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Lockdown reading : SNORT

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



This must have been the third time I’ve read The genius in by basement – The biography of a happy man by Alexander masters.

I first read it when it came out in 2011.

Then, in conjunction with Genius at play – The Curious Mind of John Horton Conway Conway’s biography by Siobhan Roberts, in july 2017, which is probably the best way to read this book.

And, then again last week, as Simon Norton‘s work pops up wherever I look, as in the previous post.

It takes some time to get used to the rather chaotic style (probably used because that’s how Masters perceives Norton), and all attempts at explaining Simon’s mathematics can better be skipped.

The book tries to find an answer as to why a child prodigy and genius like Simon Norton failed to secure a safe place in academics.

Page 328:

Simon’s second explanation of his loss of mathematical direction is heartbreaking. Now that Conway has fled to America, there is no one in the mathematical world who will work with him.

They say he is too peculiar, too shabby, too old.

His interests are fixed in mathematics that has had its day. His brilliance is frigid. His talent, perfectly suited to an extraordinary moment in algebraic history (the symmetry work at Cambridge during the early 1970s and 1980s) is out of fashion.

This may give the impression that Norton stopped doing good math after Conway left for Princeton in 1985. This is far from true.

Norton’s Wikipedia page mentions only post 1995 publications, which in itself is deplorable, as it leaves out his contributions to the ATLAS and his seminal paper with Conway on Monstrous moonshine.

Here’s Alexander Masters talking about ‘Genius in my basement’

I’ll leave you with a nice quote, comparing Monstrous Moonshine to a Sainsbury’s bag on Jupiter.

Page 334:

This much I do know: Monstrous Moonshine links the Monster to distant mathematics and the structure of space in ways that are as awe-inspiring to a man like Simon as it would be to an astronaut to step out of his space machine on Jupiter, and find a Sainsbury’s bag floating past. That’s why it’s called ‘Moonshine’, because mathematicians can even now hardly believe it.

‘I think’, said Simon, standing up from his berth and shaking crumbs and clotted blobs of oil and fish off his T-shirt onto the covers, ‘I can explain to you what Moonshine is in one sentence.’

When he really tries, Simon can be a model of clarity.

‘It is,’ he said, ‘the voice of God.’

Ps, wrt. SNORT.

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Finnegans Wake’s geometry lesson

The literary sensation that spring of 1939 no doubt was the publication of Finnegans Wake by James Joyce. On May 4th 1939 FW was published simultaneously by Faber and Faber in London and by Viking Press in New York, after seventeen years of composition.

In 1928-29, Joyce started publishing individual chapters from FW, then known as ‘Work in Progress’, including chapter II.2 ‘The Triangle’, of which a brief excerpt was already published in February 1928. The name comes from the only diagram in FW, the classical Euclidian construction of an equilateral triangle (FW, p. 293)



This Vesica piscis has multiple interpretations in FW, most of them sexual. The triangle $\Delta$ is the Sigla for Anna Livia Plurabelle throughout FW, but it also refers to the river Liffey through Dublin.

Here’s Anthony Burgess explaining some of the Sigla, the relevant part starts at 14.20 into the clip.

In fact, many of FW’s Sigla are derived from mathematical symbols, such as $\exists$ (Earwicker), $\perp$ and $\vdash$ (Issy). For more on this, please read The logic of the doodles in Finnegans Wake II.2.

Not only does the equilateral triangle $\Delta$ refer to the river Liffey, the entire Euclidian diagram can be seen as a map for Dublin and its surroundings, as emphasised by the words “Vieus Von DVbLIn” (views from Dublin) in FW right under the diagram.

Here’s Dublin with the Liffey running through it, and Phoenix Park, which also features prominently in FW, see for example Phoenix Park in Finnegans Wake.



Views of Dublin – Photo Credit

The similarity between the map and the diagram is even clearer in Joyce’s own drawing in the first draft of FW.



The Triangle – Photo Credit

There’s a lot more to say about Joyce’s uses of geometry and topography in Ulysses and Finnegans Wake, in fact Ciaran McMorran wrote an entire Glasgow Ph. D. about it, but perhaps I’ll save some of that for a future post.

But what does this have to to with the Bourbaki Code, the puzzles contained in the Bourbaki-Petard wedding announcement?



Well, I claim that Andre Weil hid the Vesica Piscis/Euclidian diagram into the ‘faire part’. The challenge is to view the wedding announcement as a partial city- map. Clearly this time, the city of Dublin should be replaced by the city of Paris. Se non e vero …

Probably, there are enough hints contained in the previous posts in this series for you to spot the triangle(s) on the map of Paris. If you do so, please leave a comment, or email me.

Meanwhile, we’ll unravel first the more obvious levels of interpretation of the wedding announcement.

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Ghost metro stations

In the strange logic of subways I’ve used a small part of the Parisian metro-map to illustrate some of the bi-Heyting operations on directed graphs.



Little did I know that this metro-map gives only a partial picture of the underground network. The Parisian metro has several ghost stations, that is, stations that have been closed to the public and are no longer used in commercial service. One of these is the Haxo metro station.



Haxo metro station – Photo Credit

The station is situated on a line which was constructed in the 1920s between Porte des Lilas (line 3bis) and Pré-Saint-Gervais (line 7bis), see light and dark green on the map above . A single track was built linking Place des Fêtes to Porte des Lilas, known as la voie des Fêtes, with one intermediate station, Haxo.



For traffic in the other direction, another track was constructed linking Porte des Lilas to Pré Saint-Gervais, with no intermediate station, called la voie navette. Haxo would have been a single-direction station with only one platform.

But, it was never used, and no access to street level was ever constructed. Occasional special enthusiast trains call at Haxo for photography.



Apart from the Haxo ‘station morte’ (dead station), these maps show another surprise, a ‘quai mort’ (dead platform) known as Porte des Lilas – Cinema. You can hire this platform for a mere 200.000 Euro/per day for film shooting.

For example, Le fabuleux destin d’Amelie Poulin has a scene shot there. In the film the metro station is called ‘Abbesses’ (3.06 into the clip)

There is a project to re-open the ghost station Haxo for public transport. From a mathematical perspective, this may be dangerous.

Remember the subway singularity?

In the famous story A subway named Mobius by A. J. Deutsch, the Boylston shuttle on the Boiston subway went into service on March 3rd, tying together the seven principal lines, on four different levels. A day later, train 86 went missing on the Cambridge-Dorchester line…

The Harvard algebraist R. Tupelo suggested the train might have hit a node, a singularity. By adding the Boylston shuttle, the connectivity of the subway system had become infinite…

Now that we know of the strange logic of subways, an alternative explanation of this accident might be that by adding the Boylston shuttle, the logic of the Boston subway changed dramatically.

This can also happen in Paris.

I know, I’ve linked already to the movie ‘Moebius’ by Gustavo Mosquera, based on Deutsch’s story, set in Buenos Aires.

But, if you have an hour to spend, here it is again.

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Lockdown reading : Bacon

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



In an attempt to raise the level of this series, I tried to get through the latest hype in high-brow literature: The Death of Francis Bacon by Max Porter.

It’s an extremely thin book, just 43 pages long, hardly a novella. My Kindle said I should be able to read it in less than an hour.

Boy, did that turn out differently. I’m a week into this book, and still struggling.



Chapter 4(?) :Three Studies for a Self-Portrait, (Francis Bacon, 1979)

A few minutes into the book I realised I didn’t know the first thing about Bacon’s death, and that the book was not going to offer me that setting. Fortunately, there’s always Wikipedia:

While holidaying in Madrid in 1992, Bacon was admitted to the Handmaids of Maria, a private clinic, where he was cared for by Sister Mercedes. His chronic asthma, which had plagued him all his life, had developed into a more severe respiratory condition and he could not talk or breathe very well.

Fine, at least I now knew where “Darling mama, sister oh Dios, Mercedes” (p.7) came from, and why every chapter ended with “Intenta descansar” (try to rest).

While I’m somewhat familiar with Bacon’s paintings, I did know too little about his life to follow the clues sprinkled throughout the book. Fortunately, there’s this excellent documentary about his life: “Francis Bacon: A Brush with Violence” (2017)

Okay, now I could place many of the characters visiting Bacon, either physically sitting on the chair he offers at the start of each chapter (“Take a seat why don’t you”), or merely as memories playing around in his head. It’s a bit unclear to me.

Then, there’s the structure of the book. Each of the seven chapters has as title the dimensions of a painting:

  • One: Oil on canvas, 60 x 46 1/2 in.
  • Two: Oil on canvas, 65 1/2 x 56 in.
  • Three: Oil on canvas, 65 x 56 in.
  • Four: Oil on canvas, 14 x 12 in.
  • Five: Oil on canvas, 78 x 58 in.
  • Six: Oil on canvas, 37 x 29 in.
  • Seven: Oil on canvas, 77 x 52 in.

Being the person I am, I hoped that if I could track down the corresponding Bacon paintings, I might begin to understand the corresponding chapter. Fortunately, Wikipedia provides a List of paintings by Francis Bacon.

Many of Bacon’s paintings are triptychs, and the dimensions refer to those of a single panel. So, even if I found the correct triptych I still had to figure out which of the three panels corresponds to the chapter.

And often, there are several possible candidates. The 14 x 12 in. panel-format Bacon often used for studies for larger works. So, chapter 4 might as well refer to his studies for a self portrait (see above), or to the three studies for a portrait of Henrietta Moraes:



Chapter 4(?) : Three studies for portrait of Henrietta Moraes (1963)

Here are some of my best guesses:



Chapter 3(?): Portrait of Henrietta Moraes (1963)



Chapter 6(?): Three Studies for Figures at the Base of a Crucifixion (1944)



Chapter 5(?): Triptych Inspired by the Oresteia of Aeschylus (1981)

No doubt, I’m just on a wild goose chase here. Probably, Max Porter is merely using existing dimensions of Bacon paintings for blank canvases to smear his words on, as explained in this erudite ArtReview What Does It Mean To Write a Painting?.

Here’s the writer Max Porter himself, explaining his book.

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Lockdown reading : Centenal

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



The Centenal Cycle is a trilogy written by Malka Older.

A Centenal is the basic political unit of a future micro-democracy. It is a neighbourhood consisting of 100.000 people which can vote for any government it wants, from anywhere in the world.

“Centenal-based microdemocracy naturally requires extensive use of technology. In my book, it’s provided through a massive international bureaucracy known as Information, which offers voters data about the thousands of possible governments and helps those governments manage what may be far-flung territories once they’re elected.” (Malka Older)

In this trilogy Malka Older draws from her own life: she obtained a Ph. D. from Sciences Po exploring the dynamics of multi-level governance and disaster response, and has more than a decade of experience in humanitarian aid and development.

The Centenal Cycle consists of these three books:

Infomocracy (2016) (link containing excerpts).



It’s been twenty years and two election cycles since Information, a powerful search engine monopoly, pioneered the switch from warring nation-states to global micro-democracy. The corporate coalition party Heritage has won the last two elections. With another election on the horizon, the Supermajority is in tight contention, and everything’s on the line.

Null States (2017).



The future of democracy is about to implode.

After the last controversial global election, the global infomocracy that has ensured thirty years of world peace is fraying at the edges. As the new Supermajority government struggles to establish its legitimacy, agents of Information across the globe strive to keep the peace and maintain the flows of data that feed the new world order.

State Tectonics (2018) (link containing excerpts).



The future of democracy must evolve or die.

The last time Information held an election, a global network outage, two counts of sabotage by major world governments, and a devastating earthquake almost shook micro-democracy apart. Five years later, it’s time to vote again, and the system that has ensured global peace for 25 years is more vulnerable than ever.

Here’s a short interview with Malka Older on Sci-Fi, AI and its possible uses in the writing process.

Here’s a longer clip in which she talks about ‘Speculative Resistance’ at the Personal Democracy Forum 2018.

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Lockdown reading : Penumbra

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



It’s difficult to admit, but Amazon’s blurb lured me into reading Mr. Penumbra’s 24-Hour Bookstore by Robin Sloan:

“With irresistible brio and dazzling intelligence, Robin Sloan has crafted a literary adventure story for the 21st century, evoking both the fairy-tale charm of Haruki Murakami and the enthusiastic novel-of-ideas wizardry of Neal Stephenson or a young Umberto Eco, but with a unique and feisty sensibility that’s rare to the world of literary fiction.” (Amazon’s blurb)

I’m a fan of Murakami’s later books (such as 1Q84 or Killing Commendatore), and Stephenson’s earlier ones (such as Snow Crash or Cryptonomicon), so if someone wrote the perfect blend, I’m in. Reading Penumbra’s bookstore, I discovered that these ‘comparisons’ were borrowed from the book itself, leaving out a few other good suggestions:

One cold Tuesday morning, he strolls into the store with a cup of coffee in one hand and his mystery e-reader in the other, and I show him what I’ve added to the shelves:

Stephenson, Murakami, the latest Gibson, The Information, House of Leaves, fresh editions of Moffat” – I point them out as I go.

(from “Mr Penumbra’s 24-Hour Bookstore”)

This trailer gives a good impression of what the book is about.

Why might you want to read this book?

  • If you have a weak spot for a bad ass Googler girl and her tecchy wizardry.
  • If you are interested in the possibilities and limitations of Google’s tools.
  • If you don’t know what a Hadoop job is or how to combine it with a Mechanical Turk to find a marker on a building somewhere in New-York.
  • If you never heard of the Gerritszoon font, preinstalled on every Mac.

As you see, Google features prominently in the book, so it is kind of funny to watch the author, Robin Sloan, give a talk at Google.

Some years later, Sloan wrote a (shorter) prequel Ajax Penumbra 1969, which is also a good read but does not involve fancy technology, unless you count tunnel construction among those.



Read it if you want to know how Penumbra ended up in his bookstore and how he recovered the last surviving copy of the book “Techne Tycheon”.

More information (together with reading suggestions) can be found at Mr Penumbra’s 24-hour bookstore: a reading map.

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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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