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Tag: Connes

The shape of languages

In the topology of dreams we looked at Sibony’s idea to view dream-interpretations as sections in a fibered space.

The ‘points’ in the base-space and fibers consisting of chunks of text, perhaps connected by links. The topology and shape of this fibered space is still shrouded in mystery.

Let’s look at a simple approach to turn a large number of texts into a topos, and define a loose metric on it.

There’s this paper An enriched category theory of language: from syntax to semantics by Tai-Danae Bradley, John Terilla and Yiannis Vlassopoulos.

Tai-Danae Bradley is an excellent communicator of everything category related, so probably it is more fun to read her own blogposts on this paper:

or to watch her Categories for AI talk: ‘Category Theory Inspired by LLMs’:

Let’s start with a collection of notes. In the paper, they consider all possible texts written in some language, but it may be a set of webpages to train a language model, or a set of recollections by someone.

Next, shred these notes into chunks of text, and point one of these to all the texts obtained by deleting some words at the start and/or end of it. For example, the note ‘a red rose’ will point to ‘a red’, ‘red rose’, ‘a’, ‘red’ and ‘rose’ (but not to ‘a rose’).

You may call this a category, to me it is just as a poset $(\mathcal{L},\leq)$. The maximal elements are the individual words, the minimal elements are the notes, or websites, we started from.

A down-set $A$ of this poset $(\mathcal{L},\leq)$ is a subset of $\mathcal{L}$ closed under taking smaller elements, that is, if $a \in A$ and $b \leq a$, then $b \in A$.

The intersection of two down-sets is again a down-set (or empty), and the union of down-sets is again a downset. That is, down-sets define a topology on our collection of text-snippets, or if you want, on language-fragments.

For example, the open determined by the word ‘red’ is the collection of all text-fragments containing this word.

The corresponding presheaf topos $\widehat{\mathcal{L}}$ is then just the category of all (set-valued) presheaves on this topological space.
As an example, the Yoneda-presheaf $\mathcal{Y}(p)$ of a text-snippet $p$ is the contra-variant functor

$$(\mathcal{L},\leq) \rightarrow \mathbf{Sets}$$

sending any $q \leq p$ to the unique map $\ast$ from $q$ to $p$, and if $q \not\leq p$ then we map it to $\emptyset$. If $A$ is a down-set (an open of over topological space) then the sections of $\mathcal{Y}(p)$ over $A$ are $\{ \ast \}$ if for all $a \in A$ we have $a \leq p$, and $\emptyset$ otherwise.

The presheaf $\mathcal{Y}(p)$ already contains some semantic information about the snippet $p$ as it gives all contexts in which $p$ appears.

Perhaps interesting is that the ‘points’ of the topos $\widehat{\mathcal{L}}$ are the notes we started from.

Recall that Connes and Gauthier-Lafaey want to construct a topos describing someone’s unconscious, and points of that topos should be the connection with that person’s consciousness.

Suppose you want to unravel your unconscious. You start by writing down a large set of notes containing all relevant facts of your life. Then you construct from these notes the above collection of snippets and its corresponding pre-sheaf topos. Clearly, you wrote your notes consciously, but probably the exact phrasing of these notes, or recurrent themes in them, or some text-combinations are ruled by your unconscious.

Ok, it’s not much, but perhaps it’s a germ of an potential approach…

(Image credit)

Now we come to the interesting part of the paper, the ‘enrichment’ of this poset.

Surely, some of these text-snippets will occur more frequently than others. For example, in your starting notes the snippet ‘red rose’ may appear ten time more than the snippet ‘red dwarf’, but this is not visible in the poset-structure. So how can we bring in this extra information?

If we have two text-snippets $p$ and $q$ and $q \leq p$, that is, $p$ is a connected sub-string of $q$. We can compute the conditional probability $\pi(q|p)$ which tells us how likely it is that if we spot an occurrence of $p$ in our starting notes, it is part of the larger sentence $q$. These numbers can be easily computed and from the rules of probability we get that for snippets $r \leq q \leq p$ we have that

$$\pi(r|p) = \pi(r|q) \times \pi(q|r)$$

so these numbers (all between $0$ and $1$) behave multiplicative along paths in the poset.

Nice in theory, but it requires an awful lot of computation. From the paper:

The reader might think of these probabilities $\pi(q|p)$ as being most well defined when $q$ is a short extension of $p$. While one may be skeptical about assigning a probability distribution on the set of all possible texts, it’s reasonable to say there is a nonzero probability that cat food will follow I am going to the store to buy a can of and, practically speaking, that probability can be estimated.

Indeed, existing LLMs successfully learn these conditional probabilities $\pi(q|p)$ using standard machine learning tools trained on large corpora of texts, which may be viewed as providing a wealth of samples drawn from these conditional probability distributions.

It may be easier to have an estimate $\mu(q|p)$ of this conditional probability for immediate successors (that is, if $q$ is obtained from $p$ by adding one word at the beginning or end of it), and then extend this measure to all arrows in the poset by taking the maximum of products along paths. In this way we have for all $r \leq q \leq p$ that

$$\mu(r|p) \geq \mu(r|q) \times \mu(q|p)$$

The upshot is that this measure $\mu$ turns our poset (or category) $(\mathcal{L},\leq)$ into a category ‘enriched’ over the unit interval $[ 0,1 ]$ (suitably made into a monoidal category).

I’ll spare you the details, just want to flash out the corresponding notion of ‘enriched presheaves’ which are the objects of the semantic category $\widehat{\mathcal{L}}^s$ in the paper, which is the enriched version of the presheaf category $\widehat{\mathcal{L}}$.

An enriched presheaf is a function (not functor)

$$F~:~\mathcal{L} \rightarrow [0,1]$$

satisfying the condition that for all text-snippets $r,q \in \mathcal{L}$ we have that

$$\mu(r|q) \leq [F(q),F(r)] = \begin{cases} \frac{F(r)}{F(q)}~\text{if $F(r) \leq F(q)$} \\ 1~\text{otherwise} \end{cases}$$

Note that the enriched (or semantic) Yoneda presheaf $\mathcal{Y}^s(p)(q) = \mu(q|p)$ satisfies this condition, and now this data not only records the contexts in which $p$ appears, but also measures how likely it is for $p$ to appear in a certain context.

Another cute application of the condition on the measure $\mu$ is that it allows us to define a ‘distance function’ (satisfying the triangle inequality) on all text-snippets in $\mathcal{L}$ by

$$d(q,p) = \begin{cases} -ln(\mu(q|p))~\text{if $q \leq p$} \\
\infty~\text{otherwise} \end{cases}$$

So, the higher $\mu(q|p)$ the closer $q$ lies to $p$, and now the snippet $p$ (example ‘red’) not only defines the open set in $\mathcal{L}$ of all texts containing $p$, but now we can structure the snippets in this open set with respect to this ‘distance’.

In this way we can turn any language, or a collection of texts in a given language, into what Lawvere called a ‘generalized metric space’.

It looks as if we are progressing slowly in our, probably futile, attempt to understand Alain Connes’ and Patrick Gauthier-Lafaye’s claim that ‘the unconscious is structured like a topos’.

Even if we accept the fact that we can start from a collection of notes, there are a number of changes we need to make to the above approach:

  • there will be contextual links between these notes
  • we only want to retain the relevant snippets, not all of them
  • between these ‘highlights’ there may also be contextual links
  • texts can be related without having to be concatenations
  • we need to implement changes when new notes are added
  • … (much more)

Perhaps, we should try to work on a specific ‘case’, and explore all technical tools that may help us to make progress.


Previously in this series:


Loading a second brain

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The topology of dreams

Last May, the meeting Lacan et Grothendieck, l’impossible rencontre? took place in Paris (see this post). Video’s of that meeting are now available online.

Here’s the talk by Alain Connes and Patrick Gauthier-Lafaye on their book A l’ombre de Grothendieck et de Lacan : un topos sur l’inconscient ? (see this post ).

Let’s quickly recall their main ideas:

1. The unconscious is structured as a topos (Jacques Lacan argued it was structured as a language), because we need a framework allowing logic without the law of the excluded middle for Lacan’s formulas of sexuation to make some sense at all.

2. This topos may differs from person to person, so we do not all share the same rules of logic (as observed in real life).

3. Consciousness is related to the points of the topos (they are not precise on this, neither in the talk, nor the book).

4. All these individual toposes are ruled by a classifying topos, and they see Lacan’s work as the very first steps towards trying to describe the unconscious by a geometrical theory (though his formulas are not first order).

Surely these are intriguing ideas, if only we would know how to construct the topos of someone’s unconscious.

Let’s go looking for clues.

At the same meeting, there was a talk by Daniel Sibony: “Mathématiques et inconscient”

Sibony started out as mathematician, then turned to psychiatry in the early 70ties. He was acquainted with both Grothendieck and Lacan, and even brought them together once, over lunch, some day in 1973. He makes a one-line appearance in Grothendieck’s Récoltes et Semailles, when G discribes his friends in ‘Survivre et Vivre’:

“Daniel Sibony (who stayed away from this group, while pursuing its evolution out of the corner of a semi-disdainful, smirking eye)”

In his talk, Sibony said he had a similar idea, 50 years before Connes and Gauthier-Lafaye (3.04 into the clip):

“At the same time (early 70ties) I did a seminar in Vincennes, where I was a math professor, on the topology of dreams. At the time I didn’t have categories at my disposal, but I used fibered spaces instead. I showed how we could interpret dreams with a fibered space. This is consistent with the Freudian idea, except that Freud says we should take the list of words from the story of the dream and look for associations. For me, these associations were in the fibers, and these thoughts on fibers and sheaves have always followed me. And now, after 50 years I find this pretty book by Alain Connes and Patrick Gauthier-Lafaye on toposes, and see that my thoughts on dreams as sheaves and fibered spaces are but a special case of theirs.”

This looks interesting. After all, Freud called dream interpretation the ‘royal road’ to the unconscious. “It is the ‘King’s highway’ along which everyone can travel to discover the truth of unconscious processes for themselves.”

Sibony clarifies his idea in the interview L’utilisation des rêves en psychothérapie with Maryse Siksou.

“The dream brings blocks of words, of “compacted” meanings, and we question, according to the good old method, each of these blocks, each of these points and which we associate around (we “unblock” around…), we let each point unfold according to the “fiber” which is its own.

I introduced this notion of the dream as fibered space in an article in the review Scilicet in 1972, and in a seminar that I gave at the University of Vincennes in 1973 under the title “Topologie et interpretation des rêves”, to which Jacques Lacan and his close retinue attended throughout the year.

The idea is that the dream is a sheaf, a bundle of fibers, each of which is associated with a “word” of the dream; interpretation makes the fibers appear, and one can pick an element from each, which is of course “displaced” in relation to the word that “produced” the fiber, and these elements are articulated with other elements taken in other fibers, to finally create a message which, once again, does not necessarily say the meaning of the dream because a dream has as many meanings as recipients to whom it is told, but which produces a strong statement, a relevant statement, which can restart the work.”

Key images in the dream (the ‘points’ of the base-space) can stand for entirely different situations in someone’s life (the points in the ‘fiber’ over an image). The therapist’s job is to find a suitable ‘section’ in this ‘sheaf’ to further the theraphy.

It’s a bit like translating a sentence from one language to another. Every word (point of the base-space) can have several possible translations with subtle differences (the points in the fiber over the word). It’s the translator’s job to find the best ‘section’ in this sheaf of possibilities.

This translation-analogy is used by Daniel Sibony in his paper Traduire la passe:

“It therefore operates just like the dream through articulated choices, from one fiber to another, in a bundle of speaking fibers; it articulates them by seeking the optimal section. In fact, the translation takes place between two fiber bundles, each in a language, but in the starting bundle the choice seems fixed by the initial text. However, more or less consciously, the translator “bursts” each word into a larger fiber, he therefore has a bundle of fibers where the given text seems after the fact a singular choice, which will produce another choice in the bundle of the other language.”

This paper also contains a pre-ChatGPT story (we’re in 1998), in which the language model fails because it has far too few alternatives in its fibers:

I felt it during a “humor festival” where I was approached by someone (who seemed to have some humor) and who was a robot. We had a brief conversation, very acceptable, beyond the conventional witticisms and knowing sighs he uttered from time to time to complain about the lack of atmosphere, repeating that after all we are not robots.

I thought at first that it must be a walking walkie-talkie and that in fact I was talking to a guy who was remote control from his cabin. But the object was programmed; the unforeseen effects of meaning were all the more striking. To my question: “Who created you?” he answered with a strange word, a kind of technical god.

I went on to ask him who he thought created me; his answer was immediate: “Oedipus”. (He knew, having questioned me, that I was a psychoanalyst.) The piquancy of his answer pleased me (without Oedipus, at least on a first level, no analyst). These bursts of meaning that we know in children, psychotics, to whom we attribute divinatory gifts — when they only exist, save their skin, questioning us about our being to defend theirs — , these random strokes of meaning shed light on the classic aftermaths where when a tile arrives, we hook it up to other tiles from the past, it ties up the pain by chaining the meaning.

Anyway, the conversation continuing, the robot asked me to psychoanalyse him; I asked him what he was suffering from. His answer was immediate: “Oedipus”.

Disappointing and enlightening: it shows that with each “word” of the interlocutor, the robot makes correspond a signifying constellation, a fiber of elements; choosing a word in each fiber, he then articulates the whole with obvious sequence constraints: a bit of readability and a certain phrasal push that leaves open the game of exchange. And now, in the fiber concerning the “psy” field, chance or constraint had fixed him on the same word, “Oedipus”, which, by repeating itself, closed the scene heavily.

Okay, we have a first potential approximation to Connes and Gauthier-Lafaye’s elusive topos, a sheaf of possible interpretation of base-words in a language.

But, the base-space is still rather discrete, or at best linearly ordered. And also in the fibers, and among the sections, there’s not much of a topology at work.

Perhaps, we should have a look at applications of topology and/or topos theory in large language models?



The shape of languages


the topos of unconsciousness

Since wednesday, as mentioned last time, the book by Alain Connes and Patrick Gauthier-Lafaye: “A l’ombre de Grothendieck et de Lacan, un topos sur l’inconscient” is available in the better bookshops.

There’s no need to introduce Alain Connes on this blog. Patrick Gauthier-Lafaye is a French psychiatrist and psycho-analyst, working in Strassbourg.

The book is a lengthy dialogue in which the authors try to find a use for topos theory in Jaques Lacan’s psycho-analytical view of the unconscious.

If you are a complete Lacanian virgin, it may be helpful to browse through “Lacan, a beginners guide” (by Lionel Bailly) first.

If this left you bewildered, for example by Lacan’s strange (ab)use of mathematics, rest assured, you’re not alone.

It is no coincidence that Lacan’s works are the first case-study in the book “Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science” by Alan Sokal (the one of the hoax) and Jean Bricmont. You can download the book from this link.

If now you feel that Sokal and Bricmont are way too harsh on Lacan, I urge you to have a go at the book “Writing the structures of the subject, Lacan and topology” by Will Greenshields.

If you don’t have the time or energy for this, let me give you one illustrative example: the topological explanation of Lacan’s formula of fantasy:

\$~\diamond~a \]

Loosely speaking this formula says “the barred subject stands within a circular relationship to the objet petit a (the object of desire), one part of which is determined by alienation, the other by separation”.

Lacan was obsessed with the immersion of the projective plane $\mathbb{P}^2(\mathbb{R})$ into $\mathbb{R}^3$ as the cross-cap. Here’s an image of it from his 1966-67 seminar on ‘Logique du fantasme’ (213 pages).

This image includes the position of the objet petit $a$ as the end point of the self-intersection curve, which itself is referred to as the ‘castration’, or the ‘phallus’, or whatever.

Brace yourself for the ‘explanation’ of $\$~\diamond~a$: if you walk twice around $a$ this divides the cross-cap into a disk and a Mobius-strip!

The mathematics is correct but I fail to see how this helps the psycho-analyst in her therapy. But hey, everyone will tell you I have absolutely no therapeutic talent.

Let’s return to the brand new book by Alain Connes and Patrick Gauthier-Lafaye: “A l’ombre de Grothendieck et de Lacan, un topos sur l’inconscient”.

It was to be expected that they would defend Lacan’s exploitation of (surface) topology by saying that he was just unfortunate not to have the more general notion of toposes available, as well as their much subtler logic. Perhaps someone should write a fictional parody on Greenshields book: “Lacan and the topos”…

Connes’ first attempt to construct the topos of unconsciousness was also not much of a surprise. According to Lacan the unconscious is ‘structured like a language’.

So, a natural approach might be to start with a ‘dictionary’-category (words and relations between them) or any other known use of a category in linguistics. A good starting point to read up on this is the blog post A new application of category theory in linguistics.

Eventually they settled for a much more ambitious project. To Connes and Gauthier-Lafaye every individual has her own topos and corresponding logic.

They don’t specify how to construct these individual toposes, but postulate that they are all connected to a classifying topos, which is their incarnation of the world of ‘myths’ and ‘fantasies’.

Surely an idea Lacan would have liked. Underlying the unconscious must be, according to Connes and Gauthier-Lafaye, a geometric theory! That is, it can be fully described by first order sentences.

Lacan himself used already some first order sequences in his teachings, such as in his logic of sexuation:

\forall x~(\Phi~x)~\quad \text{but also} \quad \exists x~\neg~(\Phi~x) \]

where $\Phi~x$ is the phallic function. Quoting from Greenshield’s book:

“While all (the sons) are subject to ($\forall x$) the law of castration ($\Phi~x$), we also learn that this law nevertheless resides upon an exception: there exists a subject ($\exists x$) that is not subject to this law ($\neg \Phi~x$). This exception is embodied by the despotic father who, not being subject to the phallic function, experiences an impossible mode of totalised jouissance (he enjoys all the women). He is, quite simply, the exception that proves the law a necessary beyond that enables the law’s geometric bounds to be defined.”

It will be quite hard (but probably great fun for psycho-analysts) to turn the whole of Lacanian theory on the unconscious into a coherent geometric theory, construct its classifying topos, and apply the Joyal-Reyes theorem to get at the individual cases/toposes.

I’m sure there are much deeper insights to be gained from Connes’ and Gauthier-Lafaye’s book, but this is what i got from a first, fast, cursory reading of it.

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Grothendieck meets Lacan

Next month, a weekend-meeting is organised in Paris on Lacan et Grothendieck, l’impossible rencontre?.

Photo from Remembering my father, Jacques Lacan

Jacques Lacan was a French psychoanalyst and psychiatrist who has been called “the most controversial psycho-analyst since Freud”.

What’s the connection between Lacan and Grothendieck? Here’s Stephane Dugowson‘s take (G-translated):

“As we know, Lacan was passionate about certain mathematics, notably temporal logic and the theory of knots, where he thought he found material for advancing the theory of psychoanalysis. For his part, Grothendieck testifies in his non-strictly mathematical writings to his passion for the psyche, as shown by many pages of his Récoltes et Semailles just published by Gallimard (in January 2022), or even, among the tens of thousands of pages discovered at his death and of which we know almost nothing, the 3700 pages of mathematics grouped under the title ‘Structure of the Psyche’.

One might therefore be surprised that the two geniuses never met. In fact, a lunch did take place in the early 1970s organized by the mathematician and psychoanalyst Daniel Sibony. But a lunch does not necessarily make a meeting, and it seems that this one unfortunately did not happen.”

As it is ‘bon ton’ these days in Parisian circles to utter the word ‘topos’, several titles of the talks given at the meeting contain that word.

There’s Stephane Dugowson‘s talk on “Logique du topos borroméen et autres logiques à trois points”.

Lacan used the Borromean link to illustrate his concepts of the Real, Symbolic, and Imaginary (RSI). For more on this, please read chapter 6 of Lionel Baily’s excellent introduction to Lacan’s work Lacan, A Beginner’s Guide.

The Borromean topos is an example of Dugowson’s toposes associated to his ‘connectivity spaces’. From his paper Définition du topos d’un espace connectif I gather that the objects in the Borromean topos consist of a triple of set-maps from a set $A$ (the global sections) to sets $A_x,A_y$ and $A_z$ (the restrictions to three disconnected ‘opens’).

\xymatrix{& A \ar[rd] \ar[d] \ar[ld] & \\ A_x & A_y & A_z} \]

This seems to be a topos with a Boolean logic, but perhaps there are other 3-point connectivity spaces with a non-Boolean Heyting subobject classifier.

There’s Daniel Sibony‘s talk on “Mathématiques et inconscient”. Sibony is a French mathematician, turned philosopher and psychoanalyst, l’inconscient is an important concept in Lacan’s work.

Here’s a nice conversation between Daniel Sibony and Alain Connes on the notions of ‘time’ and ‘truth’.

In the second part (starting around 57.30) Connes brings up toposes whose underlying logic is much subtler than brute ‘true’ or ‘false’ statements. He discusses the presheaf topos on the additive monoid $\mathbb{N}_+$ which leads to statements which are ‘one step from the truth’, ‘two steps from the truth’ and so on. It is also the example Connes used in his talk Un topo sur les topos.

Alain Connes himself will also give a talk at the meeting, together with Patrick Gauthier-Lafaye, on “Un topos sur l’inconscient”.

It appears that Connes and Gauthier-Lafaye have written a book on the subject, A l’ombre de Grothendieck et de Lacan : un topos sur l’inconscient. Here’s the summary (G-translated):

“The authors present the relevance of the mathematical concept of topos, introduced by A. Grothendieck at the end of the 1950s, in the exploration of the structure of the unconscious.”

The book will be released on May 11th.


Grothendieck stuff

January 13th, Gallimard published Grothendieck’s text Recoltes et Semailles in a fancy box containing two books.

Here’s a G-translation of Gallimard’s blurb:

“Considered the mathematical genius of the second half of the 20th century, Alexandre Grothendieck is the author of Récoltes et semailles, a kind of “monster” of more than a thousand pages, according to his own words. The mythical typescript, which opens with a sharp criticism of the ethics of mathematicians, will take the reader into the intimate territories of a spiritual experience after having initiated him into radical ecology.

In this literary braid, several stories intertwine, “a journey to discover a past; a meditation on existence; a picture of the mores of a milieu and an era (or the picture of the insidious and implacable shift from one era to another…); an investigation (almost police at times, and at others bordering on the swashbuckling novel in the depths of the mathematical megapolis…); a vast mathematical digression (which will sow more than one…); […] a diary ; a psychology of discovery and creation; an indictment (ruthless, as it should be…), even a settling of accounts in “the beautiful mathematical world” (and without giving gifts…)”.”

All literary events, great or small, are cause for the French to fill a radio show.

January 21st, ‘Le grand entretien’ on France Inter invited Cedric Villani and Jean-Pierre Bourguignon to talk about Grothendieck’s influence on mathematics (h/t Isar Stubbe).

The embedded YouTube above starts at 12:06, when Bourguignon describes Grothendieck’s main achievements.

Clearly, he starts off with the notion of schemes which, he says, proved to be decisive in the further development of algebraic geometry. Five years ago, I guess he would have continued mentioning FLT and other striking results, impossible to prove without scheme theory.

Now, he goes on saying that Grothendieck laid the basis of topos theory (“to define it, I would need not one minute and a half but a year and a half”), which is only now showing its first applications.

Grothendieck, Bourguignon goes on, was the first to envision the true potential of this theory, which we should take very seriously according to people like Lafforgue and Connes, and which will have applications in fields far from algebraic geometry.

Topos20 is spreading rapidly among French mathematicians. We’ll have to await further results before Topos20 will become a pandemic.

Another interesting fragment starts at 16:19 and concerns Grothendieck’s gribouillis, the 50.000 pages of scribblings found in Lasserre after his death.

Bourguignon had the opportunity to see them some time ago, and when asked to describe them he tells they are in ‘caisses’ stacked in a ‘libraire’.

Here’s a picture of these crates taken by Leila Schneps in Lasserre around the time of Grothendieck’s funeral.

If you want to know what’s in these notes, and how they ended up at that place in Paris, you might want to read this and that post.

If Bourguignon had to consult these notes at the Librairie Alain Brieux, it seems that there is no progress in the negotiations with Grothendieck’s children to make them public, or at least accessible.

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Do we need the sphere spectrum?

Last time I mentioned the talk “From noncommutative geometry to the tropical geometry of the scaling site” by Alain Connes, culminating in the canonical isomorphism (last slide of the talk)

Or rather, what is actually proved in his paper with Caterina Consani BC-system, absolute cyclotomy and the quantized calculus (and which they conjectured previously to be the case in Segal’s Gamma rings and universal arithmetic), is a canonical isomorphism between the $\lambda$-rings
\mathbb{Z}[\mathbb{Q}/\mathbb{Z}] \simeq \mathbb{W}_0(\overline{\mathbb{S}}) \]
The left hand side is the integral groupring of the additive quotient-group $\mathbb{Q}/\mathbb{Z}$, or if you prefer, $\mathbb{Z}[\mathbf{\mu}_{\infty}]$ the integral groupring of the multiplicative group of all roots of unity $\mathbf{\mu}_{\infty}$.

The power maps on $\mathbf{\mu}_{\infty}$ equip $\mathbb{Z}[\mathbf{\mu}_{\infty}]$ with a $\lambda$-ring structure, that is, a family of commuting endomorphisms $\sigma_n$ with $\sigma_n(\zeta) = \zeta^n$ for all $\zeta \in \mathbf{\mu}_{\infty}$, and a family of linear maps $\rho_n$ induced by requiring for all $\zeta \in \mathbf{\mu}_{\infty}$ that
\rho_n(\zeta) = \sum_{\mu^n=\zeta} \mu \]
The maps $\sigma_n$ and $\rho_n$ are used to construct an integral version of the Bost-Connes algebra describing the Bost-Connes sytem, a quantum statistical dynamical system.

On the right hand side, $\mathbb{S}$ is the sphere spectrum (an object from stable homotopy theory) and $\overline{\mathbb{S}}$ its ‘algebraic closure’, that is, adding all abstract roots of unity.

The ring $\mathbb{W}_0(\overline{\mathbb{S}})$ is a generalisation to the world of spectra of the Almkvist-ring $\mathbb{W}_0(R)$ defined for any commutative ring $R$, constructed from pairs $(E,f)$ where $E$ is a projective $R$-module of finite rank and $f$ an $R$-endomorphism on it. Addition and multiplication are coming from direct sums and tensor products of such pairs, with zero element the pair $(0,0)$ and unit element the pair $(R,1_R)$. The ring $\mathbb{W}_0(R)$ is then the quotient-ring obtained by dividing out the ideal consisting of all zero-pairs $(E,0)$.

The ring $\mathbb{W}_0(R)$ becomes a $\lambda$-ring via the Frobenius endomorphisms $F_n$ sending a pair $(E,f)$ to the pair $(E,f^n)$, and we also have a collection of linear maps on $\mathbb{W}_0(R)$, the ‘Verschiebung’-maps which send a pair $(E,f)$ to the pair $(E^{\oplus n},F)$ with
F = \begin{bmatrix} 0 & 0 & 0 & \cdots & f \\
1 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & & \vdots \\
0 & 0 & 0 & \cdots & 1 \end{bmatrix} \]
Connes and Consani define a notion of modules and their endomorphisms for $\mathbb{S}$ and $\overline{\mathbb{S}}$, allowing them to define in a similar way the rings $\mathbb{W}_0(\mathbb{S})$ and $\mathbb{W}_0(\overline{\mathbb{S}})$, with corresponding maps $F_n$ and $V_n$. They then establish an isomorphism with $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$ such that the maps $(F_n,V_n)$ correspond to $(\sigma_n,\rho_n)$.

But, do we really have the go to spectra to achieve this?

All this reminds me of an old idea of Yuri Manin mentioned in the introduction of his paper Cyclotomy and analytic geometry over $\mathbb{F}_1$, and later elaborated in section two of his paper with Matilde Marcolli Homotopy types and geometries below $\mathbf{Spec}(\mathbb{Z})$.

Take a manifold $M$ with a diffeomorphism $f$ and consider the corresponding discrete dynamical system by iterating the diffeomorphism. In such situations it is important to investigate the periodic orbits, or the fix-points $Fix(M,f^n)$ for all $n$. If we are in a situation that the number of fixed points is finite we can package these numbers in the Artin-Mazur zeta function
\zeta_{AM}(M,f) = exp(\sum_{n=1}^{\infty} \frac{\# Fix(M,f^n)}{n}t^n) \]
and investigate the properties of this function.

To connect this type of problem to Almkvist-like rings, Manin considers the Morse-Smale dynamical systems, a structural stable diffeomorphism $f$, having a finite number of non-wandering points on a compact manifold $M$.

From Topological classification of Morse-Smale diffeomorphisms on 3-manifolds

In such a situation $f_{\ast}$ acts on homology $H_k(M,\mathbb{Z})$, which are free $\mathbb{Z}$-modules of finite rank, as a matrix $M_f$ having only roots of unity as its eigenvalues.

Manin argues that this action is similar to the action of the Frobenius on etale cohomology groups, in which case the eigenvalues are Weil numbers. That is, one might view roots of unity as Weil numbers in characteristic one.

Clearly, all relevant data $(H_k(M,\mathbb{Z}),f_{\ast})$ belongs to the $\lambda$-subring of $\mathbb{W}_0(\mathbb{Z})$ generated by all pairs $(E,f)$ such that $M_f$ is diagonalisable and all its eigenvalues are either $0$ or roots of unity.

If we denote for any ring $R$ by $\mathbb{W}_1(R)$ this $\lambda$-subring of $\mathbb{W}_0(R)$, probably one would obtain canonical isomorphisms

– between $\mathbb{W}_1(\mathbb{Z})$ and the invariant part of the integral groupring $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$ for the action of the group $Aut(\mathbb{Q}/\mathbb{Z}) = \widehat{\mathbb{Z}}^*$, and

– between $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$ and $\mathbb{W}_1(\mathbb{Z}(\mathbf{\mu}_{\infty}))$ where $\mathbb{Z}(\mathbf{\mu}_{\infty})$ is the ring obtained by adjoining to $\mathbb{Z}$ all roots of unity.

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Alain Connes on his RH-project

In recent months, my primary focus was on teaching and family matters, so I make advantage of this Christmas break to catch up with some of the things I’ve missed.

Peter Woit’s blog alerted me to the existence of the (virtual) Lake Como-conference, end of september: Unifying themes in Geometry.

In Corona times, virtual conferences seem to sprout up out of nowhere, everywhere (zero costs), giving us an inflation of YouTubeD talks. I’m always grateful to the organisers of such events to provide the slides of the talks separately, as the generic YouTubeD-talk consists merely in reading off the slides.

Allow me to point you to one of the rare exceptions to this rule.

When I downloaded the slides of Alain Connes’ talk at the conference From noncommutative geometry to the tropical geometry of the scaling site I just saw a collage of graphics from his endless stream of papers with Katia Consani, and slides I’d seen before watching several of his YouTubeD-talks in recent years.

Boy, am I glad I gave Alain 5 minutes to convince me this talk was different.

For the better part of his talk, Alain didn’t just read off the slides, but rather tried to explain the thought processes that led him and Katia to move on from the results on this slide to those on the next one.

If you’re pressed for time, perhaps you might join in at 49.34 into the talk, when he acknowledges the previous (tropical) approach ran out of steam as they were unable to define any $H^1$ properly, and how this led them to ‘absolute’ algebraic geometry, meaning over the sphere spectrum $\mathbb{S}$.

Sadly, for some reason Alain didn’t manage to get his final two slides on screen. So, in this case, the slides actually add value to the talk…

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Bourbaki and Grothendieck-Serre

This time of year I’m usually in France, or at least I was before Covid. This might explain for my recent obsession with French math YouTube interviews.

Today’s first one is about Bourbaki’s golden years, the period between WW2 and 1975. Alain Connes is trying to get some anecdotes from Jean-Pierre Serre, Pierre Cartier, and Jacques Dixmier.

If you don’t have the time to sit through the whole thing, perhaps you might have a look at the discussion on whether or not to include categories in Bourbaki (starting at 51.40 into the clip).

Here are some other time-slots (typed on a qwerty keyboard, mes excuses) with some links.

  • 8.59 : Canular stupide (mort de Bourbaki)
  • 15.45 : recrutement de Koszul
  • 17.45 : recrutement de Grothendieck
  • 26.15 : influence de Serre
  • 28.05 : importance des ultra filtres
  • 35.35 : Meyer
  • 37.20 : faisceaux
  • 51.00 : Grothendieck
  • 51.40 : des categories, Gabriel-Demazure
  • 57.50 : lemme de Serre, theoreme de Weil
  • 1.03.20 : Chevalley vs. Godement
  • 1.05.26 : retraite Dieudonne
  • 1.07.05 : retraite
  • 1.10.00 : Weil vs. Serre-Borel
  • 1.13.50 : hierarchie Bourbaki
  • 1.20.22 : categories
  • 1.21.30 : Bourbaki, une secte?
  • 1.22.15 : Grothendieck C.N.R.S. 1984

The second one is an interview conducted by Alain Connes with Jean-Pierre Serre on the Grothendieck-Serre correspondence.

Again, if you don’t have the energy to sit through it all, perhaps I can tempt you with Serre’s reaction to Connes bringing up the subject of toposes (starting at 14.36 into the clip).

  • 2.10 : 2e these de Grothendieck: des faisceaux
  • 3.50 : Grothendieck -> Bourbaki
  • 6.46 : Tohoku
  • 8.00 : categorie des diagrammes
  • 9.10 : schemas et Krull
  • 10.50 : motifs
  • 11.50 : cohomologie etale
  • 14.05 : Weil
  • 14.36 : topos
  • 16.30 : Langlands
  • 19.40 : Grothendieck, cours d’ecologie
  • 24.20 : Dwork
  • 25.45 : Riemann-Roch
  • 29.30 : influence de Serre
  • 30.50 : fin de correspondence
  • 32.05 : pourquoi?
  • 33.10 : SGA 5
  • 34.50 : methode G. vs. theorie des nombres
  • 37.00 : paranoia
  • 37.15 : Grothendieck = centrale nucleaire
  • 38.30 : Clef des songes
  • 42.35 : 30.000 pages, probleme du mal
  • 44.25 : Ribenboim
  • 45.20 : Grothendieck a Paris, publication R et S
  • 48.00 : 50 ans IHES, lettre a Bourguignon
  • 50.46 : Laurant Lafforgue
  • 51.35 : Lasserre
  • 53.10 : l’humour
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RH and the Ishango bone

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

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

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

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

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

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

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

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

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

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

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

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

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

More details are given in a footnote:

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

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

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

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

Such a paper surely deserves a separate post.

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Closing in on Gabriel’s topos?

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

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

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

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

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

The second lecture can be watched here.

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

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

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

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

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

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