RH and the Ishango bone

— books, geometry, noncommutative, number theory —

“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.

the mathematician of cubism

— art, books, math —

“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.

Closing in on Gabriel’s topos?

— absolute, noncommutative, number theory —

‘Gabriel’s topos’ (see here) is the conjectural, but still elusive topos from which the validity of the Riemann hypothesis would follow.

It is the latest attempt in Alain Connes’ 20 year long quest to tackle the RH (before, he tried the tools of noncommutative geometry and later those offered by the field with one element).

For the last 5 years he hopes that topos theory might provide the missing ingredient. Together with Katia Consani he introduced and studied the geometry of the Arithmetic site, and later the geometry of the scaling site.

If you look at the points of these toposes you get horribly complicated ‘non-commutative’ spaces, such as the finite adele classes $\mathbb{Q}^*_+ \backslash \mathbb{A}^f_{\mathbb{Q}} / \widehat{\mathbb{Z}}^{\ast}$ (in case of the arithmetic site) and the full adele classes $\mathbb{Q}^*_+ \backslash \mathbb{A}_{\mathbb{Q}} / \widehat{\mathbb{Z}}^{\ast}$ (for the scaling site).

In Vienna, Connes gave a nice introduction to the arithmetic site in two lectures. The first part of the talk below also gives an historic overview of his work on the RH

The second lecture can be watched here.

However, not everyone is as optimistic about the topos-approach as he seems to be. Here’s an insightful answer on MathOverflow by Will Sawin to the question “What is precisely still missing in Connes’ approach to RH?”.

Other interesting MathOverflow threads related to the RH-approach via the field with one element are Approaches to Riemann hypothesis using methods outside number theory and Riemann hypothesis via absolute geometry.

About a month ago, from May 10th till 14th Alain Connes gave a series of lectures at Ohio State University with title “The Riemann-Roch strategy, quantizing the Scaling Site”.

The accompanying paper has now been arXived: The Riemann-Roch strategy, Complex lift of the Scaling Site (joint with K. Consani).

Especially interesting is section 2 “The geometry behind the zeros of $\zeta$” in which they explain how looking at the zeros locus inevitably leads to the space of adele classes and why one has to study this space with the tools from noncommutative geometry.

Perhaps further developments will be disclosed in a few weeks time when Connes is one of the speakers at Toposes in Como.

Archangel Gabriel will make you a topos

— books, noncommutative —

No kidding, this is the final sentence of Le spectre d’Atacama, the second novel by Alain Connes (written with Danye Chéreau (IRL Mrs. AC) and his former Ph.D. advisor Jacques Dixmier).

The book has a promising start. Armand Lafforet (IRL AC) is summoned by his friend Rodrigo to the Chilean observatory Alma in the Altacama desert. They have observed a mysterious spectrum, and need his advice.

Armand drops everything and on the flight he lectures the lady sitting next to him on proofs by induction (breaking up chocolate bars), and recalls a recent stay at the La Trappe Abbey, where he had an encounter with (the ghost of) Alexander Grothendieck, who urged him to ‘Follow the motif!’.

“Comment était-il arrivé là? Il possédait surement quelques clés. Pourquoi pas celles des songes?” (How did he get
there? Surely he owned some keys, why not those of our dreams?)

A few pages further there’s this on the notion of topos (my attempt to translate):

“The notion of space plays a central role in mathematics. Traditionally we represent it as a set of points, together with a notion of neighborhood that we call a ‘topology’. The universe of these new spaces, ‘toposes’, unveiled by Grothendieck, is marvellous, not only for the infinite wealth of examples (it contains, apart from the ordinary topological spaces, also numerous instances of a more combinatorial nature) but because of the totally original way to perceive space: instead of appearing on the main stage from the start, it hides backstage and manifests itself as a ‘deus ex machina’, introducing a variability in the theory of sets.”

So far, so good.

We have a mystery, tidbits of mathematics, and allusions left there to put a smile on any Grothendieck-aficionado’s face.

But then, upon arrival, the story drops dead.

Rodrigo has been taken to hospital, and will remain incommunicado until well in the final quarter of the book.

As the remaining astronomers show little interest in Alain’s (sorry, Armand’s) first lecture, he decides to skip the second, and departs on a hike to the ocean. There, he takes a genuine sailing ship in true Jules Verne style to the lighthouse at he end of the world.

All this drags on for at least half a year in time, and two thirds of the book’s length. We are left in complete suspense when it comes to the mysterious Atacama spectrum.

Perhaps the three authors deliberately want to break with existing conventions of story telling?

I had a similar feeling when reading their first novel Le Theatre Quantique. Here they spend some effort to flesh out their heroine, Charlotte, in the first part of the book. But then, all of a sudden, their main character is replaced by a detective, and next by a computer.

Anyway, when Armand finally reappears at the IHES the story picks up pace.

The trio (Armand, his would-be-lover Charlotte, and Ali Ravi, Cern’s computer guru) convince CERN to sell its main computer to an American billionaire with the (fake) promise of developing a quantum computer. Incidentally, they somehow manage to do this using Charlotte’s history with that computer (for this, you have to read ‘Le Theatre Quantique’).

By their quantum-computing power (Shor and quantum-encryption pass the revue) they are able to decipher the Atacame spectrum (something to do with primes and zeroes of the zeta function), send coded messages using quantum entanglement, end up in the Oval Office and convince the president to send a message to the ‘Riemann sphere’ (another fun pun), and so on, and on.

The book ends with a twist of the classic tale of the mathematician willing to sell his soul to the devil for a (dis)proof of the Riemann hypothesis:

After spending some time in purgatory, the mathematician gets a meeting with God and asks her the question “Is the Riemann hypothesis true?”.

“Of course”, God says.

“But how can you know that all non-trivial zeroes of the zeta function have real part 1/2?”, Armand asks.

And God replies:

“Simple enough, I can see them all at once. But then, don’t forget I’m God. I can see the disappointment in your face, yes I can read in your heart that you are frustrated, that you desire an explanation…

Well, we’re going to fix this. I will call archangel Gabriel, the angel of geometry, he will make you a topos!”

If you feel like running to the nearest Kindle store to buy “Le spectre d’Atacama”, make sure to opt for a package deal. It is impossible to make heads or tails of the story without reading “Le theatre quantique” first.

But then, there are worse ways to spend an idle week than by binge reading Connes…

Edit (February 28th). A short video of Alain Connes explaining ‘Le spectre d’Atacama’ (in French)

Rotations of Klein’s quartic

— groups —

The usual argument to show that the group of all orientation-preserving symmetries of the Klein quartic is the simple group $L_2(7)$ of order $168$ goes like this:

There are two families of $7$ truncated cubes on the Klein quartic. The triangles of one of the seven truncated cubes in the first family have as center the dots, all having the same colour. The triangles of one of the truncated cubes in the second family correspond to the squares all having the same colour.

If you compare the two colour schemes, you’ll see that every truncated cube in the first family is disjoint from precisely $3$ truncated cubes in the second family.

That is, we can identify the truncated cubes of the first family with the points in the Fano plane $\mathbb{P}^2(\mathbb{F}_2)$, and those of the second family with the lines in that plane.

The Klein quartic consists of $24$ regular heptagons, so its rotation symmetry group consists of $24 \times 7 = 168$ rotations,each preserving the two families of truncated cubes. This is exactly the same number as there are isomorphisms of the Fano plane, $PGL_3(\mathbb{F}_2) = L_2(7)$. Done!

For more details, check John Baez’ excellent page on the Klein quartic, or the Buckyball curve post.

Here’s another ‘look-and-see’ proof, starting from Klein’s own description of his quartic.

Look at the rotation $g$, counter-clockwise with angle $2\pi / 7$ fixing the center of the central blue heptagon, and a similar rotation $h$ fixing the center of one of the neighbouring red heptagons.

The two vertices of the edge shared by the blue and red heptagon are fixed by $g.h$ and $h.g$, respectively, so these rotations must have order three (there are $3$ heptagons meeting in the vertex).

That is, the rotation symmetry group $G$ of the Klein quartic has order $168$, and contains two elements $g$ and $h$ of order $7$, such that the subgroup generated by them contains elements of order $3$.

This is enough to prove that the $G$ must be simple and therefore isomorphic to $L_2(7)$!

The following elegant proof is often attributed to Igor Dolgachev.

If $G$ isn’t simple there is a maximal normal subgroup $N$ with $G/N$ simple .

The only non-cyclic simple group having less elements that $168$ is $A_5$ but this cannot be $G/N$ as $60$ does not divide $168$.

So, $G/N$ must be cyclic of order $2,3$ or $7$ (the only prime divisors of $168=2^3.3.7$).

Order $2$ is not possible as any group $N$ of order $84=2^2.3.7$ can just have one Sylow $7$-subgroup. Remember that the number of $7$-Sylows of $N$ must divide $2^2.3=12$ and must be equal to $1$ modulo $7$. And $G$ (and therefore $N$) has at least two different cyclic subgroups of order $7$.

Order $3$ is impossible as this would imply that the normal subgroup $N$ of order $2^3.7=56$ must contain all $7$-Sylows of $G$, and thus also an element of order $3$. But, $3$ does not divide $56$.

Order $7$ is a bit more difficult to exclude. This would mean that there is a normal subgroup $N$ of order $2^3.3=24$.

$N$ (being normal) must contain all Sylow $2$-subgroups of $G$ of which there are either $1$ or $3$ (the order of $N$ is $2^3.3=24$).

If there is just one $S$ it should be a normal subgroup with $G/S$ (of order $21$) having a (normal) unique Sylow $7$-subgroup, but then $G$ would have a normal subgroup of index $3$, which we have excluded.

The three $2$-Sylows are conjugated and so the conjugation morphism
\[
G \rightarrow S_3 \]
is non-trivial and must have image strictly larger than $C_3$ (otherwise, $G$ would have a normal subgroup of index $3$), so must be surjective.

But, $S_3$ has a normal subgroup of index $2$ and pulling this back, $G$ must also have a normal subgroup of index two, which we have excluded. Done!