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Category: books

Lockdown reading : the Carls

In this series I’ll mention some books I found entertaining, stimulating or comforting during these Corona times. Read them at your own risk.

An Absolutely Remarkable Thing (AART for the fans) by Hank Green came out in 2018, and recently I reread it when its sequel A Beautifully Foolish Endeavor appeared last summer.

“Protagonist April May discovers a large robot sculpture in Midtown Manhattan. She and her friend Andy Skampt decide to film it and post the video online, which goes viral and makes April an overnight celebrity. All over the world identical structures—known as “Carls”—have appeared in major cities at exactly the same time.” (Wikipedia)

Here’s an artist’s impression of said video, followed by the ‘Queen sequence’ (one of many puzzles in the book). On an audio fragment a faint trace of Queen’s “Don’t Stop Me Now” is heard, and April discovers the code “IAMU” after fixing a series of typos in the Wikipedia article about that song.

Three reasons why you might want to read this book now:

  1. It’s about the dangers and pitfalls of social media and online fame. (Something Hank Green is familiar with as he runs with his brother the YouTube channel Vlogbrothers.)
  2. It’s about a global pandemic. (Not caused by a virus, but by a contagious dream, containing 4096 ‘sequences’=puzzles, each resulting in an HEX-sequence to be combined into a vector-image.)
  3. It’s about the consequences of hate speech. (It’s hard not to draw parallels between ‘Peter Petrawicki’ and a former president, and between the actions of the ‘Defenders’ and the events of January 6th.)

AART doesn’t end well for April May, and it was hard to image Hank Green ever writing a sequel without doing a Bobby Ewing shower scene (showing my age here). And yes, the book ends with a two word text message from April: “Knock Knock”.

The sequel ‘A Beautifully Foolish Endeavour’ is perhaps even more enjoyable than AART. The Dream is now replaced by a Magic Book, and the storytelling (at first) no longer done by April herself but by her four evangelists (Maya, Andy, Miranda and Robin), Green’s very own Mamalujo so to speak.

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Scholze’s condensed sets and Mazzola’s path to creativity

Some months ago, Peter Scholze wrote a guest post on the Xena-blog: Liquid tensor experiment, proposing a challenge to formalise the proof of one of his results with Dustin Clausen on condensed mathematics.

Scholze and Clausen ran a masterclass in Copenhagen on condensed mathematics, which you can binge watch on YouTube starting here

Scholze also gave two courses on the material in Bonn of which the notes are available here and here.

Condensed mathematics claims that topological spaces are the wrong definition, and that one should replace them with the slightly different notion of condensed sets.

So, let’s find out what a condensed set is.

Definition: Condensed sets are sheaves (of sets) on the pro-étale site of a point.

(there’s no danger we’ll have to rewrite our undergraduate topology courses just yet…)

In his blogpost, Scholze motivates this paradigm shift by observing that the category of topological Abelian groups is not Abelian (if you put a finer topology on the same group then the identity map is not an isomorphism but doesn’t have a kernel nor cokernel) whereas the category of condensed Abelian groups is.

It was another Clausen-Scholze result in the blogpost that caught my eye.

But first, for something completely different.

In “Musical creativity”, Guerino Mazzola and co-authors introduce a seven steps path to creativity.

Here they are:

  1. Exhibiting the open question
  2. Identifying the semiotic context
  3. Finding the question’s critical sign
  4. Identifying the concept’s walls
  5. Opening the walls
  6. Displaying extended wall perspectives
  7. Evaluating the extended walls

Looks like a recipe from distant flower-power pot-infused times, no?

In Towards a Categorical Theory of Creativity for Music, Discourse, and Cognition, Mazzola, Andrée Ehresmann and co-authors relate these seven steps to the Yoneda lemma.

  1. Exhibiting the open question = to understand the object $A$
  2. Identifying the semiotic context = to describe the category $\mathcal{C}$ of which $A$ is an object
  3. Finding the question’s critical sign = $A$ (?!)
  4. Identifying the concept’s walls = the uncontrolled behaviour of the Yoneda functor
    @A~:~\mathcal{C} \rightarrow \mathbf{Sets} \qquad C \mapsto Hom_{\mathcal{C}}(C,A) \]
  5. Opening the walls = finding an objectively creative subcategory $\mathcal{A}$ of $\mathcal{C}$
  6. Displaying extended wall perspectives = calculate the colimit $C$ of a creative diagram
  7. Evaluating the extended walls = try to understand $A$ via the isomorphism $C \simeq A$.

(Actually, I first read about these seven categorical steps in another paper which might put a smile on your face: The Yoneda path to the Buddhist monk blend.)

Remains to know what a ‘creative’ subcategory is.

The creative moment comes in here: could we not find a subcategory
$\mathcal{A}$ of $\mathcal{C}$ such that the functor
Yon|_{\mathcal{A}}~:~\mathcal{C} \rightarrow \mathbf{PSh}(\mathcal{A}) \qquad A \mapsto @A|_{\mathcal{A}} \]
is still fully faithful? We call such a subcategory creative, and it is a major task in category theory to find creative categories which are as small as possible.

All the ingredients are here, but I had to read Peter Scholze’s blogpost before the penny dropped.

Let’s try to view condensed sets as the result of a creative process.

  1. Exhibiting the open question: you are a topologist and want to understand a particular compact Hausdorff space $X$.
  2. Identifying the semiotic context: you are familiar with working in the category $\mathbf{Tops}$ of all topological spaces with continuous maps as morphisms.
  3. Finding the question’s critical sign: you want to know what differentiates your space $X$ from all other topological spaces.
  4. Identifying the concept’s walls: you can probe your space $X$ with continuous maps from other topological spaces. That is, you can consider the contravariant functor (or presheaf on $\mathbf{Tops}$)
    @X~:~\mathbf{Tops} \rightarrow \mathbf{Sets} \qquad Y \mapsto Cont(Y,X) \]
    and Yoneda tells you that this functor, up to equivalence, determines the space $X$ upto homeomorphism.
  5. Opening the walls: Tychonoff tells you that among all compact Hausdorff spaces there’s a class of pretty weird examples: inverse limits of finite sets (or a bit pompous: the pro-etale site of a point). These limits form a subcategory $\mathbf{ProF}$ of $\mathbf{Tops}$.
  6. Displaying extended wall perspectives: for every inverse limit $F \in \mathbf{ProF}$ (for ‘pro-finite sets’) you can look at the set $\mathcal{X}(F)=Cont(F,X)$ of all continuous maps from $F$ to $X$ (that is, all probes of $X$ by $F$) and this functor
    \mathcal{X}=@X|_{\mathbf{ProF}}~:~\mathbf{ProF} \rightarrow \mathbf{Sets} \qquad F \mapsto \mathcal{X}(F) \]
    is a sheaf on the pre-etale site of a point, that is, $\mathcal{X}$ is the condensed set associated to $X$.
  7. Evaluating the extended walls: Clausen and Scholze observe that the assignment $X \mapsto \mathcal{X}$ embeds compact Hausdorff spaces fully faithful into condensed sets, so we can recover $X$ up to homeomorphism as a colimit from the condenset set $\mathcal{X}$. Or, in Mazzola’s terminology: $\mathbf{ProF}$ is a creative subcategory of $\mathbf{(cH)Tops}$ (all compact Hausdorff spaces).

It would be nice if someone would come up with a new notion for me to understand Mazzola’s other opus “The topos of music” (now reprinted as a four volume series).

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Grothendieck’s gribouillis (4)

Fortunately, there are a few certainties left in life:

In spring, you might expect the next instalment of Connes’ and Consani’s quest for Gabriel’s topos. Here’s the latest: $\overline{\mathbf{Spec}(\mathbb{Z})}$ and the Gromov norm.

Every half year or so, Mochizuki’s circle-of-friends tries to create some buzz announcing the next IUTeich-workshop. I’ll spare you the link, if you are still interested, follow math_jin or IUTT_bot_math_jin on Twitter.

And then, there’s the never-ending story of Grothendieck’s griboullis, kept alive by the French journalist and author Philippe Douroux.

Here are some recent links:

Alexandre Grothendieck : une mathématique en cathédrale gothique, an article (in French) by Philippe Douroux in Le Monde, May 6th (behind paywall).

L’histoire étonnante des archives du mathématicien Alexandre Grothendieck, an article (in French) on France Inter by Mathieu Vidar, based on info from Philippe Douroux.

Les archives mystérieuses de Alexandre Grothendieck, a podcast of a broadcast on France Inter on June 10th. Interesting interview (in French) with Philippe Douroux and the French mathematician Etienne Ghys (with a guest appearance by Luc Illusie).

El enigmático legado de un genio de las matemáticas, an article (in Spanish) in El Pais, May 13th, with 8 photos of some of the Gribouillis. The two pictures in this post are taken from this article.

So, what’s the latest on the 70.000+ pages left by Grothendieck?

As far as i know, the Mormoiron part of the gribouillis is still at the University of Montpellier, and has been made available online at the Grothendieck archives.

The Lasserre part of the gribouillis is still in a cellar in Paris’ Saint-Germain-des-Prés, belonging to Jean-Bernard Gillot. The French national library cannot take possession of the notes before a financial agreement is reached with Grothendieck’s children (French law does not allow children to be disinherited).

And there’s a dispute about the price to be paid. The notes were estimated at 45.000 Euros, but some prefer to believe that they may be worth several millions of dollars.

It all depends on their mathematical content.

Unfortunately, pictures claimed to be of the Lasserre notes (such as the one above) are in fact from the Mormoiron/Montpellier notes, which do indeed contain interesting mathematics.

But, it is very unlikely that the Lasserre notes contain (math) surprises. Probably, most of them look like this one

endless lists of people deported by the Nazis to extermination camps in WW2.

Or, as Philippe Douroux is quoted in the El Pais piece: “I think it’s a treasure, maybe not a mathematical one, but a human one. It’s a descent into the hell of one the best organised brains in the world.”

The film made by Catherine Aira and Yves Le Pestipon “Alexandre Grothendieck: On the Paths of a Genius” (on the quest for G’s last hideout in the French Pyrenees) can now be watched on YouTube (with English subtitles)

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RH and the Ishango bone

“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

“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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