Category: math

an einStein

Published March 28, 2023 by lievenlb

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!

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Leila Schneps on Grothendieck

Published August 20, 2022 by lievenlb

If you have neither the time nor energy to watch more than one interview or talk about Grothendieck’s life and mathematics, may I suggest to spare that privilege for Leila Schneps’ talk on ‘Le génie de Grothendieck’ in the ‘Thé & Sciences’ series at the Salon Nun in Paris.

I was going to add some ‘relevant’ time slots after the embedded YouTube-clip below, but I really think it is better to watch Leila’s interview in its entirety. Enjoy!

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Grothendieck meets Lacan

Published April 21, 2022 by lievenlb

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.

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Mamuth to Elephant (3)

Published March 13, 2022 by lievenlb

Until now, we’ve looked at actions of groups (such as the $T/I$ or $PLR$-group) or (transformation) monoids (such as Noll’s monoid) on special sets of musical elements, in particular the twelve pitch classes $\mathbb{Z}_{12}$, or the set of all $24$ major and minor chords.

Elephant-lovers recognise such settings as objects in the presheaf topos on the one-object category $\mathbf{M}$ corresponding to the group or monoid. That is, we look at contravariant functors $\mathbf{M} \rightarrow \mathbf{Sets}$.

Last time we’ve encountered the ‘Cube Dance Grap’ which depicts a particular relation among the major, minor, and augmented chords.

Recall that the twelve major chords (numbered for $1$ to $12$) are the ordered triples of tones in $\mathbb{Z}_{12}$ of the form $(n,n+4,n+7)$ (such as the triangle on the left). The twelve minor chords (numbered from $13$ to $24$) are the ordered triples $(n,n+3,n+7)$ (such as the middle triangle). The four augmented chords (numbered from $25$ to $28$) are the triples of the form $(n,n+4,n+8)$ (such as the rightmost triangle).

The Cube Dance Graph relates two of these chords when they share two tones (pitch classes) whereas the remaining tones differ by a halftone.

Picture modified from this post.

We can separate this symmetric binary relation into three sub-relations: the extension of the $P$ and $L$-operations on major and minor chords to the augmented ones (these are transformations), and the remaining relation $U$ which connects the major and minor chords to the augmented chords (and which is not a transformation).

Binary relations on the same set can be composed, so we get a monoid $\mathbf{M}$ generated by the three relations $P,L$ and $U$. The action of $\mathbf{M}$ on the $28$ chords no longer gives us an ordinary presheaf (because $U$ is not a transformation), but a relational presheaf as in the paper On the use of relational presheaves in transformational music theory by Alexandre Popoff.

That is, the action defines a contravariant functor $\mathbf{M} \rightarrow \mathbf{Rel}$ where $\mathbf{Rel}$ is the category (actually a $2$-category) of sets, but with binary relations as morphisms (that is, $Hom(X,Y)$ is all subsets of $X \times Y$), and the natural notion of composition of such relations. The $2$-morphism between relations is that of inclusion.

To compute with monoids generated by binary relations in GAP one needs to download, compile and load the package semigroups, and to represent the binary relations as partitioned binary relations as in the paper by Martin and Mazorchuk.

This is a bit more complicated than working with ordinary transformations:

But then, GAP quickly tells us that $\mathbf{M}$ is a monoid consisting of $40$ elements.

gap> M:=Semigroup([P,L,U]); gap> Size(M); 40

The Semigroups-package can also compute Green’s relations and tells us that there are seven such $R$-classes, four consisting of $6$ elements, two of four, and one of eight elements. These are also visible in the Cayley graph, exactly as last time.

Or, if you prefer the cleaner picture of the Cayley graph from the paper Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis by Popoff, Andreatta and Ehresmann.

This then allows us to compute the Heyting algebra of the subobject classifier, and all the Grothendieck topologies, at least for the ordinary presheaf topos of $\mathbf{M}$-sets, not for the relational presheaves we need here.

We can consider the same binary relation on the larger set of triads when we add the suspended triads. These are the ordered triples in $\mathbb{Z}_{12}$ of the form $(n,n+5,n+7)$, as in the rightmost triangle below.

There are twelve suspended chords (numbered from $29$ to $40$), so we now have a binary relation $T$ on a set of $40$ triads.

The relation $T$ is too coarse, and the art is to subdivide $T$ is disjoint sub-relations which are musically significant, between major and minor triads, between major/minor and augmented triads, and so on.

For each such partition we can then consider the monoids generated by these sub-relations.

In his paper, Popoff suggest relevant sub-relations $P,L,T_U,T_V$ and $T_U \cup T_V$ of $T$ which in our numbering of the $40$ chords can be represented by these PBR’s (assuming I made no mistakes…ADDED march 24th: I did make a mistake in the definition of L, see comment by Alexandre Popoff, below the corect L):

P:=PBR([[-13],[-14],[-15],[-16],[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-9],[-10],[-11],[-12],[-25],[-26],[-27],[-28],[-36],[-37],[-38],[-39],[-40],[-29],[-30],[-31],[-32],[-33],[-34],[-35]],[[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[25],[26],[27],[28],[34],[35],[36],[37],[38],[39],[40],[29],[30],[31],[32],[33]]); L:=PBR([[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-13],[-14],[-15],[-16],[-9],[ -10],[-11],[-12],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-25],[-26],[-27],[-28],[-29], [-30],[-31],[-32],[-33],[-34],[-35],[-36],[-37],[-38],[-39],[-40]],[[17], [18], [19], [ 20],[21],[22],[23],[24],[13],[14],[15],[16],[9],[10],[11],[12],[1],[2],[3],[4],[5], [6], [7],[8],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40] ]); TU:=PBR([[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-4,-8,-12,-13,-17,-21],[-1,-5,-9,-14,-18,-22],[-2,-6,-10,-15,-19,-23],[-3,-7,-11,-16,-20,-24],[],[],[],[],[],[],[],[],[],[],[],[]],[[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[25],[25],[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[4,8,12,13,17,21],[1,5,9,14,18,22],[2,6,10,15,19,23],[3,7,11,16,20,24],[],[],[],[],[],[],[],[],[],[],[],[]]); TV:=PBR([[-29],[-30],[-31],[-32],[-33],[-34],[-35],[-36],[-37],[-38],[-39],[-40],[-36],[-37],[-38],[-39],[-40],[-29],[-30],[-31],[-32],[-33],[-34],[-35],[],[],[],[],[-1,-18],[-2,-19],[-3,-20],[-4,-21],[-5,-22],[-6,-23],[-7,-24],[-8,-13],[-9,-14],[-10,-15],[-11,-16],[-12,-17]],[[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40],[36],[37],[38],[39],[40],[29],[30],[31],[32],[33],[34],[35],[],[],[],[],[1,18],[2,19],[3,20],[4,21],[5,22],[6,23],[7,24],[8,13],[9,14],[10,15],[11,16],[12,17]]); TUV:=PBR([[-26,-29],[-27,-30],[-28,-31],[-25,-32],[-26,-33],[-27,-34],[-28,-35],[-25,-36],[-26,-37],[-27,-38],[-28,-39],[-25,-40],[-25,-36],[-26,-37],[-27,-38],[-28,-39],[-25,-40],[-26,-29],[-27,-30],[-28,-31],[-25,-32],[-26,-33],[-27,-34],[-28,-35],[-4,-8,-12,-13,-17,-21],[-1,-5,-9,-14,-18,-22],[-2,-6,-10,-15,-19,-23],[-3,-7,-11,-16,-20,-24],[-1,-18],[-2,-19],[-3,-20],[-4,-21],[-5,-22],[-6,-23],[-7,-24],[-8,-13],[-9,-14],[-10,-15],[-11,-16],[-12,-17]],[[26,29],[27,30],[28,31],[25,32],[26,33],[27,34],[28,35],[25,36],[26,37],[27,38],[28,39],[25,40],[25,36],[26,37],[27,38],[28,39],[25,40],[26,29],[27,30],[28,31],[25,32],[26,33],[27,34],[28,35],[4,8,12,13,17,21],[1,5,9,14,18,22],[2,6,10,15,19,23],[3,7,11,16,20,24],[1,18],[2,19],[3,20],[4,21],[5,22],[6,23],[7,24],[8,13],[9,14],[10,15],[11,16],[12,17]]);

The resulting monoids are huge:

gap> G:=Semigroup([P,L,TU,TV]); gap> Size(G); 473293 gap> H:=Semigroup([P,L,TUV]); gap> Size(H); 994624

~~In Popoff’s paper these monoids have sizes respectively $473,293$ and $994,624$. Strangely, the offset is in both cases $144=12^2$.~~ (Added march 24: with the correct L I get the same sizes as in Popoff’s paper).

Perhaps we should try to transform such relational presheaves to ordinary presheaves.

One approach is to use the Grothendieck construction and associate to a set with such a relational monoid action a directed graph, coloured by the elements of the monoid. That is, an object in the presheaf topos of the category
\[
\xymatrix{C & E \ar[l]^c \ar@/^2ex/[r]^s \ar@/_2ex/[r]_t & V} \]
and then we should consider the slice topos over the one-vertex bouquet graph with one loop for each element in the monoid.

If you want to have more details on the musical side of things, for example if you want to know what the opening twelve chords of “Take a Bow” by Muse have to do with the Cube Dance graph, here are some more papers:

A categorical generalization of Klumpenhouwer networks, A. Popoff, M. Andreatta and A. Ehresmann.

From K-nets to PK-nets: a categorical approach, A. Popoff, M. Andreatta and A. Ehresmann.

From a Categorical Point of View: K-Nets as Limit Denotators, G. Mazzola and M. Andreatta.

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Mamuth to Elephant (2)

Published March 8, 2022 by lievenlb

Last time, we’ve viewed major and minor triads (chords) as inscribed triangles in a regular $12$-gon.

If we move clockwise along the $12$-gon, starting from the endpoint of the longest edge (the root of the chord, here the $0$-vertex) the edges skip $3,2$ and $4$ vertices (for a major chord, here on the left the major $0$-chord) or $2,3$ and $4$ vertices (for a minor chord, here on the right the minor $0$-chord).

The symmetries of the $12$-gon, the dihedral group $D_{12}$, act on the $24$ major- and minor-chords transitively, preserving the type for rotations, and interchanging majors with minors for reflections.

Mathematical Music Theoreticians (MaMuTh-ers for short) call this the $T/I$-group, and view the rotations of the $12$-gon as transpositions $T_k : x \mapsto x+k~\text{mod}~12$, and the reflections as involutions $I_k : x \mapsto -x+k~\text{mod}~12$.

Note that the elements of the $T/I$-group act on the vertices of the $12$-gon, from which the action on the chord-triangles follows.

There is another action on the $24$ major and minor chords, mapping a chord-triangle to its image under a reflection in one of its three sides.

Note that in this case the reflection $I_k$ used will depend on the root of the chord, so this action on the chords does not come from an action on the vertices of the $12$-gon.

There are three such operations: (pictures are taken from Alexandre Popoff’s blog, with the ‘funny names’ removed)

The $P$-operation is reflection in the longest side of the chord-triangle. As the longest side is preserved, $P$ interchanges the major and minor chord with the same root.

The $L$-operation is refection in the shortest side. This operation interchanges a major $k$-chord with a minor $k+4~\text{mod}~12$-chord.

Finally, the $R$-operation is reflection in the middle side. This operation interchanges a major $k$-chord with a minor $k+9~\text{mod}~12$-chord.

From this it is already clear that the group generated by $P$, $L$ and $R$ acts transitively on the $24$ major and minor chords, but what is this $PLR$-group?

If we label the major chords by their root-vertex $1,2,\dots,12$ (GAP doesn’t like zeroes), and the corresponding minor chords $13,14,\dots,24$, then these operations give these permutations on the $24$ chords:

P:=(1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) L:=(1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16) R:=(1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)

Then GAP gives us that the $PLR$-group is again isomorphic to $D_{12}$:

gap> G:=Group(P,L,R);; gap> Size(G); 24 gap> IsDihedralGroup(G); true

In fact, if we view both the $T/I$-group and the $PLR$-group as subgroups of the symmetric group $Sym(24)$ via their actions on the $24$ major and minor chords, these groups are each other centralizers! That is, the $T/I$-group and $PLR$-group are dual to each other.

For more on this, there’s a beautiful paper by Alissa Crans, Thomas Fiore and Ramon Satyendra: Musical Actions of Dihedral Groups.

What does this new MaMuTh info learns us more about our Elephant, the Topos of Triads, studied by Thomas Noll?

Last time we’ve seen the eight element triadic monoid $T$ of all affine maps preserving the three tones $\{ 0,4,7 \}$ of the major $0$-chord, computed the subobject classified $\Omega$ of the corresponding topos of presheaves, and determined all its six Grothendieck topologies, among which were these three:

Why did we label these Grothendieck topologies (and corresponding elements of $\Omega$) by $P$, $L$ and $R$?

We’ve seen that the sheafification of the presheaf $\{ 0,4,7 \}$ in the triadic topos under the Grothendieck topology $j_P$ gave us the sheaf $\{ 0,3,4,7 \}$, and these are the tones of the major $0$-chord together with those of the minor $0$-chord, that is the two chords in the $\langle P \rangle$-orbit of the major $0$-chord. The group $\langle P \rangle$ is the cyclic group $C_2$.

For the sheafication with respect to $j_L$ we found the $T$-set $\{ 0,3,4,7,8,11 \}$ which are the tones of the major and minor $0$-,$4$-, and $8$-chords. Again, these are exactly the six chords in the $\langle P,L \rangle$-orbit of the major $0$-chord. The group $\langle P,L \rangle$ is isomorphic to $Sym(3)$.

The $j_R$-topology gave us the $T$-set $\{ 0,1,3,4,6,7,9,10 \}$ which are the tones of the major and minor $0$-,$3$-, $6$-, and $9$-chords, and lo and behold, these are the eight chords in the $\langle P,R \rangle$-orbit of the major $0$-chord. The group $\langle P,R \rangle$ is the dihedral group $D_4$.

More on this can be found in the paper Commuting Groups and the Topos of Triads by Thomas Fiore and Thomas Noll.

The operations $P$, $L$ and $R$ on major and minor chords are reflexions in one side of the chord-triangle, so they preserve two of the three tones. There’s a distinction between the $P$ and $L$ operations and $R$ when it comes to how the third tone changes.

Under $P$ and $L$ the third tone changes by one halftone (because the corresponding sides skip an even number of vertices), whereas under $R$ the third tone changes by two halftones (a full tone), see the pictures above.

The $\langle P,L \rangle = Sym(3)$ subgroup divides the $24$ chords in four orbits of six chords each, three major chords and their corresponding minor chords. These orbits consist of the

$0$-, $4$-, and $8$-chords (see before)
$1$-, $5$-, and $9$-chords
$2$-, $6$-, and $10$-chords
$3$-, $7$-, and $11$-chords

and we can view each of these orbits as a cycle tracing six of the eight vertices of a cube with one pair of antipodal points removed.

These four ‘almost’ cubes are the NE-, SE-, SW-, and NW-regions of the Cube Dance Graph, from the paper Parsimonious Graphs by Jack Douthett and Peter Steinbach.

To translate the funny names to our numbers, use this dictionary (major chords are given by a capital letter):

The four extra chords (at the N, E, S, and P places) are augmented triads. They correspond to the triads $(0,4,8),~(1,5,9),~(2,6,10)$ and $(3,7,11)$.

That is, two triads are connected by an edge in the Cube Dance graph if they share two tones and differ by an halftone in the third tone.

This graph screams for a group or monoid acting on it. Some of the edges we’ve already identified as the action of $P$ and $L$ on the $24$ major and minor triads. Because the triangle of an augmented triad is equilateral, we see that they are preserved under $P$ and $L$.

But what about the edges connecting the regular triads to the augmented ones? If we view each edge as two directed arrows assigned to the same operation, we cannot do this with a transformation because the operation sends each augmented triad to six regular triads.

Alexandre Popoff, Moreno Andreatta and Andree Ehresmann suggest in their paper Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis that one might use a monoid generated by relations, and they show that there is such a monoid with $40$ elements acting on the Cube Dance graph.

Popoff claims that usual presheaf toposes, that is contravariant functors to $\mathbf{Sets}$ are not enough to study transformational music theory. He suggest to use instead functors to $\mathbf{Rel}$, that is Sets with as the morphisms binary relations, and their compositions.

Another Elephant enters the room…

(to be continued)

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From Mamuth to Elephant

Published March 7, 2022 by lievenlb

Here, MaMuTh stands for Mathematical Music Theory which analyses the pitch, timing, and structure of works of music.

The Elephant is the nickname for the ‘bible’ of topos theory, Sketches of an Elephant: A Topos Theory Compendium, a two (three?) volume book, written by Peter Johnstone.

How can we get as quickly as possible from the MaMuth to the Elephant, musical illiterates such as myself?

What Mamuth-ers call a pitch class (sounds that are a whole number of octaves apart), is for us a residue modulo $12$, as an octave is usually divided into twelve (half)tones.

We’ll just denote them by numbers from $0$ to $11$, or view them as the vertices of a regular $12$-gon, and forget the funny names given to them, as there are several such encodings, and we don’t know a $G$ from a $D\#$.

Our regular $12$-gon has exactly $24$ symmetries. Twelve rotations, which they call transpositions, given by the affine transformations
\[
T_k~:~x \mapsto x+k~\text{mod}~12 \]
and twelve reflexions, which they call involutions, given by
\[
I_k~:~x \mapsto -x+k~\text{mod}~12 \]
What for us is the dihedral group $D_{12}$ (all symmetries of the $12$-gon), is for them the $T/I$-group (for transpositions/involutions).

Let’s move from individual notes (or pitch classes) to chords (or triads), that is, three notes played together.

Not all triples of notes sound nice when played together, that’s why the most commonly played chords are among the major and minor triads.

A major triad is an ordered triple of elements from $\mathbb{Z}_{12}$ of the form
\[
(n,n+4~\text{mod}~12,n+7~\text{mod}~12) \]
and a minor triad is an ordered triple of the form
\[
(n,n+3~\text{mod}~12,n+7~\text{mod}~12) \]
where the first entry $n$ is called the root of the triad (or chord) and its funny name is then also the name of that chord.

For us, it is best to view a triad as an inscribed triangle in our regular $12$-gon. The triangles of major and minor triads have edges of different lengths, a small one, a middle, and a large one.

Starting from the root, and moving clockwise, we encounter in a major chord-triangle first the middle edge, then the small edge, and finally the large edge. For a minor chord-triangle, we have first the small edge, then the middle one, and finally the large edge.

On the left, two major triads, one with root $0$, the other with root $6$. On the right, two minor triads, also with roots $0$ and $6$.

(Btw. if you are interested in the full musical story, I strongly recommend the alpof blog by Alexandre Popoff, from which the above picture is taken.)

Clearly, there are $12$ major triads (one for each root), and $12$ minor triads.

From the shape of the triad-triangles it is also clear that rotations (transpositions) send major triads to major triads (and minors to minors), and that reflexions (involutions) interchange major with minor triads.

That is, the dihedral group $D_{12}$ (or if you prefer the $T/I$-group) acts on the set of $24$ major and minor triads, and this action is transitive (an element stabilising a triad-triangle must preserve its type (so is a rotation) and its root (so must be the identity)).

Can we hear the action of the very special group element $T_6$ (the unique non-trivial central element of $D_{12}$) on the chords?

This action is not only the transposition by three full tones, but also a point-reflexion with respect to the center of the $12$-gon (see two examples in the picture above). This point reflexion can be compositionally meaningful to refer to two very different upside-down worlds.

In It’s $T_6$-day, Alexandre Popoff gives several examples. Here’s one of them, the Ark theme in Indiana Jones – Raiders of the Lost Ark.

“The $T_6$ transformation is heard throughout the map room scene (in particular at 2:47 in the video): that the ark is a dreadful object from a very different world is well rendered by the $T_6$ transposition, with its inherent tritone and point reflection.”

Let’s move on in the direction of the Elephant.

We saw that the only affine map of the form $x \mapsto \pm x + k$ fixing say the major $0$-triad $(0,4,7)$ is the identity map.

But, we can ask for the collection of all affine maps $x \mapsto a x + b$ fixing this major $0$-triad set-wise, that is, such that
\[
\{ b, 4a+b~\text{mod}~12, 7a+b~\text{mod}~2 \} \subseteq \{ 0,4,7 \} \]

A quick case-by-case analysis shows that there are just eight such maps: the identity and the constant maps
\[
x \mapsto x,~x \mapsto 0,~x \mapsto 4, ~x \mapsto 7 \]
and the four maps
\[
\underbrace{x \mapsto 3x+7}_a,~\underbrace{x \mapsto 8x+4}_b,~x \mapsto 9x+4,~x \mapsto 4x \]

Compositions of such maps again preserve the set $\{ 0,4,7 \}$ so they form a monoid, and a quick inspection with GAP learns that $a$ and $b$ generate this monoid.

gap> a:=Transformation([10,1,4,7,10,1,4,7,10,1,4,7]);; gap> b:=Transformation([12,8,4,12,8,4,12,8,4,12,8,4]);; gap> gens:=[a,b];; gap> T:=Monoid(gens); gap> Size(T); 8

The monoid $T$ is the triadic monoid of Thomas Noll’s paper The topos of triads.

The monoid $T$ can be seen as a one-object category (with endomorphisms the elements of $T$). The corresponding presheaf topos is then the category of all sets equipped with a right $T$-action.

Actually, Noll considers just one such presheaf (and its sub-presheaves) namely $\mathcal{F}=\mathbb{Z}_{12}$ with the action of $T$ by affine maps described before.

He is interested in the sheafifications of these presheaves with respect to Grothendieck topologies, so we have to describe those.

For any monoid category, the subobject classifier $\Omega$ is the set of all right ideals in the monoid.

Using the GAP sgpviz package we can draw its Cayley graph (red coloured vertices are idempotents in the monoid, the blue vertex is the identity map).

gap> DrawCayleyGraph(T);

The elements of $T$ (vertices) which can be connected by oriented paths (in both ways) in the Cayley graph, such as here $\{ 2,4 \}$, $\{ 3,7 \}$ and $\{ 5,6,8 \}$, will generate the same right ideal in $T$, so distinct right ideals are determined by unidirectional arrows, such as from $1$ to $2$ and $3$ or from $\{ 2,4 \}$ to $5$, or from $\{ 3,7 \}$ to $6$.

This gives us that $\Omega$ consists of the following six elements:

$0 = \emptyset$
$C = \{ 5,6,8 \} = a.T \wedge b.T$
$L = \{ 2,4,5,6,8 \}=a.T$
$R = \{ 3,7,5,6,8 \}=b.T$
$P = \{ 2,3,4,5,6,7,8 \}=a.T \vee b.T$
$1 = T$

As a subobject classifier $\Omega$ is itself a presheaf, so wat is the action of the triad monoid $T$ on it? For all $A \in \Omega$, and $s \in T$ the action is given by $A.s = \{ t \in T | s.t \in A \}$ and it can be read off from the Cayley-graph.

$\Omega$ is a Heyting algebra of which the inclusions, and logical operations can be summarised in the picture below, using the Hexboards and Heytings-post.

In this case, Grothendieck topologies coincide with Lawvere-Tierney topologies, which come from closure operators $j~:~\Omega \rightarrow \Omega$ which are order-increasing, idempotent, and compatible with the $T$-action and with the $\wedge$, that is,

if $A \leq B$, then $j(A) \leq j(B)$
$j(j(A)) = j(A)$
$j(A).t=j(A.t)$
$j(A \wedge B) = j(A) \wedge j(B)$

Colouring all cells with the same $j$-value alike, and remaining cells $A$ with $j(A)=A$ coloured yellow, we have six such closure operations $j$, that is, Grothendieck topologies.

The triadic monoid $T$ acts via affine transformations on the set of pitch classes $\mathbb{Z}_{12}$ and we’ve defined it such that it preserves the notes $\{ 0,4,7 \}$ of the major $(0,4,7)$-chord, that is, $\{ 0,4,7 \}$ is a subobject of $\mathbb{Z}_{12}$ in the topos of $T$-sets.

The point of the subobject classifier $\Omega$ is that morphisms to it classify subobjects, so there must be a $T$-equivariant map $\chi$ making the diagram commute (vertical arrows are the natural inclusions)
\[
\xymatrix{\{ 0,4,7 \} \ar[r] \ar[d] & 1 \ar[d] \\ \mathbb{Z}_{12} \ar[r]^{\chi} & \Omega} \]

What does the morphism $\chi$ do on the other pitch classes? Well, it send an element $k \in \mathbb{Z}_{12} = \{ 1,2,\dots,12=0 \}$ to

$1$ iff $k \in \{ 0,4,7 \}$
$P$ iff $a(k)$ and $b(k)$ are in $\{ 0,4,7 \}$
$L$ iff $a(k) \in \{ 0,4,7 \}$ but $b(k)$ is not
$R$ iff $b(k) \in \{ 0,4,7 \}$ but $a(k)$ is not
$C$ iff neither $a(k)$ nor $b(k)$ is in $\{ 0,4,7 \}$

Remember that $a$ and $b$ are the transformations (images of $(1,2,\dots,12)$)
a:=Transformation([10,1,4,7,10,1,4,7,10,1,4,7]);; b:=Transformation([12,8,4,12,8,4,12,8,4,12,8,4]);;
so we see that

$0,1,4$ are mapped to $1$
$3$ is mapped to $P$
$8,11$ are mapped to $L$
$1,6,9,10$ are mapped to $R$
$2,5$ are mapped to $C$

Finally, we can compute the sheafification of the sub-presheaf $\{ 0,4,7 \}$ of $\mathbb{Z}$ with respect to a Grothendieck topology $j$: it consists of the set of those $k \in \mathbb{Z}_{12}$ such that $j(\chi(k)) = 1$.

The musically interesting Grothendieck topologies are $j_P, j_L$ and $j_R$ with corresponding sheaves:

For $j_P$ we get the sheaf $\{ 0,3,4,7 \}$ which Mamuth-ers call a Major-Minor Mixture as these are the notes of both the major and minor $0$-triads
For $j_L$ we get $\{ 0,3,4,7,8,11 \}$ which is an example of an Hexatonic scale (six notes), here they are the notes of the major and minor $0,~4$ and $8$-triads
For $j_R$ we get $\{ 0,1,3,4,6,7,9,10 \}$ which is an example of an Octatonic scale (eight notes), here they are the notes of the major and minor $0,~3,~6$ and $9$-triads

We could have played the same game starting with the three notes of any other major triad.

Those in the know will have noticed that so far I’ve avoided another incarnation of the dihedral $D_{12}$ group in music, namely the $PLR$-group, which explains the notation for the elements of the subobject classifier $\Omega$, but this post is already way too long.

(to be continued…)

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Hexboards and Heytings

Published February 26, 2022 by lievenlb

A couple of days ago, Peter Rowlett posted on The Aperiodical: Introducing hexboard – a LaTeX package for drawing games of Hex.

Hex is a strategic game with two players (Red and Blue) taking turns placing a stone of their color onto any empty space. A player wins when they successfully connect their sides together through a chain of adjacent stones.

Here’s a short game on a $5 \times 5$ board (normal play uses $11\times 11$ boards), won by Blue, drawn with the LaTeX-package hexboard.

As much as I like mathematical games, I want to use the versability of the hexboard-package for something entirely different: drawing finite Heyting algebras in which it is easy to visualise the logical operations.

Every full hexboard is a poset with minimal cell $0$ and maximal cell $1$ if cell-values increase if we move horizontally to the right or diagonally to the upper-right. With respect to this order, $p \vee q$ is the smallest cell bigger than both $p$ and $q$, and $p \wedge q$ is the largest cell smaller than $p$ and $q$.

The implication $p \Rightarrow q$ is the largest cell $r$ such that $r \wedge p \leq q$, and the negation $\neg p$ stands for $p \Rightarrow 0$. With these operations, the full hexboard becomes a Heyting algebra.

Now the fun part. Every filled area of the hexboard, bordered above and below by a string of strictly increasing cells from $0$ to $1$ is also a Heyting algebra, with the induced ordering, and with the logical operations defined similarly.

Note that this mustn’t be a sub-Heyting algebra as the operations may differ. Here, we have a different value for $p \Rightarrow q$, and $\neg p$ is now $0$.

If you’re in for an innocent “Where is Wally?”-type puzzle: $W = (\neg \neg p \Rightarrow p)$.

Click on the image to get the solution.

The downsets in these posets can be viewed as the open sets of a finite topology, so these Heyting algebra structures come from the subobject classifier of a topos.

There are more interesting toposes with subobject classifier determined by such hex-Heyting algebras.

For example, the Topos of Triads of Thomas Noll in music theory has as its subobject classifier the hex-Heyting algebra (with cell-values as in the paper):

Note to self: why not write a couple of posts on this topos?

Another example: the category of all directed graphs is the presheaf topos of the two object category ($V$ for vertices, and $E$ for edges) with (apart from the identities) just two morphisms $s,t : V \rightarrow E$ (for start- and end-vertex of a directed edge).

The subobject classifier $\Omega$ of this topos is determined by the two Heyting algebras $\Omega(E)$ and $\Omega(V)$ below.

These ‘hex-Heyting algebras’ are exactly what Eduardo Ochs calls ‘planar Heyting algebras’.

Eduardo has a very informative research page, containing slides and handouts of talks in which he tries to explain topos theory to “children” (using these planar Heyting algebras) including:

Perhaps now is a good time to revive my old sga4hipsters-project.

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Learners’ logic

Published February 22, 2022 by lievenlb

In the Learners and Poly-post we’ve seen that learners from $A$ to $B$ correspond to set-valued representations of a directed graph $G$ and therefore form a presheaf topos.

Any topos comes with its Mitchell-Benabou language, allowing us to speak of formulas, propositions and their truth values. Two objects play a special role in this: the terminal object $\mathbf{1}$, and the subobject classifier $\mathbf{\Omega}$. It is a fun exercise to determine these special learners.

$T$ is the free rooted tree with branches sprouting from every node $n \in T_0$ for each element in $A \times B$. $C$ will be our set of colours, one for each element of $Maps(A,B) \times Maps(A \times B,A)$.

For every map $\lambda : T_0 \rightarrow C$ we get a coloured rooted tree $T_{\lambda}$, and for each branch $(a,b)$ from the root we get another rooted sub-tree $T_{\lambda}(a,b)$ which is again of the form $T_{\mu}$ for a certain map $\mu : T_0 \rightarrow C$.

The directed graph $G$ has a vertex $v_{\lambda} \in V$ for each coloured rooted tree $T_{\lambda}$ and a directed edge $v_{\lambda} \rightarrow v_{\mu}$ if $T_{\mu}$ is the isomorphism class of coloured rooted trees of the subtree $T_{\lambda}(a,b)$ for some $(a,b) \in A \times B$.

There are exactly $\# A \times B$ directed edges leaving every vertex in $G$, but there may be (many) more incoming edges. We can colour each vertex $v_{\lambda}$ with the colour of the root of $T_{\lambda}$.

The coloured directed graph $G$ depicts the learning process in a neural network, being trained to find a suitable map $A \rightarrow B$. The colour of a vertex $v_{\lambda}$ gives a map $f \in Maps(A,B)$ (and a request function). If the network now gives as output $b \in B$ for a given input $a \in A$, we can move on to the end-vertex $v_{\mu}$ of the directed edge labeled $(a,b)$ out of $v_{\lambda}$. The colour of $v_{\mu}$ gives us a new (hopefully improved) map $f_{new} \in Maps(A,B)$ (and a new request function). A new training data $(a’,b’)$ brings us to a new vertex and map, and so on.

Clearly, some parts of $G$ are more efficient to find the desired map than others, and the aim of the game is to distinguish efficient from inefficient learners. A first hint that Grothendieck topologies and their corresponding sheafifications will turn out to be important.

We’ve seen that a learner, that is a morphism $Py^P \rightarrow C y^{A \times B}$ in $\mathbf{Poly}$, assigns a set $P_{\lambda}$ to every vertex $v_{\lambda}$ (this set may be empty) and a map $P_{\lambda} \rightarrow P_{\mu}$ to every directed edge $v_{\lambda} \rightarrow v_{\mu}$ in $G$.

The terminal object $\mathbf{1}$ in this setting assigns to each vertex a singleton $\{ \ast \}$, and the obvious maps for each directed edge. In $\mathbf{Poly}$-speak, the terminal object is the morphism
\[
\mathbf{1}~:~V y^V \rightarrow C y^{A \times B} \]
which sends each vertex $v_{\lambda} \in V$ to its colour $c \in C$, and where the backtrack map $\varphi^{\#}_{v_{\lambda}}[c]$ maps $(a,b)$ to $v_{\mu}$ if this is the end-vertex of the edge labelled $(a,b)$ out of $v_{\lambda}$. That is, $\mathbf{1}$ contains all information about the coloured directed graph $G$.

The subobject classifier $\mathbf{\Omega}$ assigns to each vertex $v_{\lambda}$ the set $\mathbf{\Omega}(v_{\lambda})$ of all subsets $S$ of directed paths in $G$, starting at $v_{\lambda}$, such that if $p \in S$ then also all prolongated paths belong to $S$. Note that the emptyset $\emptyset$ satisfies this requirement, so is an element of this vertex set. Another special element in $\mathbf{\Omega}(v_{\lambda})$ is the set $\mathbf{1}_{\lambda}$ of all oriented paths starting at $v_{\lambda}$.

$\mathbf{\Omega}(v_{\lambda})$ is an Heyting algebra with $1=\mathbf{1}_{\lambda}$, $0 = \emptyset$, partially ordered via inclusion, and logical operations $\wedge$ (intersection), $\vee$ (union), $\neg$ (with $\neg S$ the largest $S’ \in \mathbf{\Omega}(v_{\lambda})$ disjoint from $S$) and $\Rightarrow$ defined by $S \Rightarrow S’$ is the union of all $S” \in \mathbf{\Omega}(v_{\lambda})$ such that $S” \cap S \subseteq S’$.

$S \wedge \neg S$ is not always equal to $1$. Here, the union misses the left edge from the root. So, we will not be able to prove things by contradiction.

If $v_{\lambda} \rightarrow v_{\mu}$ is the directed edge labeled $(a,b)$, then the corresponding map $\mathbf{\Omega}(v_{\lambda}) \rightarrow \mathbf{\Omega}(v_{\mu})$ takes an $S \in \mathbf{\Omega}(v_{\lambda})$, drops all paths which do not pass through $v_{\mu}$ and removes from those who do the initial edge $(a,b)$. If no paths in $S$ pass through $v_{\mu}$ then $S$ is mapped to $\emptyset \in \mathbf{\Omega}(v_{\mu})$.

If $\Omega = \bigsqcup_{\lambda} \mathbf{\Omega}(v_{\lambda})$ then the subobject classifier is the morphism in $\mathbf{Poly}$
\[
\mathbf{\Omega}~:~\Omega y^{\Omega} \rightarrow C y^{A \times B} \]
sending a path starting in $v_{\lambda}$ to the colour of $v_{\lambda}$ and the backtrack map of $(a,b)$ the image of the path under the map $\mathbf{\Omega}(v_{\lambda}) \rightarrow \mathbf{\Omega}(v_{\mu})$.

Ok, let’s define the Learner’s Mitchell-Benabou language.

We’ll view a learner $Py^P \rightarrow C y^{A \times B}$ as a set-valued representation $P$ of the directed graph $G$ with vertex set $P_{\lambda}$ placed at vertex $v_{\lambda}$.

A formula $\phi(p)$ of the language with a free variable $p$ is a morphism (of representations of $G$) from a learner $P$ to the subobject classifier
\[
\phi~:~P \rightarrow \mathbf{\Omega} \]
Such a morphism determines a sub-representation of $P$ which we can denote $\{ p | \phi(p) \}$ with vertex sets
\[
\{ p | \phi(p) \}_{\lambda} = \{ p \in P_{\lambda}~|~\phi(v_{\lambda})(p) = \mathbf{1}_{\lambda} \} \]

On formulas we can apply logical connectives to get more formulas. For example, the formula $\phi(p) \Rightarrow \psi(q)$ is the composition
\[
P \times Q \rightarrow^{\phi \times \psi} \mathbf{\Omega} \times \mathbf{\Omega} \rightarrow^{\Rightarrow} \mathbf{\Omega} \]

By quantifying all free variables we get a formula without free variables, and those correspond to morphisms $\mathbf{1} \rightarrow \mathbf{\Omega}$, that is, to sub-representations of the terminal object $\mathbf{1}$.

For example, if $\phi(p)$ is the formula with free variable $p$ corresponding to the morphism $\phi : P \rightarrow \mathbf{\Omega}$, then we have
\[
\forall p : \phi(p) = \{ v_{\lambda} \in V~|~\{ p | \phi(p) \}_{\lambda} = P_{\lambda} \} \]
and
\[
\exists p : \phi (p) = \{ v_{\lambda} \in V~|~\{ p | \phi(p) \}_{\lambda} \not= \emptyset \} \]

Sub-representations of $\mathbf{1}$ again form a Heyting-algebra in the obvious way, so we can assign a “truth-value” to a formula without free variables as that sub-object of $\mathbf{1}$.

There’s a lot more to say, so perhaps this will be continued.

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Yet more topos news

Published February 18, 2022 by lievenlb

Every topos has its own internal language, the so called Mitchell-Bénabou language, allowing us to speak about formulas and their truth values.

Sadly, Jean Bénabou died last week.

Here’s a nice interview with Bénabou (in French) on category theory, Grothendieck, logic, and a rant on plagiarism among topos theorists (starting at 1:00:16).

Yesterday, France Culture’s ‘La methode scientifique’ hosted Alain Connes, Laurent Lafforgue and Olivia Caramello in a special programme Grothendieck: la moisson (Grothendieck, the harvest), dedicated to the recent publication of ‘Récoltes et Semailles’.

An interesting item is ‘le reportage du jour’ by Céline Loozen in which she manages to have a look at the 60.000 pages of Grothendieck’s Lasserre notes, stocked in the cellars of the Librairie Alain Brieux, and talks to Jean-Bernard Gillot who is commissioned by Grothendieck’s son to appraise the work (starts at 36:40).

Perhaps the publication of ‘Récoltes et Semailles’ is part of a deal with the family to make these notes available, at last.

Towards the end of the programme Connes, Caramello and Lafforgue lament that topos theory is still not taken seriously by the mathematical community at large, whereas it is welcomed warmly by the engineers of Huawei.

In more topos news, I learn from the blog of Olivia Caramello, that Laurent Lafforgue is going to give an online course on toposes as ‘bridges’ at the University of Warwick, the first talk starts today at 14hrs London time.

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The hype cycle of an idea

Published February 13, 2022 by lievenlb

These three ideas (re)surfaced over the last two decades, claiming to have potential applications to major open problems:

(2000) $\mathbb{F}_1$-geometry tries to view $\mathbf{Spec}(\mathbb{Z})$ as a curve over the field with one element, and mimic Weil’s proof of RH for curves over finite fields to prove the Riemann hypothesis.
(2012) IUTT, for Inter Universal Teichmuller Theory, the machinery behind Mochizuki’s claimed proof of the ABC-conjecture.
(2014) topos theory : Connes and Consani redirected their RH-attack using arithmetic sites, while Lafforgue advocated the use of Caramello’s bridges for unification, in particular the Langlands programme.

It is difficult to voice an opinion about the (presumed) current state of such projects without being accused of being either a believer or a skeptic, resorting to group-think or being overly critical.

We lack the vocabulary to talk about the different phases a mathematical idea might be in.

Such a vocabulary exists in (information) technology, the five phases of the Gartner hype cycle to represent the maturity, adoption, and social application of a certain technology :

Technology Trigger
Peak of Inflated Expectations
Trough of Disillusionment
Slope of Enlightenment
Plateau of Productivity

This model can then be used to gauge in which phase several emerging technologies are, and to estimate the time it will take them to reach the stable plateau of productivity. Here’s Gartner’s recent Hype Cycle for emerging Artificial Intelligence technologies.

Picture from Gartner Hype Cycle for AI 2021

What might these phases be in the hype cycle of a mathematical idea?

Technology Trigger: a new idea or analogy is dreamed up, marketed to be the new approach to that problem. A small group of enthusiasts embraces the idea, and tries to supply proper definitions and the very first results.
Peak of Inflated Expectations: the idea spreads via talks, blogposts, mathoverflow and twitter, and now has enough visibility to justify the first conferences devoted to it. However, all this activity does not result in major breakthroughs and doubt creeps in.
Trough of Disillusionment: the project ran out of steam. It becomes clear that existing theories will not lead to a solution of the motivating problem. Attempts by key people to keep the idea alive (by lengthy papers, regular meetings or seminars) no longer attract new people to the field.
Slope of Enlightenment: the optimistic scenario. One abandons the original aim, ditches the myriad of theories leading nowhere, regroups and focusses on the better ideas the project delivered.
A negative scenario is equally possible. Apart for a few die-hards the idea is abandoned, and on its way to the graveyard of forgotten ideas.
Plateau of Productivity: the polished surviving theory has applications in other branches and becomes a solid tool in mathematics.

It would be fun so see more knowledgable people draw such a hype cycle graph for recent trends in mathematics.

Here’s my own (feeble) attempt to gauge where the three ideas mentioned at the start are in their cycles, and here’s why:

IUTT: recent work of Kirti Joshi, for example this, and this, and that, draws from IUTT while using conventional language and not making exaggerated claims.
$\mathbb{F}_1$: the preliminary programme of their seminar shows little evidence the $\mathbb{F}_1$-community learned from the past 20 years.
Topos: Developing more general theory is not the way ahead, but concrete examples may carry surprises, even though Gabriel’s topos will remain elusive.

Clearly, you don’t agree, and that’s fine. We now have a common terminology, and you can point me to results or events I must have missed, forcing me to redraw my graph.

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