total of all the primes between 1 and 1000” and “if she multiplied 60, the total number of lines in one side column, by 168, the grand total of lines, she’d get 10080,…,not such a far guess from 9592, the actual total of primes between 1 and 100000.” (page 139-140)

The bone is believed to be more than 20000 years old, prime numbers were probably not understood until about 500 BC…



More interesting than these speculations on the nature of the Ishango bone is the description of the tools Amiele thinks to need to tackle the Riemann Hypothesis:

“These included algebraic geometry (which combines commutative algebra with the language and problems of geometry); noncommutative geometry (concerned with the geometric approach to associative algebras, in which multiplication is not commutative, that is, for which $x$ times $y$ does not always equal $y$ times $x$); quantum field theory on noncommutative spacetime, and mathematical aspects of quantum models of consciousness, to name a few.” (page 115)

The breakthrough came two years later when Amiele was giving a lecture on Grothendieck’s dessins d’enfant.

“Dessin d’enfant, or ‘child’s drawing’, which Amiele had discovered in Grothendieck’s work, is a type of graph drawing that seemed technically simple, but had a very strong impression on her, partly due to the familiar nature of the objects considered. (…) Amiele found subtle arithmetic invariants associated with these dessins, which were completely transformed, again, as soon as another stroke was added.” (page 116)

Amiele’s ‘disproof’ of RH is outlined on pages 122-124 of “The Ishango Bone” and is a mixture of recognisable concepts and ill-defined terms.

“Her final result proved that Riemann’s Hypothesis was false, a zero must fall to the east of Riemann’s critical line whenever the zeta function of point $q$ with momentum $p$ approached the aelotropic state-vector (this is a simplification, of course).” (page 123)

More details are given in a footnote:

“(…) a zero must fall to the east of Riemann’s critical line whenever:

\[
\zeta(q_p) = \frac{( | \uparrow \rangle + \Psi) + \frac{1}{2}(1+cos(\Theta))\frac{\hbar}{\pi}}{\int(\Delta_p)} \]

(…) The intrepid are invited to try the equation for themselves.” (page 124)

Wilson’s “The Ishango Bone” was published in 2012. A fair number of topics covered (the Ishango bone, dessin d’enfant, Riemann hypothesis, quantum theory) also play a prominent role in the 2015 paper/story by Michel Planat “A moonshine dialogue in mathematical physics”, but this time with additional story-line: monstrous moonshine…

Such a paper surely deserves a separate post.



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the mathematician of cubism

Published December 1, 2018 by lieven

“Pythagorean Crimes” by Tefcros Michaelides is a murder mystery set at the beginning of the 20th century. It starts with Hilbert’s address at the 1900 ICM in Paris (in which he gives his list of problems, such as the 2nd, his program for a finitistic proof of the consistency of the axioms of arithmetic) and ends in the early 1930ties (perhaps you can by now already guess which theorem will play a crucial role in the plot?).

It depicts beautifully daily (or better, nightly) life in mathematical and artistic circles, especially in Paris between 1900 and 1906.

Bricard, Caratheodory, Dedekind, Dehn, De la Vallee-Poussin, Frege, Godel, Hadamard, Hamel, Hatzidakis, Hermite, Hilbert, Klein, Lindemann, Minkowski, Peano, Poincare, Reynaud, Russell and Whitehead all make a brief appearance, as do Appollinaire, Casagemas, Cezanne, Degas, Derain, Max Jacob, Jacobides, Lumiere, Matisse, Melies, Pallares, Picasso, Renoir, Salmon, Toulouse-Lautrec, Utrillo, Zola.



Both lists contain names I had never heard of. But the biggest surprise, to me, was to discover the name of Maurice Princet, “le mathématicien du cubisme”.

Princet (1875-1973) was a mathematician who frequented the group around Pablo Picasso at the Bateau-Lavoir in Montmartre (at least until 1907 when his wife left him for the painter Derain).

Princet introduced the group to the works of Poincare and the concept of the 4-th dimension. He gave Picasso the book “Traité élémentaire de géométrie à quatre dimensions” by Jouffret, describing hyper-cubes and other polyhedra in 4 dimensions and ways to project them dowm to the 2 dimensions of the canvas.



This book appears to have been influential in the genesis of Picasso’s Les Demoiselles d’Avignon (the painting also appears, in an unfinished state, in “Pythagorean Crimes”).



Some other painters tried to capture movement with projections from the 4-th dimension. A nice example is Nude descending a staircase by Marcel Duchamp (mostly known for his urinoir…).



Maurice Princet loved to get the artists interested in the new views on space. Duchamp told Pierre Cabanne, “We weren’t mathematicians at all, but we really did believe in Princet”.

I don’t know whether Duchamp liked Princet’s own attempts at painting. Here’s a cubistic work by Maurice Princet himself.



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