neverendingbooks
The following are the titles of recent articles syndicated from neverendingbooks
Add this feed to your friends list for news aggregation, or view this feed's syndication information.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.

[ << Previous 20 ]
Tuesday, September 19th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
3:01 pm
A projective plain (plane) of order ten

A projective plane of order $n$ is a collection of $n^2+n+1$ lines and $n^2+n+1$ points satisfying:

  • every line contains exactly $n+1$ points
  • every point lies on exactly $n+1$ lines
  • any two distinct lines meet at exactly one point
  • any two distinct points lie on exactly one line

Clearly, if $q=p^k$ is a pure prime power, then the projective plane over $\mathbb{F}_q$, $\mathbb{P}^2(\mathbb{F}_q)$ (that is, all nonzero triples of elements from the finite field $\mathbb{F}_q$ up to simultaneous multiplication with a non-zero element from $\mathbb{F}_q$) is a projective plane of order $q$.

The easiest example being $\mathbb{P}^2(\mathbb{F}_2)$ consisting of seven points and lines

But, there are others. A triangle is a projective plane of order $1$, which is not of the above form, unless you believe in the field with one element $\mathbb{F}_1$…

And, apart from $\mathbb{P}^2(\mathbb{F}_{3^2})$, there are three other, non-isomorphic, projective planes of order $9$.

It is clear then that for all $n < 10$, except perhaps $n=6$, a projective plane of order $n$ exists.

In 1938, Raj Chandra Bose showed that there is no plane of order $6$ as there cannot be $5$ mutually orthogonal Latin squares of order $6$, when even the problem of two orthogonal squares of order $6$ (see Euler’s problem of the $36$ officers) is impossible.

Yeah yeah Bob, I know it has a quantum solution.

Anyway by May 1977, when Lenstra’s Festschrift ‘Een pak met een korte broek’ (a suit with shorts) was published, the existence of a projective plane of order $10$ was still wide open.

That’s when Andrew Odlyzko (probably known best for his numerical work on the Riemann zeta function) and Neil Sloane (probably best known as the creator of the On-Line Encyclopedia of Integer Sequences) joined forces to publish in Lenstra’s festschrift a note claiming (jokingly) the existence of a projective plane of order ten, as they were able to find a finite field of ten elements.



Here’s a transcript:

A PROJECTIVE PLAIN OF ORDER TEN

A. M. Odlyzko and N.J.A. Sloane

This note settles in the affirmative the notorious question of the existence of a projective plain of order ten.

It is well-known that if a finite field $F$ is given containing $n$ elements, then the projective plain of order $n$ can be immediately constructed (see M. Hall Jr., Combinatorial Theory, Blaisdell, Waltham, Mass. 1967 and D.R. Hughes and F.C. Piper, Projective Planes, Springer-Verlag, N.Y., 1970).

For example, the points of the plane are represented by the nonzero triples $(\alpha,\beta,\gamma)$ of elements of $F$, with the convention that $(\alpha,\beta,\gamma)$ and $(r\alpha, r\beta, r\gamma)$ represent the same point, for all nonzero $r \in F$.

Furthermore this plain even has the desirable property that Desargues’ theorem holds there.

What makes this note possible is our recent discovery of a field containing exactly ten elements: we call it the digital field.

We first show that this field exists, and then give a childishly simple construction which the reader can easily verify.

The Existence Proof

Since every real number can be written in the decimal system we conclude that

\[
\mathbb{R} = GF(10^{\omega}) \]

Now $\omega = 1.\omega$, so $1$ divides $\omega$. Therefore by a standard theorem from field theory (e.g. B. L. van der Waerden, Modern Algebra, Ungar, N.Y., 1953, 2nd edition, Volume 1, p. 117) $\mathbb{R}$ contains a subfield $GF(10)$. This completes the proof.

The Construction

The elements of this digital field are shown in Fig. 1.

They are labelled $Left_1, Left_2, \dots, Left_5, Right_1, \dots, Right_5$ in the natural ordering (reading from left to right).



Addition is performed by counting, again in the natural way. An example is shown in Fig. 2, and for further details the reader can consult any kindergarten student.

In all digital systems the rules for multiplication can be written down immediately once addition has been defined; for example $2 \times n = n+n$. The reader will easily verify the rest of the details.

Since this field plainly contains ten elements (see Fig. 1) we conclude that there is a projective plain of order ten.

So far, the transcript.

More seriously now, the non-existence of a projective plane of order ten was only established in 1988, heavily depending on computer-calculations. A nice account is given in

C. M. H. Lam, “The Search for a Finite Projective Plane of Order 10”.

Now that recent iPhones nearly have the computing powers of former Cray’s, one might hope for easier proofs.

Fortunately, such a proof now exists, see A SAT-based Resolution of Lam’s Problem by Curtis Bright, Kevin K. H. Cheung, Brett Stevens, Ilias Kotsireas, Vijay Ganesh

David Roberts, aka the HigherGeometer, did a nice post on this
No order-10 projective planes via SAT
.

Monday, September 18th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
2:32 pm
A suit with shorts

I’m retiring in two weeks so I’m cleaning out my office.

So far, I got rid of almost all paper-work and have split my book-collection in two: the books I want to take with me, and those anyone can grab away.

Here’s the second batch (math/computer books in the middle, popular science to the right, thrillers to the left).



If you’re interested in some of these books (click for a larger image, if you want to zoom in) and are willing to pay the postage, leave a comment and I’ll try to send them if they survive the current ‘take-away’ phase.

Here are two books I definitely want to keep. On the left, an original mimeographed version of Mumford’s ‘Red Book’.

On the right, ‘Een pak met een korte broek’ (‘A suit with shorts’), a collection of papers by family and friends, presented to Hendrik Lenstra on the occasion of the defence of his Ph.D. thesis on Euclidean number-fields, May 18th 1977.

If the title intrigues you, a photo of young Hendrik in suit and shorts is included.

This collection includes hilarious ‘papers’ by famous people including

  • ‘A headache-causing problem’ by Conway (J.H.), Paterson (M.S.), and Moscow (U.S.S.R.)
  • ‘A projective plain of order ten’ by A.M. Odlyzko and N.J.A. Sloane
  • ‘La chasse aux anneaux principaux non-Euclidiens dans l’enseignement’ by Pierre Samuel
  • ‘On time-like theorems’ by Michiel Hazewinkel
  • ‘She loves me, she loves me not’ by Richard K. Guy
  • ‘Theta invariants for affine root systems’ by E.J.N. Looijenga
  • ‘The prime of primes’ by F. Lenstra and A.J. Oort
  • (and many more, most of them in Dutch)

Perhaps I can do a couple of posts on some of these papers. It might break this clean-up routine.

Sunday, September 17th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
11:38 am
the L-game

In 1982, the BBC ran a series of 10 weekly programmes entitled de Bono’s Thinking Course. In the book accompanying the series Edward de Bono recalls the origin of his ‘L-Game’:


Image

Many years ago I was sitting next to the famous mathematician, Professor Littlewood, at dinner in Trinity College. We were talking about getting computers to play chess. We agreed that chess was difficult because of the large number of pieces and different moves. It seemed an interesting challenge to design a game that was as simple as possible and yet could be played with a degree of skill.

As a result of that challenge I designed the ‘L-Game’, in which each player has only one piece (the L-shape piece). In turn he moves this to any new vacant position (lifting up, turning over, moving across the board to a vacant position, etc.). After moving his L-piece he can – if he wishes – move either one of the small neutral pieces to any new position. The object of the game is to block your opponent’s L-shape so that no move is open to it.

It is a pleasant exercise in symmetry to calculate the number of possible L-game positions.

The $4 \times 4$ grid has $8$ symmetries, making up the dihedral group $D_8$: $4$ rotations and $4$ reflections.

An L-piece breaks all these symmetries, that is, it changes in form under each of these eight operations. That is, using the symmetries of the $4 \times 4$-grid we can put one of the L-pieces (say the Red one) on the grid as a genuine L, and there are exactly 6 possibilities to do so.

For each of these six positions one can then determine the number of possible placings of the Blue L-piece. This is best done separately for each of the 8 different shapes of that L-piece.

Here are the numbers when the red L is placed in the left bottom corner:



In total there are thus 24 possibilities to place the Blue L-piece in that case. We can repeat the same procedure for the remaining Red L-positions. Here are the number of possibilities for Blue in each case:



That is, there are 82 possibilities to place the two L-pieces if the Red one stands as a genuine L on the board.

But then, the L-game has exactly $18368 = 8 \times 82 \times 28$ different positions, where the factor

  • $8$ gives the number of symmetries of the square $4 \times 4$ grid.
  • Using these symmetries we can put the Red L-piece on the grid as a genuine $L$ and we just saw that this leaves $82$ possibilities for the Blue L-piece.
  • This leaves $8$ empty squares and so $28 = \binom{8}{2}$ different choices to place the remaining two neutral pieces.

The $2296 = 82 \times 28$ positions in which the red L-piece is placed as a genuine L can then be analysed by computer and the outcome is summarised in Winning Ways 2 pages 384-386 (with extras on pages 408-409).

Of the $2296$ positions only $29$ are $\mathcal{P}$-positions, meaning that the next player (Red) will loose. Here are these winning positions for Blue




Here, neutral piece(s) should be put on the yellow square(s). A (potential) remaining neutral piece should be placed on one of the coloured squares. The different colours indicate the remoteness of the $\mathcal{P}$-position:

  • Pink means remoteness $0$, that is, Red has no move whatsoever, so mate in $0$.
  • Orange means remoteness $2$: Red still has a move, but will be mated after Blue’s next move.
  • Purple stands for remoteness $4$, that is, Blue mates Red in $4$ moves, Red starting.
  • Violet means remoteness $6$, so Blue has a mate in $6$ with Red starting
  • Olive stands for remoteness $8$: Blue mates within eight moves.

Memorising these gives you a method to spot winning opportunities. After Red’s move image a board symmetry such that Red’s piece is a genuine L, check whether you can place your Blue piece and one of the yellow pieces to obtain one of the 29 $\mathcal{P}$-positions, and apply the reverse symmetry to place your piece.

If you don’t know this, you can run into trouble very quickly. From the starting position, Red has five options to place his L-piece before moving one of the two yellow counters.



All possible positions of the first option loose immediately.



For example in positions $a,b,c,d,f$ and $l$, Blue wins by playing


Image

Here’s my first attempt at an opening repertoire for the L-game. Question mark means immediate loss, question mark with a number means mate after that number of moves, x means your opponent plays a sensible strategy.









Surely I missed cases, and made errors in others. Please leave corrections in the comments and I’ll try to update the positions.

Monday, August 21st, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
9:54 am
9 Bourbaki founding members, really?

The Clique (Twenty Øne Piløts fanatic fanbase) is convinced that the nine Bishops of Dema were modelled after the Bourbaki-group.

It is therefore of no surprise to see a Photoshopped version circulating of this classic picture of some youthful Bourbaki-members (note Jean-Pierre Serre poster-boying for Elon Musk’s site),

replacing some of them with much older photos of other members. Crucial seems to be that there are just nine of them.

I don’t know whether the Clique hijacked Bourbaki’s Wikipedia page, or whether they were inspired by its content to select those people, but if you look at that Wikipedia page you’ll see in the right hand column:

Founders

  • Henri Cartan
  • Claude Chevalley
  • Jean Coulomb
  • Jean Delsarte
  • Jean Dieudonné
  • Charles Ehresmann
  • René de Possel
  • André Weil

Really? Come on.

We know for a fact that Charles Ehresmann was brought in to replace Jean Leray, and Jean Coulomb to replace Paul Dubreil. Surely, replacements can’t be founders, can they?

Well, unfortunately it is not quite that simple. There’s this silly semantic discussion: from what moment on can you call someone a Bourbaki-member…

The collective name ‘Nicolas Bourbaki’ was adopted only at the Bourbaki-congress in Besse in July 1935 (see also this post).

But, before the Besse-meeting there were ten ‘proto-Bourbaki’ meetings, the first one on December 10th, 1934 in Cafe Capoulade. These meetings have been described masterly by Liliane Beaulieu in A Parisian Cafe and Ten Proto-Bourbaki Meetings (1934-35) (btw. if you know a direct link to the pdf, please drop it in the comments).

During these early meetings, the group called itself ‘The Committee for the Treatise on Analysis’, and not yet Bourbaki, whence the confusion.

Do we take the Capoulade-1934 meeting as the origin of the Bourbaki group (in which case the founding-members would be Cartan, Chevalley, De Possel, Delsarte, Dieudonne, and Weil), or was the Bourbaki-group founded at the Besse-congress in 1935 (when Cartan, Chevalley, Coulomb, De Possel, Dieudonne, Mandelbrojt, and Weil were present)?

Here’s a summary of which people were present at all meetings from December 1934 until the second Chancay-congress in September 1939, taken from Gatien Ricotier ‘Projets collectifs et personnels autour de Bourbaki dans les années 1930 à 1950′:

07-1935 is the Besse-congress, 09-1936 is the ‘Escorial’-congress (or Chancay 1) and 09-1937 is the second Chancay-congress. The ten dates prior to July 1935 are the proto-Bourbaki meetings.

Even though Delsarte was not present at the Besse-1935 congress, and De Possel moved to Algiers and left Bourbaki in 1941, I assume most people would agree that the six people present at the first Capoulade-meeting (Cartan, Chevalley, De Possel, Delsarte, Dieudonne, and Weil) should certainly be counted among the Bourbaki founding members.

What about the others?

We can safely eliminate Dubreil: he was present at just one proto-Bourbaki meeting and left the group in April 1935.

Also Leray’s case is straightforward: he was even excluded from the Besse-meeting as he didn’t contribute much to the group, and later he vehemently opposed Bourbaki, as we’ve seen.

Coulomb’s role seems to restrict to securing a venue for the Besse-meeting as he was ‘physicien-adjoint’ at the ‘Observatoire Physique du Globe du Puy-de-Dome’.



Because of this he could rarely attend the Julia-seminar or Bourbaki-meetings, and his interest in mathematical physics was a bit far from the themes pursued in the seminar or by Bourbaki. It seems he only contributed one small text, in the form of a letter. Due to his limited attendance, even after officially been asked to replace Dubreil, he can hardly be counted as a founding member.

This leaves Szolem Mandelbrojt and Charles Ehresmann.

We’ve already described Mandelbrojt as the odd-man-out among the early Bourbakis. According to the Bourbaki archive he only contributed one text. On the other hand, he also played a role in organising the Besse-meeting and in providing financial support for Bourbaki. Because he was present already early on (from the second proto-Bourbaki meeting) until the Chancay-1937 meeting, some people will count him among the founding members.

Personally I wouldn’t call Charles Ehresmann a Bourbaki founding member because he joined too late in the process (March 1936). Still, purists (those who argue that Bourbaki was founded at Besse) will say that at that meeting he was put forward to replace Jean Leray, and later contributed actively to Bourbaki’s meetings and work, and for that reason should be included among the founding members.

What do you think?

How many Bourbaki founding members are there? Six (the Capoulade-gang), seven (+Mandelbrojt), eight (+Mandelbrojt and Ehresmann), or do you still think there were nine of them?

Friday, August 11th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
3:34 pm
TØP PhotoShop mysteries

Suppose you’re writing a book, and for the duration of that project you keep a certain photo as your desktop-background. I guess we might assume that picture to be inspirational for your writing process.

If you PhotoShopped it to add specific elements, might we assume these extra bits to play a crucial role in your story?

Now, let’s turn to Twenty One Pilots and the creation process of their album Trench, released on October 5, 2018

We know from this tweet (from August 19th, 2018) that Tyler Joseph’s desktop-background picture was a photoshopped version of the classic Bourbaki-1938 photo on the left below, given it Trench-yellow, and added a bearded man in the doorway (photo on the right)




And we know from this interview (from September 5th, 2018) that, apart from the bearded man, he also replaced in the lower left corner the empty chair by a sitting person (lower photo).

The original photo features on the Wikipedia page on Nicolas Bourbaki, and as Tyler Joseph has revealed that Blurryface‘s real name is Nicolas Bourbaki (for whatever reason), and that he appears in the lyrics of Morph on Trench, this may make some sense.

But, of the seven people in the picture only three were founding members of Bourbaki: Weil, Diedonne and Delsarte. Ehresmann entered later, replacing Jean Leray, and Pison and Chabauty were only guinea pigs at that moment (they later entered Bourbaki, Chabauty briefly and Pison until 1950), and finally, Simonne Weil never was a member.

There’s another strange thing about the original picture. All of them, but Andre and Simone Weil, look straight into the camera, the Weil’s seem to be more focussed on something happening to the right.

Now, TØP has something with the number 9. There are nine circles on the cover of Blurryface (each representing one of a person’s insecurities, it seems), there are nine towers in the City of Dema, nine Bishops, etc.



So, from their perspective it makes sense to Photoshop two extra people in, and looking at the original there are two obvious places to place them: in the empty doorway, and on the empty chair.

But, who are they, and what is their significance?

1. The bearded man in the doorway

As far as I know, nobody knows who he is. From a Bourbaki point of view it can only be one person: Elie Cartan.

We know he was present at the 1938 Bourbaki Dieulefit/Beauvallon meeting, and that he was kind of a father figure to Bourbaki. Among older French mathematicians he was one of few (perhaps the only one) respected by all of Bourbaki.

But, bearded man is definitely not Elie Cartan…

If bearded man exists and has a Wikipedia page, the photo should be on that page. So, if you find him, please leave a comment.

Previous in this series I made a conjecture about him, but I’m not at all sure.

2. Why, of all people, Szolem Mandelbrojt?

We know from this Twentyonepilots subReddit post that the man sitting on the previously empty chair in none other than Bourbaki founding member Szolem Mandelbrojt, shopped in from this other iconic early Bourbaki-photo from the 1937 Chancay-meeting.

Let me tell you why this surprises me.

Szolem Mandelbrojt was atypical among the first Bourbaki-gang in many ways: he was the only one who didn’t graduate from the ENS, he was a bit older than the rest, he was the only one who was a full Professor (at Clermont-Ferrand) whereas the others were ‘maitre de conference’, he was the only one who didn’t contribute actively in the Julia seminar (the proto-Bourbaki seminar) nor much to the Bourbaki-congresses either for that matter, etc. etc.

Most of all, I don’t think he would feel particularly welcome at the 1938 congress. Here’s why.



(Jacques Hadamard (left), and Henri Lebesgue (right))

From Andre Weil’s autobiography (page 120):

Hadamard’s retirement left his position open. I thought myself not unworthy of succeeding him; my friends, especially Cartan and Delsarte, encouraged me to a candidate. It seemed to me that Lebesgue, who was the only mathematician left at the College de France, did not find my candidacy out of place. He even let me know that it was time to begin my ‘campaign visits’.

But the Bourbaki-campaign against a hierarchy of scientific prizes instituted by Jean Perrin (the so called ‘war of the medals’) interfered with his personal campaign. (Perhaps more important was that Mandelbrojt did his Ph.D. under Hadamard…)

Again from Weil’s autobiography (page 121):

Finally Lebesque put an end to my visits by telling me that he had decided on Mandelbrojt. It seemed to me that my friends were more disappointed than I at this outcome.

In the spring of 1938, Mandelbrojt succeeded Hadamard at the College de France.

There’s photographic evidence that Mandelbrojt was present at the 1935 Besse-congress and clearly at the 1937-Chancay meeting, but I don’t know that he was even present at Chancay-1936.

The only picture I know of that meeting is the one below. Standing on bench: Chevalley’s nephews, seated Andre Weil and Chevalley’s mother; standing, left to right: Ninette Ehresmann, Rene de Possel, Claude Chavalley, Jacqueline Chavalley, Mirles, Jean Delsarte and Charles Ehresmann.

Of all possible people, Szolem Mandelbrojt would be the miscast at the 1938-meeting. So, why did they shop him in?

– convenience: they had an empty chair in the original picture, another Bourbaki-photo with a guy sitting on such a chair, so why not shop him in?

– mistaken identity: in the subReddit post the sitting guy was mistakenly identified as Claude Chevalley. Now, there is a lot to say about wishing to add Chevalley to the original. He is by far the most likeable of all Bourbakis, so if these nine were ever supposed to be the nine Bishops of Dema, he most certainly would be Keons. But, Chevalley was already in the US at that time, and was advised by the French consul to remain there in view of the situation in Europe. As a result, Chevalley could not obtain a French professorship before the early 50ties.

– a deep hidden clue: remember all that nonsense about Josh Dun’s ‘alma mater’ being that Ukrainian building where Nico and the niners was shot? Well, Szolem Mandelbrojt’s alma mater was the University of Kharkiv in Ukraine. See this post for more details.

3. Is it all about Simone Weil?

If you super-impose the two photographs, pinning Mandelbrojt in both, the left border of the original 1938-picture is an almost perfect mirror for both appearances of Simone Weil. Can she be more important in all of this than we think?

Previous in the Bourbaki&TØP series:

Friday, August 4th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
12:38 pm
Dema2Trench, AND REpeat

There’s this band Twenty One Pilots and they’ve woven a complicated story around some of their albums, notably Blurryface, Trench, and Scaled and Icy.

Since Trench, an important component of that story is the Bourbaki group, so I’m just curious whether the few things I know about them can help to clarify parts in the TØP- storyline.

Pretty pointless, I know, as no artistic project will follow blindly historical facts. But hey, as long as I discover new things I’ll keep going.

The story’s about a City of Dema, ruled by nine Bishops installing a terror regime called Vialism, and a land outside the city walls, called Trench, to which citizens would like to escape.

Think of Dema as an extremely toxic environment from which you need to escape to a safe place, let’s call it Trench.

Sadly, too often survivors from abusive settings later on create their own toxic environment, abusive to others.

So, Dema-escapees to Trench should always be wary of the danger of creating a new Dema for others.

It is very hard to break these Dema2Trench cycles of violence. That’s probably why the map of the City of Dema is circular.

Let’s start with these two photos not (yet) in the Dema-lore:



Both pictures are of the French mathematician Gaston Julia.

Julia graduated from the ENS in 1914, so was among the worst victims of the military regime Lavisse installed at the ENS. He was mobilised but hadn’t yet completed his second year of military training. That was shortened to just 5 months, after which we has send as a second lieutenant to the war.

In January 1915 he was seriously wounded in his face, had to undergo a series of operations and for the rest of his life he resigned himself to wearing a leather strap around the area where his nose had been.

He ran a weekly seminar from 1933 till 1939, the Seminaire Julia, to which the Bourbaki core members contributed a vast number of lectures.

Until 1937-38 (so just before the Dieulefit Bourbaki congress) the Bourbakis felt happy citizens of Julia’s Seminar/Dema. But then they discovered his political agenda and were expelled from it, or escaped from it depending on the version.

Jean Leray convinced Julia that it was a terrible mistake to let his seminar run by Bourbaki, and that things would go much better if he ran it. Julia expelled Bourbaki from the seminar, changed its name to ‘ Cercle mathématique de l’École normale supérieure’ and moved the venue from the IHP to the ENS. The attendants of this seminar were younger and less international that in the preceding years, hence more malleable to his political ideas.

Another reason for the break-up between Bourbaki and Julia was that they reproached him of attending in June 1937 the festivities of the bicentennial of the University of Gottingen, which were seen as pure propaganda for the Nazi-regime.

During WW2, Julia collaborated with the occupying Nazi-regime in that he tried to find French mathematicians to contribute to the Zentralblatt. After the war he was briefly suspended for this.

Much more on the Julia seminar and the break-up with Bourbaki can be read in the thesis by Gatien Ricotier ‘Projets collectifs et personnels autour de Bourbaki dans les années 1930 à 1950’, and Michele Audin’s book on the Julia Seminar.

Let us compare Julia’s photographs to these two in Dema-lore:



Is it a coincidence that Clancy in Trench has a scar on his nose? Is it a coincidence that the black paint on some of the Bishop’s faces looks a lot like Julia’s mask?

Can it be that victims of one Dema-era become Bishops in a next era?

This repetitiveness of Dema-environments also indicates the importance of Bishop Andre. Recall that all the Bishops’ names (except for Nico) come from concatenations of word-parts in the lyrics of the songs on the Blurryface album.

ANDRE comes from ‘..AND REpeat’ in Fairly local:

Tomorrow I’ll keep a beat
And repeat yesterday’s dance

In view of this, let’s have another look at the two Bourbaki-related photographs that appeared in the run up to the Trench-album:



On the left is the photo of the Dieulefit/Beauvallon 1938 meeting, which is on the Bourbaki Wikipedia page, and was on the desktop of Tyler Joseph.

On the right a photo of Andre Weil together with a girl, according to Wikipedia the picture dates from 1956. I’m pretty certain it was taken in the summer of 1957, and that the girl is Mireille Cartan, the second youngest daughter of Henri Cartan. Not that any of this matters, TØP-wise. A clipping of the girl was among the material originally posted at the dmaorg.info site.

In 1938, Andre Weil was a victim of Lavisse’s Dema. His year was the last one getting a military training to become reserve officers in the French infantry/artillery (as were Cartan, Dieudonne and Delsarte).

When France would mobilise they were forced to return to Dema (military service) and lead their bataljons as second lieutenants into war. All of them, except for Weil, did this.
Weil escaped to Trench (Finland), and was taken back to Dema, and imprisonment.

In 1957, Bourbaki dominated much of French mathematical life, and certainly its influence in Paris was suffocating for aspiring math-students. A good read on this is Jacques Roubaud’s Mathematique.

Bourbaki has turned French mathematics (and beyond) into its own Dema, and Andre Weil certainly was one of the more important Bishops of it.

Previous in the Bourbaki&TØP series:

Wednesday, July 26th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
1:00 pm
Weil photos used in Dema-lore

On April 20th of 2018, twenty one pilots updated their store page to include a video with a hidden message at the end of it.

and with a bit of sleuthing it led to a page on the dmaorg.info site containing:



This was immediately identified as part of the photo on the right, which is on the French Wikipedia page for Andre Weil.

The photo is clipped in such a way one cannot be certain whether the child is a boy or girl, so a logical explanation is that this is supposed to be the nine year old Clancy, shielding his eyes from the violence (vialism) he just discovered in Dema.

The full picture suggests that Clancy’s struggles might mirror some in Andre Weil’s life.

Andre Weil was born May 6th, 1906, so ‘in his ninth year’ World War 1 breaks out in 1914.

Last time we’ve seen that Bourbaki’s Dema = Ecole Normal Superieure in Paris during WW1, Vialism = militant patriotism sending ENS-graduates as trained reserve second lieutenants in the infantry to the trenches, and there getting killed ‘pour la patrie’ and the glory of the ENS and its director Ernest Lavisse, “L’instituteur national”.

Here’s a G-translation of his letter to the young French, published September 23rd 1914:

Dear children of France, You will be old one day, and, like the old, you will like to remember times past. There will come evenings when your little children, seeing you dreaming, will say to you: Tell us, grandfather. And you will tell. It will be a few episodes of the war, a long march, an alert, a bayonet assault, a cavalry charge, the feat of a battery of 75, the strewn enemy dead on the plain, or else, in the streets of a city, the serried ranks of corpses left standing for lack of room to fall; and then the death of comrades, the terrible losses of your company and your regiment, your wounds received in Belgium, in Champagne, on the banks of the Rhine, beyond the Rhine; but the joy of victories, the poles knocked down on too narrow frontiers, triumphal entries.

On those evenings, after the amazed children have gone to sleep, you will open a drawer where you will have collected precious objects, a bullet extracted from a wound, a piece of shell, a cloth where your blood will have turned pale, a cross of honour, I hope, or a military medal, at the very least a medal from the 1914 war, on the ribbon of which the silver clasps will bear the names of immortal battles. And whatever your life, happy or unhappy, you will be able to say: I lived great days such as the history of men had not yet seen. And you will be right to be proud of your youth, because you are sublime young people!

I have read your letters; I have spoken with the wounded. Through you, I know what heroism is. I had heard a lot about it, being a historian by profession, but now I see it, I touch it, and how beautiful your heroism is, embellished with grace and smiling in the French way! Young soldiers if you were given one chevron per battle, your march would not be enough to accommodate them, because at the end of the war you would count more chevrons than years;

Young soldiers you are glorious old warriors.

Oh! Thanks thanks! Thank you for the beautiful end of life that you give to the elderly who, for forty-four years have suffered so much from the abasement of the fatherland.

The 44 years refers to the Franco-Prussian war of 1870 in which Bourbaki (the general) played a dramatic role.

The next cycle of militant patriotism occurred in the years leading up to the second world war. Here, Andre Weil’s experiences mirror those of Clancy. He tried several times to escape, first from military action (although he too was a reserve officer in the French army), then from France itself. He was captured in Finland, brought back to France to face trial and imprisonment, was released on the condition that he did active military duty, escaped with the French army to England, there demobilised he refused to join de Gaulle’s troops, left England on a boat to Marseille, from where he escaped to the US.

All this, and much more, you can read in his autobiography The Apprenticeship of a Mathematician, especially Chapter VI, The War and I: A Comic Opera in Six Acts.



(for TØP-ers, note the Bishop-red cover…)

Comic or not, the book tries to ‘explain’ his actions in those years, but failed to convince the French from offering him a professorship at a French university after the war.

Perhaps it may be worth looking into a comparison between Weil’s autobiography and the collected Clancy letters.

I guess that’s the best I can do to explain the use of that Weil photo by TØP. Surely they didn’t search any deeper as to where and when this picture was taken, or who the girl was next to Weil.

In case anyone might be interested, I’ll be happy to explain my own theory about this in another post.

I’m sure the full photograph ended up in the ‘Trench-bible’, given to the director of their clip-movies. The scenery is used at the end of the Jumpsuit video when ‘Clancy’ takes out a jumpsuit from the burning car and walks away along a road very similar to that in the photo.



The boy/girl shielding his/her eyes for the violence, should have been used at about minute one into the Outside video



Now, there’s another Weil (or rather Bourbaki) photograph we know did inspire Twenty One Pilots, the classic picture at the Dieulefit/Beauvallon 1938 Bourbaki-congress



which was photoshopped in order to get Szolem Mandelbrojt in from the Chancay (quite similar to Clancy now that i type this) 1937 Bourbaki congress



Now, these were the only two Bourbaki-meetings Simone Weil (Andre’s sister) attended, and she features prominently in both pictures.

Probably this brother/sister thing struct a chord with Twenty One Pilots. But then, you quickly end up with this iconic picture of both of them, taken in the summer of 1922, just before Andre entered the Ecole Normale (he entered the ENS at age 16…)



I’d love to be send a copy of the ‘Trench bible’ because I’m fairly certain also this photograph is in it. At the end of the Nico and the niners-video you see this boy and girl (who may be around age 9 and discover the truth about Dema) finding a jumpsuit with the Bishops approaching



and they reappear a bit older at the end of the Outside-video, with a burning Dema in the background.



Previous in the Bourbaki&TØP series:

Saturday, July 15th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
3:17 pm
Where’s Bourbaki’s Dema?

In this series I’m trying to figure out why the Bourbaki-group was an inspiration for the storyline of Trench, the fifth studio album by American musical duo Twenty One Pilots (or TØP).

Trench-lore centers around the city of Dema, ruled by nine Bishops enforcing Vialism (a fake religion asking people to take their own lives to glorify Dema).

It is an unfortunate coincidence that the city of Dema in the movie-clip of Nico and the Niners was inspired, and shot in Kyiv and Charkiv, Ukraine (the clip is from 2018).

I’ve just found out that Dema is situated in my country… (NATN was made in Kyiv and Charkiv, Ukraine)
by u/Sasha0503 in twentyonepilots

Here’s the corresponding ‘beyond the movie’-clip, from which we learn (or rather are told) that the movie’s ‘city of Dema’ was shot in ‘the former Ukranian high-school’ of Josh Dun (he even calls it his ‘Alma Mater’), all in quotes because I don’t buy any of it, but take it as a desperate hint to identify Dema.

In the Trench movie-clips, Josh Dun is always cast as a Bandito, and last time we saw that also the Bourbaki-gang is likely to be close to the Banditos. Dema is supposed to be the Alma Mater of (at least some of) the Banditos. Hold that thought.

As for the connection between the City of Dema and the Bourbaki-group, we only have one piece of solid information:

That the Bourbaki-group named themselves after Nico=Nicolas Bourbaki clearly resonates with Twenty Øne Pilots who got their name from the 1947 play ‘All My Sons’ by American playwright Arthur Miller.

But the crucial info is: “The story of Dema happened before them”, so the story of Dema with the Bishops and Vialism happened before the Bourbaki-group. An extra piece of evidence that there is no way the Bourbaki-group are the nine Bishops of Dema.

So, what happened before the Bourbaki-group?

Mathematically, their direct predecessors were David Hilbert, Emil Artin, Emmy Noether and her boys, in short German mathematicians from the 1920’s and early 1930’s.

Several of the Bourbaki founding members studied in Germany (Weil in Gottingen in 1927, Chevalley in Hamburg in 1931 and Marburg in 1932, and Ehresmann in Gottingen in 1930).

They were inspired by Hilbert’s program (We must know, we will know), and set out to introduce the German style of writing and doing mathematics in France.

So, a first candidate (also given the Bauhaus-like architecture of Dema) might be ‘German Mathematicians’, or in German, DEutscher MAthematiker = DEMA.

But one can hardly argue that there was a self-destructive attitude (like Vialism) present among that group, quite the opposite.

Still, one can ask why German mathematics was that strong in the 1920’s, compared to the French. France and Germany took different approaches with their intelligentsia during WW1: while Germany protected its young students and scientists, France instead committed them to the front, owing to the French culture of egalitarianism.

Remember that the album is called Trench, and the dirtiest trench-war in all of human history was WW1. Hold that thought.

But, how does this help us in identifying Dema.

A few months before the release of Trench, a website was launched containing letters (from a character named Clancy) and some photos (including part of a photo of Andre Weil). That website’s URL still is dmaorg.info.

On the rear of the boat in the movie-clip Saturday, we see ‘030904 DMA ORG’ (the 030904 is simplistic code for CID, believed to mean ‘Clancy Is Dead’).

Compare this to the official email address of the ‘Association des collaborateurs de Nicolas Bourbaki’ which is: bourbaki@dma.ens.fr.

Here, ‘dma’ stands for ‘Département de mathématiques et applications’, one of the fifteen departments of the ENS, the Ecole Normal Superieure in Paris, Rue d’Ulm.

Remember the inspirational, photoshopped photo of the Bourbaki 1938 congress in Dieulefit/Deauvallon:




All seven people in the original picture are ‘normaliens’, that is, their ‘Alma Mater’ is the Ecole Normale Superieure’. All but Simone Weil graduated from the DEpartement de MAthematiques=DEMA, as DMA was called then.

Whence the hypothesis: Bourbaki’s Dema = ENS before and during WW1

It is a conglomerate of buildings and its central courtyard forms a kind of secular cloister around its basin. This space is called “la Cour aux Ernests” in reference to a former director, Ernest Bersot. He had placed red (!) fish (=the Ernests in ENS-slang) in the basin, which have become one of the symbols of the school.

More important to us is that the pond of the Ernests is reached by crossing the “aquarium”, where the ENS war memorial is located, commemorating the 239 (former) ENS-students killed in WW1 on a tatal of about 1400 of them drafted…

From the Wikipedia-page on Nicolas Bourbaki:

“The deaths of ENS students resulted in a lost generation in the French mathematical community; the estimated proportion of ENS mathematics students (and French students generally) who died in the war ranges from one-quarter to one-half, depending on the intervals of time (c. 1900–1918, especially 1910–1916) and populations considered.”

A chilling, and very detailed account of the circumstances that lead to the deaths of about 50% of ENS-student from the period 1910-1913 can be found in the paper Pourquoi les normaliens sont-ils morts en masse en 1914-1918 ? Une explication structurale (Why the normaliens suffered mass-death in 1914-18, a structural explanation) by Nicolas (!) Mariot. Here’s the abstract:

“The École Normale Supérieure d’Ulm is always mentioned when historians summarize the ravages of World War I in France: the institution embodies the commitment of intellectuals at the front. The article offers an interpretation of mortality rates of the School which allows to understand why it is primarily students during schooling (Classes 1910-1913) that are heavily affected. Rather than basing the interpretation on the single assumption of sacrifice, it puts forward arguments pertaining to the history of the school in the immediate pre-war, including the institution of military training after the reform service in 1905, competition with the École Polytechnique to retain the best scientific students, and finally the forced commitment of the ENS students in the infantry.”

Crucial in this is the role of Ernest Lavisse who was the director of the ENS from 1903 till 1919.



Before, normaliens had to serve 12 months in the army, just like all other students. In 1905 the law changed, and under Lavisse’s influence ENS-students were given a heavy military training. From the paper:

“From now on, normalien students, like those of other major military schools, are subject to a two-year service: a first year before their actual entry into rue d’Ulm, which they had to perform in an infantry regiment; a second on leaving, which they can finish as a reserve second lieutenant, but always in the infantry, if they pass the tests. And there is more: because between these two years, the students also follow a fairly heavy military preparation, including theoretical and physical exercises, even on Sundays, organised by two officers seconded full-time for this mission within the walls of the School.”

He also instilled in the ENS-student a radical sense of patriotism, and was a fervent propagandist for l’Union sacrée. From the paper:

“Intellectual mobilisation crystallised in the figures of Lavisse and Durkheim via their famous Letters to all French people distributed in millions of copies across the country. In the fall of 1914, the two founded and took control, respectively as director and secretary of the Committee for Studies and Documents on the War, a propaganda organ for the country’s executives.

Alongside them, eight of the nine other members of the Committee are former students of the School who have become professors at the Sorbonne (Charles Andler, Charles Seignobos, Émile Boutroux, Ernest Denis, Gustave Lanson) or at the Collège de France (Joseph Bédier, Henri Bergson, Jacques Hadamard), without even mentioning the role of editorial secretary held by Lucien Herr, legendary counterpart of Paul Dupuy at the Rue d’Ulm library.”

I would have liked that there were only nine members of the committee, but there were eleven of them…

Anyway, Lavisse and his eight professors created an extremely patriotic environment at the ENS during WW1, encouraging students to go to (the) Trench(es) and give their life for France and the glory of the Ecole. The ENS-monument is the equivalent of the Neon Gravestones in Dema-lore.

Did you spot it too? LAVIsse is almost a perfect anagram for VIALism.

Se non e vero…

Concluding, the best theory I can come up with in order to include the Bourbaki-group in Dema-lore is that their Dema is the ENS in WW1 and preceding years, and that Vialism is the regime installed by Lavisse and the other members from the committee.

Previous in the Bourbaki&TØP series:

Wednesday, July 5th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
3:58 pm
Bourbaki = Bishops or Banditos?

In this series I’m trying to make sense of the inclusion of Nicolas Bourbaki in the storyline of Trench, the fifth studio album by American musical duo Twenty One Pilots (or TØP).

That story is about the walled city of Dema, ruled by nine bishops dressed in red cloaks. They enforce their religion, called Vialism, on the inhabitants of Dema. The end-goal in Vialism is that you take your own life, having maximal impact (think: suicide bombers).



Outside the walls of Dema is the land of Trench, where a group of fighters (the Banditos) set up camp. They are dressed in grey trenchcoats adorned with yellow ribbons, and their goal is to help people escape from Dema, via its east side.

What on earth has the Bourbaki group to do with any of this?

We know from a tweet by Tyler Joseph that one Bourbaki-photograph (or at least a PhotoShopped version of it) played an important role in the creative process, as he used it as the background image on his Mac while producing the album.

It can also be seen, blurred in the background, in the opening seconds of his interview with Zane Lowe on Apple Music:

Here are the three versions: the original photo, overlayed with the tweet-image, and a screenshot from the interview:




In the original photo, taken at the 1938 Dieulefit Bourbaki congress, we see seven people, from left to right

In the overlayed image an additional eight person, a bearded man, is PhotoShopped in the doorway, and in the screenshot we see that apart from this mystery man the empty chair in the lower left corner is replaced by a ninth person.

Common belief among the ‘clique’ (TØP’s fanbase name) is that the nine people in the desktop image are the nine bishops of Dema, with the bearded man being Nico (the head-bishop), Andre Weil bishop Andre, and Simone Weil bishop Sacarver (the only known female bishop).

But who are these two additional persons?

In a remarkable tour de force, Reddit user ‘banditosleepers’ was able to find the man sitting in the lower left corner (though he identified him wrongly)

I found one of the people shopped into Tyler’s desktop (2018)
by u/banditosleepers in twentyonepilots

There’s this other famous pre-WW2 photo of a Bourbaki congress, this time the 1937 congress in Chancay:

Here we see, from left to right:

That is, the ninth person, sitting in the left lower corner of the screenshot is Szolem Mandelbrojt (and not Claude Chevalley, as claimed in the Reddit-post).

There’s another photo, taken at about the same moment, showing that also Jean Dieudonne (sitting on the bank in front next to Andre Weil) and Charles Ehresmann (sitting on the bank on the right, next to Jean Delsarte and Claude Chevalley) were present.

Several people (myself included) have wasted too many hours trying to identify the bearded man in the doorway, starting from the assumption that he too might have some connection with Bourbaki (either the group or the general), without success.

There’s one problem with the ‘bearded man = Nico’ hypothesis. Nico is supposed to be the tallest of the bishops, and bearded man’s head is level with Andre Weil’s head, who stand on the first step, so bearded man must at least be 15cm smaller than Andre, who was already rather short.

Here’s a theory.

The original Dieulefit 1938 photo is the first one on the Wikipedia page for Nicolas Bourbaki, and if you click on it you get a link to one of my own blogposts Bourbaki and the miracle of silence.

In that post (from 2010) I found the exact location where that photo was taken: the Ecole de Beauvallon, founded in 1929 by Marguerite Soubeyran and Catherine Krafft, which was the first ‘modern’ boarding school in France for both boys and girls having behavioural problems. From 1936 on the school’s director was Simone Monnier.

The lower picture is taken at the Ecole de Beauvallon in 1943, the woman in the middle is Marguerite Soubeyran.

From 1936 on, about 20 refugees from the Spanish civil war found a home at the school, and in the ‘pension’, next to the school. When WW2 started, about 1500 people were hidden from the German occupation in Dieulefit (having a total population of 3500) : Jewish children, intellectuals, artists, trade union leaders, many in the Ecole and the Pension. None were betrayed to the Germans, and this is called Le miracle de silence à Dieulefit.



Given this historical context, there is only one possible way to include this Bourbaki-photo in the Trench-storyline: the Ecole de Beauvallon is the equivalent of the Bandito-camp, and the Bourbakis are Banditos (or at the very least, refugees having found a safe place in the camp).

In hindsight this was already given away by Tyler Joseph in that he gave the photo the color yellow, specifically Bandito-yellow 0xFCE300, which the bishops are unable to see (they see it as grey).

How does this help in identifying the bearded man in the doorway?

Standing in the doorway, he’s the one feeling at home there, gathering around him his guests or fellow fighters. My conjecture is that he is the leader of the bandito-camp, as seen at 2.12 in the movieclip of Levitate.

Granted, it is not a perfect match. I think ‘bearded man’ is how Tyler Joseph envisioned the camp-leader, and included his picture in the ‘Trench Bible’, the 60 page booklet given to the movie-director, who did the casting some months later.



Next time we will try to find City of Dema, or at least what it should be if you take the inclusion of Bourbaki in the Trench storyline seriously.

Previous in the Bourbaki&TØP series:

Thursday, June 22nd, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
7:28 pm
Bourbaki and TØP : East is up

Somehow I missed all the excitement, five years ago. From Bourbaki’s Wikipedia page.

In 2018, the American musical duo Twenty One Pilots released a concept album named Trench. The album’s conceptual framework was the mythical city of “Dema” ruled by nine “bishops”; one of the bishops was named “Nico”, short for Nicolas Bourbaki. Another of the bishops was named Andre, which may refer to André Weil. Following the album’s release, there was a spike in internet searches for “Nicolas Bourbaki”.

Google Trends for Nicolas Bourbaki
by u/HiLlBiLlYjOeL_ in twentyonepilots

With summer and retirement coming up, I’m all in for another Bourbaki riddle.

So, what’s going on?

Tyler Joseph and Josh Dun have weaved complicated storylines around the different albums of their band Twenty One Pilots (or TØP for short), each referring to a distinct era (the Blurryface-era, the Trench-era, the Scaled and Icy-ra, etc.), each having a different color scheme, characters and so on.

You can easily get lost forever in their sub-Reddit, or the numerous YouTube-clips and blogposts made by the ‘clique’ (as their fanbase calls itself). Perhaps the quickest intro in the TØP-world is this site.

The Bourbaki-group is important to the Trench-era, yet there are very few direct references in the songs. There’s the song “Morph” (lol!) containing:

He’ll always try to stop me, that Nicolas Bourbaki
He’s got no friends close, but those who know him most know
He goes by Nico
He told me I’m a copy
When I’d hear him mock me, that’s almost stopped me

So Nico=Nicolas Bourbaki, and there’s the song “Nico and the niners”

starting off with:

East is up
I’m fearless when I hear this on the low
East is up
I’m careless when I wear my rebel clothes
East is up
When Bishops come together they will know that
Dema don’t control us, Dema don’t control
East is up

and that’s about it.

We’ll cover Dema and the Bishops in later posts, but for now remember the mantra “East is up”, which supposedly indicates the direction of escape from the Bishops and the city of Dema.

A few months before the release of Trench, a mysterious website appeared, containing letters from someone called ‘Clancy’ and some pictures and gifs. One of these pictures was soon found out to be part of an iconic photo of Andre Weil.




This caught the attention of the ‘clique’ because another picture indicated that the name of one of the Bishops was Andre.

Poor Andre was credited for just two things he managed to do : he founded a secret group of mathematicians, called Nicolas Bourbaki (important because another Bishop’s name was ‘Nico’) and he invented the symbol $\emptyset$ for the empty set (important because TØP used it since the Blurryface-era). I guess most mathematicians will remember Andre Weil for other things.

The clique-consensus seems to be that the girl next to Andre (some even thought it was a boy) is his daughter Sylvie Weil.

If you ever read her novel Chez les Weil you’ll remember that Sylvie did have from a very young age the same exuberant hairstyle as her aunt Simone Weil. So no, she’s definitely not Sylvie.

I’ll save my theory as to where and when this photo was taken, who the girl next to Andre is, and how this picture was used later on in TØP-iconography, for another post.

For now, I just want to point out one tiny detail: the girl is shielding her eyes from the blistering summer-sun, and shadows are falling from right to left.

Got it? Yes: East is up!

The Bourbaki-hype intensified when Tyler Joseph tweeded on August 19th, indicating that a new Album called ‘Trench’ was coming up:

Again, there’s a lot more to say about this tweet, but for now look at the desktop-image. It’s part of one of the most known Bourbaki images of all time (also featuring on their Wikipedia page): the Dieulefit 1938-congress (which we discovered to be taken at Beauvallon).

(Left to right: Simone Weil, C. Pisot, Andre Weil, Jean Dieudonne (seated), Claude Chabauty, Charles Ehresmann and Jean Delsarte)

Ah, you spotted it too? We’ll come back to this, and the clique made even more surprising discoveries wrt this picture.

You see, we’ll have a lot of ground to cover, so let’s stick to the “East is up” motto,for now.

Via Google maps you can check that the exit-door in the picture is located to the East side of the main building of the Ecole de Beauvallon.

All Bourbakie-congress followers are outside, so does this mean they’ve escaped Dema? Are they now Banditos (whence the Yellow-background-color)?

If you’ve never heard about Banditos or te relevance of the color Yellow, we’ll cover that too.

There’s another ‘East is up’-side to the Beauvallon-story. For this we have to recall some of the history of the spiritual father of the B-gang, General Charles-Denis Bourbaki.



In the Franco-Prussian war of 1870-71, he was given the command of ‘armee de l’Est (yes, the ‘East’-army!), a ramshackle of ill-trained men.

After some initial successes they suffered defeat in the battle of Lizaine and were forced to escape to Switzerland (Yes: East!) where they were disarmed, and treated for their injuries (this was one of the first cases of the International Red Cross, and is remembered in the Bourbaki Panorama in Luzern).

For clique-people: Red cross & Red architecture of the Boubaki-panorama = color of the Bishops.

Anyway, the important fact is that General Bourbaki had to escape to the East.

During the Bourbaki-congress in Beauvallon in 1938 a similar situation occurred. From Andre Weil’s The Apprenticeship of a Mathematician (page 123-124):

In 1938, Bourbaki held a congress in Dieulefit, where Chabauty, who had joined the ranks of the Master’s collaborators, had familie ties. Elie Cartan graciously joined us and took part in some of our discussions.

This was precisely the time of the Munich conference. There were sinister forebodings in the air. We devoured the newspapers and huddled over the radio: this was one Bourbaki congress where hardly any real work was accomplished.

By the time I had resolved that, if war broke out, I would refuse to serve. In the middle of the congress, after confiding in Delsarte, I thought up some pretext or other and left for Switzerland.

But the immediate threat of war soon seemed to have dissipated, so I returned after two days.

So, there’s a remarkable analogy between General Bourbaki’s escape to the East in 1871, and Andre and Simone Weil’s flight to the East at the time of the Munich agreement.

Clearly, the ‘East is up’ mantra is not the only reason why Tyler Joseph used these two Bourbaki-related photos in his narrative, but it illustrates that none of these choices is arbitrary.

I think Tyler knows a lot about Bourbaki. His knowledge about them goes certainly deeper than that of the average clique-member (who state that Bourbaki was a group of mathematicians trying to prove God’s existence, or that there where exactly nine Bourbaki founders, corresponding to the nine Bishops of Dema).

But then, TØP never corrects erroneous clique-statements, every fan-theory is correct to them. In fact, they see the interactions with their fanbase as a collective work in storytelling: they pose a riddle, the clique proposes various possible solutions, and afterwards they may use one of these proposals in their further work.

Here’s an interview making clear that Tyler knows a lot more about Bourbaki than most people (1.50 till 4.40)

Interviewer: “How far are you into a Wikipedia wormhole when you come across this? (the Bourbaki group)”
Tyler: “No, no THEY named their group after Nicolas Bourbaki”
Interviewer: “but there is no Nicolas Bourbaki, right?”
Tyler: They named their group after Blurryface.”
Interviewer: “Even though it was the 1930″s?”
Tyler: “Yeah.”
Interviewer: “So how does it relate?”
Tyler: “its EVERYTHING and at the same time, has nothing to do with it”
Interviewer: “See I’m no good at math, this is difficult for me.”
Tyler: “Math?! *laughs* Math has nothing to do with it… and yet it has everything to do with it.”

Okay, in the next couple of posts I’ll use the little I know about the Bourbaki group trying to make sense of the Trench-era narrative.

Thursday, May 4th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
2:12 pm
Grothendieck’s gribouillis (5)

After the death of Grothendieck in November 2014, about 30.000 pages of his writings were found in Lasserre.



Since then I’ve been trying to follow what happened to them:

So, what’s new?

In December last year, there was the official opening of the Istituto Grothendieck in the little town of Mondovi in Northern Italy.The videos of the talks given at that meeting are now online.

The Institute houses two centres, the Centre for topos theory and its applications with mission statement:

The Centre for Topoi Theory and its Applications carries out highly innovative research in the field of Grothendieck’s topos theory, oriented towards the development of the unifying role of the concept of topos across different areas of mathematics.

Particularly relevant to these aims is the theory of topos-theoretic ‘bridges’ of Olivia Caramello, coordinator of the Centre and principal investigator of the multi-year project “Topos theory and its applications”.

and the Centre for Grothendiecian studies with mission:

The Centre for Grothendiecian Studies is dedicated to honoring the memory of Alexander Grothendieck through extensive work to valorize his work and disseminate his ideas to the general public.

In particular, the Centre aims to carry out historical/philosophical and editorial work to promote the publication of the unpublished works of A. Grothendieck, as well as to promote the production of translations of already published works in various languages.

No comment on the first. You can look up the Institute’s Governance page, contemplate recent IHES-events, and conjure up your own story.

More interesting is the Centre of Grothendiec(k)ian studies. Here’s the YouTube-clip of the statement made by Johanna Grothendieck (daughter of) at the opening.

She hopes for two things: to find money and interested persons to decrypt and digitalise Grothendieck’s Lasserre gribouillis, and to initiate the re-edition of the complete mathematical works of Grothendieck.

So far, Grothendieck’s family was withholding access to the Lasserre writings. Now they seem to grant access to the Istituto Grothendieck and authorise it to digitalise the 30.000 pages.

Further good news is that a few weeks ago Mateo Carmona was appointed as coordinator of the Centre of grothendieckian studies.



You may know Mateo from his Grothendieck Github Archive. A warning note on that page states: “This site no longer updates (since Feb. 2023) and has been archived. Please visit [Instituto Grothendieck] or write to Mateo Carmona at mateo.carmona@csg.igrothendieck.org”. So probably the site will be transferred to the Istituto.

Mateo Carmona says:

As Coordinator of the CSG, I will work tirelessly to ensure that the Centre provides comprehensive resources for scholars, students, and enthusiasts interested in Grothendieck’s original works and modern scholarship. I look forward to using my expertise to coordinate and supervise the work of the international group of researchers and volunteers who will promote Grothendieck’s scientific and cultural heritage through the CSG.

It looks as if Grothendieck’s gribouillis are in good hands, at last.

Friday, April 28th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
2:09 pm
A newish toy in town

In a recent post I recalled Claude Levy-Strauss’ observation “In Paris, intellectuals need a new toy every 15 years”, and gave a couple of links showing that the most recent IHES-toy has been spreading to other Parisian intellectual circles in recent years.

At the time (late sixties), Levy-Strauss was criticising the ongoing Foucault-hype. It appears that, since then, the frequency of a hype cycle is getting substantially shorter.

Ten days ago, the IHES announced that Dustin Clausen (of condensed math fame) is now joining the IHES as a permanent professor.

To me, this seems like a sensible decision, moving away from (too?) general topos theory towards explicit examples having potential applications to arithmetic geometry.

On the relation between condensed sets and toposes, here’s Dustin Clausen talking about “Toposes generated by compact projectives, and the example of condensed sets”, at the “Toposes online” conference, organised by Alain Connes, Olivia Caramello and Laurent Lafforgue in 2021.

Two days ago, Clausen gave another interesting (inaugural?) talk at the IHES on “A Conjectural Reciprocity Law for Realizations of Motives”.

Thursday, April 20th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
9:37 am
The forests of the unconscious

We start from a large data-set $V=\{ k,l,m,n,\dots \}$ (texts, events, DNA-samples, …) with a suitable distance-function ($d(m,n) \geq 0~d(k,l)+d(l,m) \geq d(k.m)$) which measures the (dis)similarity between individual samples.

We’re after a set of unknown events $\{ p,q,r,s,\dots \}$ to explain the distances between the observed data. An example: let’s assume we’ve sequenced the DNA of a set of species, and computed a Hamming-like distance to measures the differences between these sequences.



(From Geometry of the space of phylogenetic trees by Billera, Holmes and Vogtmann)

Biology explains these differences from the fact that certain species may have had more recent common ancestors than others. Ideally, the measured distances between DNA-samples are a tree metric. That is, if we can determine the full ancestor-tree of these species, there should be numbers between ancestor-nodes (measuring their difference in DNA) such that the distance between two existing species is the sum of distances over the edges of the unique path in this phylogenetic tree connecting the two species.

Last time we’ve see that a necessary and sufficient condition for a tree-metric is that for every quadruple $k,l,m,n \in V$ we have that the maximum of the sum-distances

$$\{ d(k,l)+d(m,n),~d(k,m)+d(l,n),~d(k,n)+d(l,m) \}$$

is attained at least twice.

In practice, it rarely happens that the measured distances between DNA-samples are a perfect fit to this condition, but still we would like to compute the most probable phylogenetic tree. In the above example, there will be two such likely trees:



(From Geometry of the space of phylogenetic trees by Billera, Holmes and Vogtmann)

How can we find them? And, if the distances in our data-set do not have such a direct biological explanation, is it still possible to find such trees of events (or perhaps, a forest of event-trees) explaining our distance function?

Well, tracking back these ancestor nodes looks a lot like trying to construct colimits.

By now, every child knows that if their toy category $T$ does not allow them to construct all colimits, they can always beg for an upgrade to the presheaf topos $\widehat{T}$ of all contravariant functors from $T$ to $Sets$.

But then, the child can cobble together too many crazy constructions, and the parents have to call in the Grothendieck police who will impose one of their topologies to keep things under control.

Can we fall back on this standard topos philosophy in order to find these forests of the unconscious?



(Image credit)

We have a data-set $V$ with a distance function $d$, and it is fashionable to call this setting a $[0,\infty]$-‘enriched’ category. This is a misnomer, there’s not much ‘category’ in a $[0,\infty]$-enriched category. The only way to define an underlying category from it is to turn $V$ into a poset via $n \geq m$ iff $d(n,m)=0$.

Still, we can define the set $\widehat{V}$ of $[0,\infty]$-enriched presheaves, consisting of all maps
$$p : V \rightarrow [0,\infty] \quad \text{satisfying} \quad \forall m,n \in V : d(m,n)+p(n) \geq p(m)$$
which is again a $[0,\infty]$-enriched category with distance function
$$\hat{d}(p,q) = \underset{m \in V}{max} (q(m) \overset{.}{-} p(m)) \quad \text{with} \quad a \overset{.}{-} b = max(a-b,0)$$
so $\widehat{V}$ is a poset via $p \geq q$ iff $\forall m \in V : p(m) \geq q(m)$.

The good news is that $\widehat{V}$ contains all limits and colimits (because $[0,\infty]$ has sup’s and inf’s) and that $V$ embeds isometrically in $\widehat{V}$ via the Yoneda-map
$$m \mapsto y_m \quad \text{with} \quad y_m(n)=d(n,m)$$
The mental picture of a $[0,\infty]$-enriched presheaf $p$ is that of an additional ‘point’ with $p(m)$ the distance from $y_m$ to $p$.

But there’s hardly a subobject classifier to speak of, and so no Grothendieck topologies nor internal logic. So, how can we select from the abundance of enriched presheaves, the nodes of our event-forest?

We can look for special properties of the ancestor-nodes in a phylogenetic tree.



For any ancestor node $p$ and any $m \in V$ there is a unique branch from $p$ having $m$ as a leaf (picture above,left). Take another branch in $p$ and a leaf vertex $n$ of it, then the combination of these two paths gives the unique path from $m$ to $n$ in the phylogenetic tree, and therefore
$$\hat{d}(y_m,y_n) = d(m,n) = p(m)+p(n) = \hat{d}(p,y_m) + \hat{d}(p,y_n)$$
In other words, for every $m \in V$ there is another $n \in V$ such that $p$ lies on the geodesic from $m$ to $n$ (identifying elements of $V$ with their Yoneda images in $\widehat{V}$).

Compare this to Stephen Wolfram’s belief that if we looked properly at “what ChatGPT is doing inside, we’d immediately see that ChatGPT is doing something “mathematical-physics-simple” like following geodesics”.

Even if the distance on $V$ is symmetric, the extended distance function on $\widehat{V}$ is usually far from symmetric. But here, as we’re dealing with a tree-distance, we have for all ancestor-nodes $p$ and $q$ that $\hat{d}(p,q)=\hat{d}(q,p)$ as this is just the som of the weights of the edges on the unique path from $p$ and $q$ (picture above, on the right).

Right, now let’s look at a non-tree distance function on $V$, and let’s look at those elements in $\widehat{V}$ having similar properties as the ancestor-nodes:

$$T_V = \{ p \in \widehat{V}~:~\forall n \in V~:~p(n) = \underset{m \in V}{max} (d(m,n) \overset{.}{-} p(m)) \}$$

Then again, for every $p \in T_V$ and every $n \in V$ there is an $m \in V$ such that $p$ lies on a geodesic from $n$ to $m$.

The simplest non-tree example is $V = \{ a,b,c,d \}$ with say

$$d(a,c)+d(b,d) > max(d(a,b)+d(c,d),d(a,d)+d(b,c))$$

In this case, $T_V$ was calculated by Andreas Dress in Trees, Tight Extensions of Metric Spaces, and the Cohomological Dimension of Certain Groups: A Note on Combinatorial Properties of Metric Spaces. Note that Dress writes $mn$ for $d(m,n)$.



If this were a tree-metric, $T_V$ would be the tree, but now we have a $2$-dimensional cell $T_0$ consisting of those presheaves lying on a geodesic between $a$ and $c$, and one between $b$ and $d$. Let’s denote this by $T_0 = \{ a—c,b—d \}$.

$T_V$ has eight $1$-dimensional cells, and with the same notation we have



Let’s say that $V= \{ a,b,c,d \}$ are four DNA-samples of species but failed to satisfy the tree-metric condition by an error in the measurements, how can we determine likely phylogenetic trees for them? Well, given the shape of the cell-complex $T_V$ there are four spanning trees (with root in $f_a,f_b,f_c$ or $f_d$) having the elements of $V$ as their only leaf-nodes. Which of these is most likely the ancestor-tree will depend on the precise distances.

For an arbitrary data-set $V$, the structure of $T_V$ has been studied extensively, under a variety of names such as ‘Isbell’s injective hull’, ‘tight span’ or ‘tropical convex hull’, in slightly different settings. So, in order to use results one sometimes have to intersect with some (un)bounded polyhedron.

It is known that $T_V$ is always a cell-complex with dimension of the largest cell bounded by half the number of elements of $V$. In this generality it will no longer be the case that there is a rooted spanning tree of teh complex having the elements of $V$ as its only leaves, but we can opt for the best forest of rooted trees in the $1$-skeleton having all of $V$ as their leaf-nodes. Theses are the ‘forests of the unconscious’ explaining the distance function on the data-set $V$.

Apart from the Dress-paper mentioned above, I’ve found these papers informative:

So far, we started from a data-set $V$ with a symmetric distance function, but for applications in LLMs one might want to drop that condition. In that case, Willerton proved that there is a suitable replacement of $T_V$, which is now called the ‘directed tight span’ and which coincides with the Isbell completion.

Recently, Simon Willerton gave a talk at the African Mathematical Seminar called ‘Looking at metric spaces as enriched categories’:

Willerton also posts a series(?) on this at the n-category cafe, starting with Metric spaces as enriched categories I.

(tbc?)

Previously in this series:

Wednesday, April 12th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
8:41 am
Against toposes

The French anthropologist and ethnologist Claude Levi-Strauss once observed

“In Paris, intellectuals need a new toy every 15 years.”

Some pointers to applications of their toy of choice for the past ten years:

How do Parisian mathematicians with a lifelong interest in topos theory react to this hype?

With humour!

Here’s an ‘exposé parodique’ (parodical lecture) by Stéphane Dugowson on “Contre les topos” (against toposes).

Monday, April 10th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
1:34 pm
The tropical brain-forest

If machine learning, AI, and large language models are here to stay, there’s this inevitable conclusion:


At the start of this series, the hope was to find the topos of the unconscious. Pretty soon, attention turned to the shape of languages and LLMs.

In large language models all syntactic and semantic information is encoded is huge arrays of numbers and weights. It seems unlikely that $\mathbf{Set}$-valued presheaves will be useful in machine learning, but surely Huawei will prove me wrong.

$[0,\infty]$-enriched categories (aka generalised metric spaces) and associated $[0,\infty]$-enriched presheaves may be better suited to understand existing models.

But, as with ordinary presheaves, there are just too many $[0,\infty]$-enriched ones, So, how can we weed out the irrelevant ones?

For inspiration, let’s turn to evolutionary biology and their theory of phylogenetic trees. They want to trace back common (extinguished) ancestors of existing species by studying overlaps in the DNA.



(A tree of life, based on completely sequenced genomes, from Wikipedia)

The connection between phylogenetic trees and tropical geometry is nicely explained in the paper Tropical mathematics by David Speyer and Bernd Sturmfels.

The tropical semi-ring is the set $(-\infty,\infty]$, equipped with a new addition $\oplus$ and multiplication $\odot$

$$a \oplus b = min(a,b), \quad \text{and} \quad a \odot b = a+b$$

Because tropical multiplication is ordinary addition, a tropical monomial in $n$ variables

$$\underbrace{x_1 \odot \dots \odot x_1}_{j_1} \odot \underbrace{x_2 \odot \dots \odot x_2}_{j_2} \odot \dots$$

corresponds to the linear polynomial $j_1 x_1 + j_2 x_2 + \dots \in \mathbb{Z}[x_1,\dots,x_n]$. But then, a tropical polynomial in $n$ variables

$$p(x_1,\dots,x_n)=a \odot x_1^{i_1}\dots x_n^{i_n} \oplus b \odot x_1^{j_1} \dots x_n^{j_n} \oplus \dots$$

gives the piece-wise linear function on $p : \mathbb{R}^n \rightarrow \mathbb{R}$

$$p(x_1,\dots,x_n)=min(a+i_1 x_1 + \dots + i_n x_n,b+j_1 x_1 + \dots + j_n x_n, \dots)$$

The tropical hypersurface $\mathcal{H}(p)$ then consists of all points of $v \in \mathbb{R}^n$ where $p$ is not linear, that is, the value of $p(v)$ is attained in at least two linear terms in the description of $p$.

Now, for the relation to phylogenetic trees: let’s sequence the genomes of human, mouse, rat and chicken and compute the values of a suitable (necessarily symmetric) distance function between them:



Image

From these distances we want to trace back common ancestors and their difference in DNA-profile in a consistent manner, that is, such that the distance between two nodes in the tree is the sum of the distances of the edges connecting them.

In this example, such a tree is easily found (only the weights of the two edges leaving the root can be different, with sum $0.8$):



In general, let’s sequence the genomes of $n$ species and determine their distance matrix $D=(d_{ij})_{i,j}$. Biology asserts that this distance must be a tree-distance, and those can be characterised by the condition that for all $1 \leq i,j,k,l \leq n$, among the three numbers

$$d_{ij}+d_{kl},~d_{ik}+d_{jl},~d_{il}+d_{jk}$$

the maximum is attained at least twice.

What has this to do with tropical geometry? Well, $D$ is a tree distance if and only if $-D$ is a point in the tropical Grassmannian $Gr(2,n)$.

Here’s why: let $e_{ij}=-d_{ij}$ then the above condition is that the minimum of

$$e_{ij}+e_{kl},~e_{ik}+e_{jl},~e_{il}+e_{jk}$$

is attained at least twice, or that $(e_{ij})_{i,j}$ is a point of the tropical hypersurface

$$\mathcal{H}(x_{ij} \odot x_{kl} \oplus x_{ik} \odot x_{jl} \oplus x_{il} \odot x_{jk})$$

and we recognise this as one of the defining quadratic Plucker relations of the Grassmannian $Gr(2,n)$.

More on this can be found in another paper by Speyer and Sturmfels The tropical Grassmannian, and the paper Geometry of the space of phylogenetic trees by Louis Billera, Susan Holmes and Karen Vogtmann.

What’s the connection with $[0,\infty]$-enriched presheaves?

The set of all species $V=\{ m,n,\dots \}$ , together with the distance function $d(m,n)$ between their DNA-sequences is a $[0,\infty]$-category. Recall that a $[0,\infty]$-enriched presheaf on $V$ is a function $p : V \rightarrow [0,\infty]$ satisfying for all $m,n \in V$

$$d(m,n)+p(n) \geq p(m)$$

For an ancestor node $p$ we can take for every $m \in V$ as $p(m)$ the tree distance from $p$ to $m$, so every ancestor is a $[0,\infty]$-enriched presheaf.

We also defined the distance between such $[0,\infty]$-enriched presheaves $p$ and $q$ to be

$$\hat{d}(p,q) = sup_{m \in V}~max(q(m)-p(m),0)$$

and this distance coincides with the tree distance between the nodes.

So, all ancestors nodes in a phylogenetic tree are very special $[0,\infty]$-enriched presheaves, optimal for the connection with the underlying $[0,\infty]$-enriched category (the species and their differences in genome).

We would like to garden out such exceptional $[0,\infty]$-enriched presheaves in general, but clearly the underlying distance of a generalised metric space, even when it is symmetric, is not a tree metric.

Still, there might be regions in the space where we can do the above. So, in general we might expect not one tree, but a forest of trees formed by the $[0,\infty]$-enriched presheaves, optimal for the metric we’re exploring.

If we think of the underlying $[0,\infty]$-category as the conscious manifestations, then this forest of presheaves are the underlying brain-states (or, if you want, the unconscious) leading up to these.

That’s why I like to call this mental picture the tropical brain-forest.



(Image credit)

Where’s the tropical coming from?

Well, I think that in order to pinpoint these ‘optimal’ $[0,\infty]$-enriched presheaves a tropical-like structure on these, already mentioned by Simon Willerton in Tight spans, Isbell completions and semi-tropical modules, will be relevant.

For any two $[0,\infty]$-enriched presheaves we can take $p \oplus q = p \wedge q$, and for every $s \in [0,\infty]$ we can define

$$s \odot p : V \rightarrow [0,\infty] \qquad m \mapsto max(p(m)-s,0)$$

and check that this is again a $[0,\infty]$-presheaf. The mental idea of $s \odot p$ is that of a fat point centered at $p$ with size $s$.

(tbc)

Previously in this series:

Tuesday, April 4th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
12:05 pm
Stephen Wolfram on ChatGPT

A month ago, Stephen Wolfram put out a little booklet (140 pages) What Is ChatGPT Doing … and Why Does It Work?.



It gives a gentle introduction to large language models and the architecture and training of neural networks.

The entire book is freely available:

The advantage of these online texts is that you can click on any of the images, copy their content into a Mathematica notebook, and play with the code.

This really gives a good idea of how an extremely simplified version of ChatGPT (based on GPT-2) works.

Downloading the model (within Mathematica) uses about 500Mb, but afterwards you can complete any prompt quickly, and see how the results change if you turn up the ‘temperature’.

You should’t expect too much from this model. Here’s what it came up with from the prompt “The major results obtained by non-commutative geometry include …” after 20 steps, at temperature 0.8:


NestList[StringJoin[#, model[#, {"RandomSample", "Temperature" -> 0.8}]] &,
"The major results obtained by non-commutative geometry include ", 20]

The major results obtained by non-commutative geometry include vernacular accuracy of math and arithmetic, a stable balance between simplicity and complexity and a relatively low level of violence.

Lol.

In the more philosophical sections of the book, Wolfram speculates about the secret rules of language that ChatGPT must have found if we want to explain its apparent succes. One of these rules, he argues, must be the ‘logic’ of languages:

But is there a general way to tell if a sentence is meaningful? There’s no traditional overall theory for that. But it’s something that one can think of ChatGPT as having implicitly “developed a theory for” after being trained with billions of (presumably meaningful) sentences from the web, etc.

What might this theory be like? Well, there’s one tiny corner that’s basically been known for two millennia, and that’s logic. And certainly in the syllogistic form in which Aristotle discovered it, logic is basically a way of saying that sentences that follow certain patterns are reasonable, while others are not.

Something else ChatGPT may have discovered are language’s ‘semantic laws of motion’, being able to complete sentences by following ‘geodesics’:

And, yes, this seems like a mess—and doesn’t do anything to particularly encourage the idea that one can expect to identify “mathematical-physics-like” “semantic laws of motion” by empirically studying “what ChatGPT is doing inside”. But perhaps we’re just looking at the “wrong variables” (or wrong coordinate system) and if only we looked at the right one, we’d immediately see that ChatGPT is doing something “mathematical-physics-simple” like following geodesics. But as of now, we’re not ready to “empirically decode” from its “internal behavior” what ChatGPT has “discovered” about how human language is “put together”.

So, the ‘hidden secret’ of successful large language models may very well be a combination of logic and geometry. Does this sound familiar?

If you prefer watching YouTube over reading a book, or if you want to see the examples in action, here’s a video by Stephen Wolfram. The stream starts about 10 minutes into the clip, and the whole lecture is pretty long, well over 3 hours (about as long as it takes to read What Is ChatGPT Doing … and Why Does It Work?).

Tuesday, March 28th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
1:54 pm
an einStein

On March 20th, David Smith, Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss announced on the arXiv that they’d found an ein-Stein (a stone), that is, one piece to tile the entire plane, in uncountably many different ways, all of them non-periodic (that is, the pattern does not even allow a translation symmetry).

This einStein, called the ‘hat’ (some prefer ‘t-shirt’), has a very simple form : you take the most symmetric of all plane tessellations, $\ast 632$ in Conway’s notation, and glue sixteen copies of its orbifold (or if you so prefer, eight ‘kites’) to form the gray region below:



(all images copied from the aperiodic monotile paper)

Surprisingly, you do not even need to impose gluing conditions (unlike in the two-piece aperiodic kite and dart Penrose tilings), but you’ll need flipped hats to fill up the gaps left.

A few years ago, I wrote some posts on Penrose tilings, including details on inflation and deflation, aperiodicity, uncountability, Conway worms, and more:

To prove that hats tile the plane, and do so aperiodically, the authors do not apply inflation and deflation directly on the hats, but rather on associated tilings by ‘meta-tiles’ (rough outlines of blocks of hats). To understand these meta-tiles it is best to look at a large patch of hats:



Here, the dark-blue hats are the ‘flipped’ ones, and the thickened outline around the central one gives the boundary of the ’empire’ of a flipped hat, that is, the collection of all forced tiles around it. So, around each flipped hat we find such an empire, possibly with different orientation. Also note that most of the white hats (there are also isolated white hats at the centers of triangles of dark-blue hats) make up ‘lines’ similar to the Conway worms in the case of the Penrose tilings. We can break up these ‘worms’ into ‘propeller-blades’ (gray) and ‘parallelograms’ (white). This gives us four types of blocks, the ‘meta-tiles’:



The empire of a flipped hat consists of an H-block (for Hexagon) made of one dark-blue (flipped) and three light-blue (ordinary) hats, one P-block (for Parallelogram), one F-block (for Fylfot, a propellor blade), and one T-block (for Triangle) for the remaining hat.



The H,T and P blocks have rotational symmetries, whereas the underlying block of hats does not. So we mark the intended orientation of the hats by an arrow, pointing to the side having two or three hat-pieces sticking out.

Any hat-tiling gives us a tiling with the meta-tile pieces H,T,P and F. Conversely, not every tiling by meta-tiles has an underlying hat-tiling, so we have to impose gluing conditions on the H,T,P and F-pieces. We can do this by using the boundary of the underlying hat-block, cutting away and adding hat-parts. Then, any H,T,P and F-tiling satisfying these gluing conditions will come from an underlying hat-tiling.

The idea is now to devise ‘inflation’- and ‘deflation’-rules for the H,T,P and F-pieces. For ‘inflation’ start from a tiling satisfying the gluing (and orientation) conditions, and look for the central points of the propellors (the thick red points in the middle picture).



These points will determine the shape of the larger H,T,P and F-pieces, together with their orientations. The authors provide an applet to see these inflations in action.

Choose your meta-tile (H,T,P or F), then click on ‘Build Supertiles’ a number of times to get larger and larger tilings, and finally unmark the ‘Draw Supertiles’ button to get a hat-tiling.

For ‘deflation’ we can cut up H,T,P and F-pieces into smaller ones as in the pictures below:



Clearly, the hard part is to verify that these ‘inflated’ and ‘deflated’ tilings still satisfy the gluing conditions, so that they will have an underlying hat-tiling with larger (resp. smaller) hats.

This calls for a lengthy case-by-case analysis which is the core-part of the paper and depends on computer-verification.

Once this is verified, aperiodicity follows as in the case of Penrose tilings. Suppose a tiling is preserved under translation by a vector $\vec{v}$. As ‘inflation’ and ‘deflation’ only depend on the direct vicinity of a tile, translation by $\vec{v}$ is also a symmetry of the inflated tiling. Now, iterate this process until the diameter of the large tiles becomes larger than the length of $\vec{v}$ to obtain a contradiction.

Siobhan Roberts wrote a fine article Elusive ‘Einstein’ Solves a Longstanding Math Problem for the NY-times on this einStein.

It would be nice to try this strategy on other symmetric tilings: break the symmetry by gluing together a small number of its orbifolds in such a way that this extended tile (possibly with its reversed image) tile the plane, and find out whether you discovered a new einStein!

Thursday, March 23rd, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
2:44 pm
The super-vault of missing notes

Last time we’ve constructed a wide variety of Jaccard-like distance functions $d(m,n)$ on the set of all notes in our vault $V = \{ k,l,m,n,\dots \}$. That is, $d(m,n) \geq 0$ and for each triple of notes we have a triangle inequality

$$d(k,l)+d(l,m) \geq d(k,m)$$

By construction we had $d(m,n)=d(n,m)$, but we can modify any of these distances by setting $d'(m,n)= \infty$ if there is no path of internal links from note $m$ to note $n$, and $d'(m,n)=d(m,n)$ otherwise. This new generalised distance is no longer symmetric, but still satisfies the triangle inequality, and turns $V$ into a Lawvere space.

$V$ becomes an enriched category over the monoidal category $[0,\infty]=\mathbb{R}_+ \cup \{ \infty \}$ (the poset-category for the reverse ordering ($a \rightarrow b$ iff $a \geq b$) with $+$ as ‘tensor product’ and $0$ as unit). The ‘enrichment’ is the map

$$V \times V \rightarrow [0,\infty] \qquad (m,n) \mapsto d(m,n)$$

Writers (just like children) have always loved colimits. They want to morph their notes into a compelling story. Sadly, such colimits do not always exist yet in our vault category. They are among too many notes still missing from it.



(Image credit)

For ordinary categories, the way forward is to ‘upgrade’ your category to the presheaf category. In it, ‘the child can cobble together crazy constructions to his heart’s content’. For our ‘enriched’ vault $V_d$ we should look at the (enriched) category of enriched presheaves $\widehat{V_d}$. In it, the writer will find inspiration on how to cobble together her texts.

An enriched presheaf is a map $p : V \rightarrow [0,\infty]$ such that for all notes $m,n \in V$ we have

$$d(m,n) + p(n) \geq p(m)$$

Think of $p(n)$ as the distance (or similarity) of the virtual note $p$ to the existing note $n$, then this condition is just an extension of the triangle inequality. The lower the value of $p(n)$ the closer $p$ resembles $n$.

Each note $n \in V$ determines its Yoneda presheaf $y_n : V \rightarrow [0,\infty]$ by $m \mapsto d(m,n)$. By the triangle inequality this is indeed an enriched presheaf in $\widehat{V_d}$.

The set of all enriched presheaves $\widehat{V_d}$ has a lot of extra structure. It is a poset (note the reversal of ordering due to the poset structure on $[0,\infty]$)

$$p \leq q \qquad \text{iff} \qquad \forall n \in V : q(n) \leq p(n)$$

with minimal element $0 : \forall n \in V, 0(n)=\infty$, and maximal element $1 : \forall n \in V, 1(n)=0$.

It is even a lattice with $p \wedge q(n) = max(p(n),q(n))$ and $p \vee q(n)=min(p(n),q(n))$. It is easy to check that $p \wedge q$ and $p \vee q$ are again enriched presheaves.

Here’s $\widehat{V_d}$ when the vault consists of just two notes $V=\{ m,n \}$ of non-zero distance to each other (whether symmetric or not) as a subset of $[0,\infty] \times [0,\infty]$.



This vault $\widehat{V_d}$ of all missing (and existing) notes is again enriched over $[0,\infty]$ via

$$\widehat{d} : \widehat{V_d} \times \widehat{V_d} \rightarrow [0,\infty] \qquad \widehat{d}(p,q) = max(0,\underset{n \in V}{sup} (q(n)-p(n)))$$

The triangle inequality follows because the definition of $\widehat{d}(p,q)$ is equivalent to $\forall m \in V : \widehat{d}(p,q)+p(m) \geq q(m)$. Even if we start from a symmetric distance function $d$ on $V$, it is clear that this extended distance $\widehat{d}$ on $\widehat{V_d}$ is far from symmetric. The Yoneda map

$$y : V_d \rightarrow \widehat{V_d} \qquad n \mapsto y_n$$

is an isometry and the enriched version of the Yoneda lemma says that for all $p \in \widehat{V_d}$

$$p(n) = \widehat{d}(y_n,p)$$

Indeed, taking $m=n$ in $\widehat{d}(y_n,f)+y_n(m) \geq p(m)$ gives $\widehat{d}(y_n,p) \geq p(n)$. Conversely,
from the presheaf condition $d(m,n)+p(n) \geq p(m)$ for all $m,n$ follows

$$p(n) \geq max(0,\underset{m \in V}{sup}(p(m)-d(m,n)) = \widehat{d}(y_n,p)$$

In his paper Taking categories seriously, Bill Lawvere suggested to consider enriched presheaves $p \in \widehat{V_d}$ as ‘refined’ closed set of the vault-space $V_d$.

For every subset of notes $X \subset V$ we can consider the presheaf (use triangle inequality)

$$p_X : V \rightarrow [0,\infty] \qquad m \mapsto \underset{n \in X}{inf}~d(m,n)$$

then its zero set $Z(p_X) = \{ m \in V~:~p_X(m)=0 \}$ can be thought of as the closure of $X$, and the collection of all such closed subsets define a topology on $V$.

In our simple example of the two note vault $V=\{ m,n \}$ this is just the discrete topology, but we can get more interesting spaces. If $d(n,m)=0$ but $d(m,n) > 0$



we get the Sierpinski space: $n$ is the only closed point, and lies in the closure of $m$. Of course, if your vault contains thousands of notes, you might get more interesting topologies.

In the special case when $V_d$ is a poset-category, as was the case in the shape of languages post, this topology is the down-set (or up-set) topology.

Now, what is this topology when you start with the Lawvere-space $\widehat{V_d}$? From the definitions we see that

$$\widehat{d}(p,q) = 0 \quad \text{iff} \quad \forall n \in V~:~p(n) \geq q(n) \quad \text{iff} \quad p \leq q$$

So, all presheaves in the down-set $\downarrow_p$ lie in the closure of $p$, and $p$ lies in the closure of all everything in the up-set $\uparrow_p$ of $p$. So, this time the topology has as its closed sets all up-sets of the poset $\widehat{V_d}$.



What’s missing is a good definition for the implication $p \Rightarrow q$ between two enriched presheaves $p,q \in \widehat{V_d}$. In An enriched category theory of language: from syntax to semantics it is said that this should be, perhaps only to be used in their special poset situation (with adapted notations)

$$p \Rightarrow q : V \rightarrow [0,\infty] \qquad \text{where} \quad (p \Rightarrow q)(n) = \widehat{d}(y_n \wedge p,q)$$

but I can’t even show that this is a presheaf. I may be horribly wrong, but in their proof of this (lemma 5) they seem to use their lemma 4, but with the two factors swapped.

If you have suggestions, please let me know. And if you trow Kelly’s Basic concepts of enriched category theory at me, please add some guidelines on how to use it. I’m just a passer-by.

Probably, I should also read up on Isbell duality, as suggested by Lawvere in his paper Taking categories seriously, and worked out by Simon Willerton in Tight spans, Isbell completions and semi-tropical modules

(tbc)

Previously in this series:

Friday, March 17th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
11:29 am
Stella Maris (Cormac McCarthy)

This week, I was hit hard by synchronicity.

Lately, I’ve been reading up a bit on psycho-analysis, tried to get through Grothendieck’s La clef des songes (the key to dreams) and I’m in the process of writing a series of blogposts on how to construct a topos of the unconscious.

And then I read Cormac McCarthy‘s novels The passenger and Stella Maris, and got hit.



Stella Maris is set in 1972, when the math-prodigy Alicia Western, suffering from hallucinations, admits herself to a psychiatric hospital, carrying a plastic bag containing forty thousand dollars. The book consists entirely of dialogues, the transcripts of seven sessions with her psychiatrist Dr. Cohen (nomen est omen).

Alicia is a doctoral candidate at the University Of Chicago who got a scholarship to visit the IHES to work with Grothendieck on toposes.

During the psychiatric sessions, they talk on a wide variety of topics, including the nature of mathematics, quantum mechanics, music theory, dreams, and the unconscious (and its role in doing mathematics).

The core question is not how you do math but how does the unconscious do it. How it is that it’s demonstrably better at it than you are? You work on a problem and then you put it away for a while. But it doesnt go away. It reappears at lunch. Or while you’re taking a shower. It says: Take a look at this. What do you think? Then you wonder why the shower is cold. Or the soup. Is this doing math? I’m afraid it is. How is it doing it? We dont know. How does the unconscious do math? (page 99)

Before going to the IHES she had to send Grothendieck a paper (‘It was an explication of topos theory that I thought he probably hadn’t considered.’ page 136, and ‘while it proved three problems in topos theory it then set about dismantling the mechanism of the proofs.’ page 151). At the IHES ‘I met three men that I could talk to: Grothendieck, Deligne, and Oscar Zariski.’ (page 136).

I don’t know whether Zariski visited the IHES in the early 70ties, and while most historical allusions (to Grothendieck’s life, his role in Bourbaki etc.) are correct, Alicia mentions the ‘Langlands project’ (page 66) which may very well have been the talk of town at the IHES in 1972, but the mention of Witten ‘Grothendieck writes everything down. Witten nothing.’ (page 100) raised an eyebrow.

The book also contains these two nice attempts to capture some of the essence of topos theory:

When you get to topos theory you are at the edge of another universe.
You have found a place to stand where you can look back at the world from nowhere. It’s not just some gestalt. It’s fundamental. (page 13)

You asked me about Grothendieck. The topos theory he came up with is a witches’ brew of topology and algebra and mathematical logic.
It doesnt even have a clear identity. The power of the theory is still speculative. But it’s there.
You have a sense that it is waiting quietly with answers to questions that nobody has asked yet. (page 68)

I did read ‘The passenger’ first, which is probably better as then you’d know already some of the ghosts haunting Alicia, but it’s not a must if you are only interested in their discussions about the nature of mathematics. Be warned that it is a pretty dark book, better not read when you’re already feeling low, and it should come with a link to a suicide prevention line.

Here’s a more considered take on Stella Maris:

Wednesday, March 15th, 2023
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
5:20 pm
[ << Previous 20 ]

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.