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Category: geometry

The $\mathbb{F}_1$ World Seminar

For some time I knew it was in the making, now they are ready to launch it:

The $\mathbb{F}_1$ World Seminar, an online seminar dedicated to the “field with one element”, and its many connections to areas in mathematics such as arithmetic, geometry, representation theory and combinatorics. The organisers are Jaiung Jun, Oliver Lorscheid, Yuri Manin, Matt Szczesny, Koen Thas and Matt Young.

From the announcement:

“While the origins of the “$\mathbb{F}_1$-story” go back to attempts to transfer Weil’s proof of the Riemann Hypothesis from the function field case to that of number fields on one hand, and Tits’s Dream of realizing Weyl groups as the $\mathbb{F}_1$ points of algebraic groups on the other, the “$\mathbb{F}_1$” moniker has come to encompass a wide variety of phenomena and analogies spanning algebraic geometry, algebraic topology, arithmetic, combinatorics, representation theory, non-commutative geometry etc. It is therefore impossible to compile an exhaustive list of topics that might be discussed. The following is but a small sample of topics that may be covered:

Algebraic geometry in non-additive contexts – monoid schemes, lambda-schemes, blue schemes, semiring and hyperfield schemes, etc.
Arithmetic – connections with motives, non-archimedean and analytic geometry
Tropical geometry and geometric matroid theory
Algebraic topology – K-theory of monoid and other “non-additive” schemes/categories, higher Segal spaces
Representation theory – Hall algebras, degenerations of quantum groups, quivers
Combinatorics – finite field and incidence geometry, and various generalizations”

The seminar takes place on alternating Wednesdays from 15:00 PM – 16:00 PM European Standard Time (=GMT+1). There will be room for mathematical discussion after each lecture.

The first meeting takes place Wednesday, January 19th 2022. If you want to receive abstracts of the talks and their Zoom-links, you should sign up for the mailing list.

Perhaps I’ll start posting about $\mathbb{F}_1$ again, either here, or on the dormant $\mathbb{F}_1$ mathematics blog. (see this post for its history).

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Huawei and topos theory

Apart from the initiatives I mentioned last time, Huawei set up a long term collaboration with the IHES, the Huawei Young Talents Program.

“Every year, the Huawei Young Talents Program will fund on average 7 postdoctoral fellowships that will be awarded by the Institute’s Scientific Council, only on the basis of scientific excellence. The fellows will collaborate with the Institute’s permanent professors and work on topics of their interest.”

Over the next ten years, Huawei will invest 5 million euros in this program, and an additional 1 million euros goes into the creation of the ‘Huawei Chair in Algebraic Geometry’. It comes as no particular surprise that the first chairholder is Laurent Lafforgue.

At the launch of this Young Talents Program in November 2020, Lafforgue gave a talk on The creative power of categories: History and some new perspectives.

The latter part of the talk (starting at 47:50) clarifies somewhat Huawei’s interest in topos theory, and what Lafforgue (and others) hope to get out of their collaboration with the telecom company.

Clearly, Huawei is interested in deep neural networks, and if you can convince them your expertise is useful in that area, perhaps they’ll trow some money at you.

Jean-Claude Belfiore, another mathematician turned Huaweian, is convinced topos theory is the correct tool to study DNNs. Here’s his Huawei-clip from which it is clear he was originally hired to improve Huawei’s polar code.

At the 2018 IHES-Topos conference he gave the talk Toposes for Wireless Networks: An idea whose time has come, and recently he arXived the paper Topos and Stacks of Deep Neural Networks, written jointly with Daniel Bennequin. Probably, I’ll come back to this paper another time, for now, the nForum has this page on it.

Towards the end of his talk, Lafforgue suggests the idea of creating an institute devoted to toposes and their applications, endorsed by IHES and supported by Huawei. Surely he knows that the Topos Institute already exists.

And, if you wonder why Huawei trows money at IHES rather than your university, I leave you with Lafforgue’s parting words:

“IHES professors are able to think and evaluate for themselves, whereas most mathematicians just follow ‘group thinking'”

Ouch!

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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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Imagination and the Impossible

Two more sources I’d like to draw from for this fall’s maths for designers-course:

1. Geometry and the Imagination

A fantastic collection of handouts for a two week summer workshop entitled ’Geometry and the Imagination’, led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 1991, based on a course ‘Geometry and the Imagination’ they taught twice before at Princeton.

Among the goodies a long list of exercises in imagining (always useful to budding architects) and how to compute curvature by peeling potatoes and other vegetables…

The course really shines in giving a unified elegant classification of the 17 wallpaper groups, the 7 frieze groups and the 14 families of spherical groups by using Thurston’s concept of orbifolds.

If you think this will be too complicated, have a look at the proof that the orbifold Euler characteristic of any symmetry pattern in the plane with bounded fundamental domain is zero :

Take a large region in the plane that is topologically a disk (i.e. without holes). Its Euler characteristic is $1$. This is approximately equal to $N$ times the orbifold Euler characteristic for some large $N$, so the orbifold Euler characteristic must be $0$.

This then leads to the Orbifold Shop where they sell orbifold parts:

  • a handle for 2 Euros,
  • a mirror for 1 Euro,
  • a cross-cap for 1 Euro,
  • an order $n$ cone point for $(n-1)/n$ Euro,
  • an order $n$ corner reflector for $(n-1)/2n$ Euro, if you have the required mirrors to install this piece.



Here’s a standard brick wall, with its fundamental domain and corresponding orbifold made from a mirror piece (1 Euro), two order $2$ corner reflectors (each worth $.25$ Euro), and one order $2$ cone point (worth $.5$ Euro). That is, this orbifold will cost you exactly $2$ Euros.

If you spend exactly $2$ Euros at the Orbifold Shop (and there are $17$ different ways to do this), you will have an orbifold coming from a symmetry pattern in the plane with bounded fundamental domain, that is, one of the $17$ wallpaper patterns.

For the mathematicians among you desiring more details, please read The orbifold notation for two-dimensional groups by Conway and Daniel Huson, from which the above picture was taken.

2. On the Cohomology of Impossible Figures by Roger Penrose

The aspiring architect should be warned that some constructions are simply not possible in 3D, even when they look convincing on paper, such as Escher’s Waterfall.



M.C. Escher, Waterfall – Photo Credit

In his paper, Penrose gives a unified approach to debunk such drawings by using cohomology groups.

Clearly I have no desire to introduce cohomology, but it may still be possible to get the underlying idea across. Let’s take the Penrose triangle (all pictures below taken from Penrose’s paper)



The idea is to break up such a picture in several parts, each of which we do know to construct in 3D (that is, we take a particular cover of our figure). We can slice up the Penrose triangle in three parts, and if you ever played with Lego you’ll know how to construct each one of them.



Next, position the constructed pieces in space as in the picture and decide which of the two ends is closer to you. In $Q_1$ it is clear that point $A_{12}$ is closer to you than $A_{13}$, so we write $A_{12} < A_{13}$.

Similarly, looking at $Q_2$ and $Q_3$ we see that $A_{23} < A_{21}$ and that $A_{31} < A_{32}$.

Next, if we try to reassemble our figure we must glue $A_{12}$ to $A_{21}$, that is $A_{12}=A_{21}$, and similarly $A_{23}=A_{32}$ and $A_{31}=A_{13}$. But, then we get
\[
A_{13}=A_{31} < A_{32}=A_{23} < A_{21}=A_{12} < A_{13} \] which is clearly absurd.

Once again, if you have suggestions for more material to be included, please let me know.

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Designer Maths

This fall, I’ll be teaching ‘Mathematics for Designers’ to first year students in Architecture.

The past few weeks I’ve been looking around for topics to be included in such as course, relevant to architects/artists (not necessarily to engineers/mathematicians).

One of the best texts I’ve found on this (perhaps in need of a slight update) is the 1986-paper by Jay Kappraff: A course in the mathematics of design. He suggests the following list of topics:

  • graph theory
  • polyhedra
  • tilings of the plane
  • three dimensional packings
  • proportion and the golden mean
  • transformations
  • symmetry
  • vectors

We all know that an awful lot of math and computation is needed to design a building, but today all of the hardcore use of vectors, equations and transformations is conveniently hidden from the architect’s view by digital design platforms and CAD-programs.

These computational tools offer new creative possibilities, as illustrated in the beautiful book The new mathematics of architecture by Jane Burry and Mark Burry, also available in Dutch with a cover picture of the Möbius bridge in Bristol



In this book, about 50 recent architectural projects are clustered around these topics:

  • mathematical surfaces and seriality
  • chaos, complexity, emergence
  • packings and tilings
  • optimization
  • topology
  • datascapes and multi-dimensionality

In the description of the projects, cool math-topics are (sadly only) touched, including

It will take me some time to find a balance between these two approaches. Common themes clearly are

  • Shapes : what is possible/impossible in 2D and 3D, and how can mathematics help us to find new exciting shapes (think minimal and Seifert surfaces, knot complements, etc.)
  • Symmetry : what is possible/impossible in 2D and 3D, and what can mathematics tell us about new symmetries (think emerging symmetries from aperiodic tilings and quasicrystals)

Over the coming months I’ll be writing the course notes and may post about it here. For this reason I’ve included a new category DesignerMaths.

If you have suggestions, please let me know.

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a monstrous unimodular lattice

An integral $n$-dimensional lattice $L$ is the set of all integral linear combinations
\[
L = \mathbb{Z} \lambda_1 \oplus \dots \oplus \mathbb{Z} \lambda_n \]
of base vectors $\{ \lambda_1,\dots,\lambda_n \}$ of $\mathbb{R}^n$, equipped with the usual (positive definite) inner product, satisfying
\[
(\lambda, \mu ) \in \mathbb{Z} \quad \text{for all $\lambda,\mu \in \mathbb{Z}$.} \]
But then, $L$ is contained in its dual lattice $L^* = Hom_{\mathbb{Z}}(L,\mathbb{Z})$, and if $L = L^*$ we say that $L$ is unimodular.

If all $(\lambda,\lambda) \in 2 \mathbb{Z}$, we say that $L$ is an even lattice. Even unimodular lattices (such as the $E_8$-lattice or the $24$ Niemeier lattices) are wonderful objects, but they can only live in dimensions $n$ which are multiples of $8$.

Just like the Conway group $Co_0 = .0$ is the group of rotations of the Leech lattice $\Lambda$, one might ask whether there is a very special lattice on which the Monster group $\mathbb{M}$ acts faithfully by rotations. If such a lattice exists, it must live in dimension at least $196883$.



Simon Norton (1952-2019) – Photo Credit

A first hint of such a lattice is in Conway’s original paper A simple construction for the Fischer-Griess monster group (but not in the corresponding chapter 29 of SPLAG).

Conway writes that Simon Norton showed ‘by a very simple computations that does not even require knowledge of the conjugacy classes, that any $198883$-dimensional representation of the Monster must support an invariant algebra’, which, after adding an identity element $1$, we now know as the $196884$-dimensional Griess algebra.

Further, on page 529, Conway writes:

Norton has shown that the lattice $L$ spanned by vectors of the form $1,t,t \ast t’$, where $t$ and $t’$ are transposition vectors, is closed under the algebra multiplication and integral with respect to the doubled inner product $2(u,v)$. The dual quotient $L^*/L$ is cyclic of order some power of $4$, and we believe that in fact $L$ is unimodular.

Here, transposition vectors correspond to transpositions in $\mathbb{M}$, that is, elements of conjugacy class $2A$.

I only learned about this lattice yesterday via the MathOverflow-post A lattice with Monster group symmetries by Adam P. Goucher.

In his post, Adam considers the $196883$-dimensional lattice $L’ = L \cap 1^{\perp}$ (which has $\mathbb{M}$ as its rotation symmetry group), and asks for the minimal norm (squared) of a lattice point, which he believes is $448$, and for the number of minimal vectors in the lattice, which might be
\[
2639459181687194563957260000000 = 9723946114200918600 \times 27143910000 \]
the number of oriented arcs in the Monster graph.

Here, the Monster graph has as its vertices the elements of $\mathbb{M}$ in conjugacy class $2A$ (which has $9723946114200918600$ elements) and with an edge between two vertices if their product in $\mathbb{M}$ again belongs to class $2A$, so the valency of the graph must be $27143910000$, as explained in that old post the monster graph and McKay’s observation.

When I asked Adam whether he had more information about his lattice, he kindly informed me that Borcherds told him that the Norton lattice $L$ didn’t turn out to be unimodular after all, but that a unimodular lattice with monstrous symmetry had been constructed by Scott Carnahan in the paper A Self-Dual Integral Form of the Moonshine Module.



Scott Carnahan – Photo Credit

The major steps (or better, the little bit of it I could grasp in this short time) in the construction of this unimodular $196884$-dimensional monstrous lattice might put a smile on your face if you are an affine scheme aficionado.

Already in his paper Vertex algebras, Kac-Moody algebras, and the Monster, Richard Borcherds described an integral form of any lattice vertex algebra. We’ll be interested in the lattice vertex algebra $V_{\Lambda}$ constructed from the Leech lattice $\Lambda$ and call its integral form $(V_{\Lambda})_{\mathbb{Z}}$.

One constructs the Moonshine module $V^{\sharp}$ from $V_{\Lambda}$ by a process called ‘cyclic orbifolding’, a generalisation of the original construction by Frenkel, Lepowsky and Meurman. In fact, there are now no less than 51 constructions of the moonshine module.

One starts with a fixed point free rotation $r_p$ of $\Lambda$ in $Co_0$ of prime order $p \in \{ 2,3,5,7,13 \}$, which one can lift to an automorphism $g_p$ of the vertex algebra $V_{\Lambda}$ of order $p$ giving an isomorphism $V_{\Lambda}/g_p \simeq V^{\sharp}$ of vertex operator algebras over $\mathbb{C}$.

For two distinct primes $p,p’ \in \{ 2,3,5,7,13 \}$ if $Co_0$ has an element of order $p.p’$ one can find one such $r_{pp’}$ such that $r_{pp’}^p=r_{p’}$ and $r_{pp’}^{p’}=r_p$, and one can lift $r_{pp’}$ to an automorphism $g_{pp’}$ of $V_{\Lambda}$ such that $V_{\Lambda}/g_{pp’} \simeq V_{\Lambda}$ as vertex operator algebras over $\mathbb{C}$.

Problem is that these lifts of automorphisms and the isomorphisms are not compatible with the integral form $(V_{\Lambda})_{\mathbb{Z}}$ of $V_{\Lambda}$, but ‘essentially’, they can be performed on
\[
(V_{\Lambda})_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Z}[\frac{1}{pp’},\zeta_{2pp’}] \]
where $\zeta_{2pp’}$ is a primitive $2pp’$-th root of unity. These then give a $\mathbb{Z}[\tfrac{1}{pp’},\zeta_{2pp’}]$-form on $V^{\sharp}$.

Next, one uses a lot of subgroup information about $\mathbb{M}$ to prove that these $\mathbb{Z}[\tfrac{1}{pp’},\zeta_{2pp’}]$-forms of $V^{\sharp}$ have $\mathbb{M}$ as their automorphism group.

Then, using all his for different triples in $\{ 2,3,5,7,13 \}$ one can glue and use faithfully flat descent to get an integral form $V^{\sharp}_{\mathbb{Z}}$ of the moonshine module with monstrous symmetry and such that the inner product on $V^{\sharp}_{\mathbb{Z}}$ is positive definite.

Finally, one looks at the weight $2$ subspace of $V^{\sharp}_{\mathbb{Z}}$ which gives us our Carnahan’s $196884$-dimensional unimodular lattice with monstrous symmetry!

Beautiful as this is, I guess it will be a heck of a project to deduce even the simplest of facts about this wonderful lattice from running through this construction.

For example, what is the minimal length of vectors? What is the number of minimal length vectors? And so on. All info you might have is very welcome.

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Escher’s stairs

Stairways feature prominently in several drawings by Maurits Cornelis (“Mauk”) Escher, for example in this lithograph print Relativity from 1953.



Relativity (M. C. Escher) – Photo Credit

From its Wikipedia page:

In the world of ‘Relativity’, there are three sources of gravity, each being orthogonal to the two others.
Each inhabitant lives in one of the gravity wells, where normal physical laws apply.
There are sixteen characters, spread between each gravity source, six in one and five each in the other two.
The apparent confusion of the lithograph print comes from the fact that the three gravity sources are depicted in the same space.
The structure has seven stairways, and each stairway can be used by people who belong to two different gravity sources.

Escher’s inspiration for “Relativity” (h/t Gerard Westendorp on Twitter) were his recollections of the staircases in his old secondary school in Arnhem, the Lorentz HBS.
The name comes from the Dutch physicist and Nobel prize winner Hendrik Antoon Lorentz who attended from 1866 to 1869, the “Hogere Burger School” in Arnhem, then at a different location (Willemsplein).



Stairways Lorentz HBS in Arnhem – Photo Credit

Between 1912 and 1918 Mauk Escher attended the Arnhem HBS, located in the Schoolstraat and build in 1904-05 by the architect Gerrit Versteeg. The school building is constructed around a monumental central stairway.



Arnhem HBS – G. Versteeg 1904-05 – Photo Credit



Plan HBS-Arnhem by G. Versteeg – Photo Credit

If you flip the picture below in the vertical direction, the two side-stairways become accessible to figures living in an opposite gravitation field.



Central staircase HBS Arnhem – Photo Credit

There’s an excellent post on the Arnhem-years of Mauk Escher by Pieter van der Kuil. Unfortunately (for most of you) in Dutch, but perhaps Google translate can do its magic here.

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Ghost metro stations

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



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



Haxo metro station – Photo Credit

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



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

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



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

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

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

Remember the subway singularity?

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

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

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

This can also happen in Paris.

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

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

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Know thy neighbours

Two lattices $L$ and $L’$ in the same vector space are called neighbours if their intersection $L \cap L’$ is of index two in both $L$ and $L’$.

In 1957, Martin Kneser gave a method to find all unimodular lattices (of the same dimension and signature) starting from one such unimodular lattice, finding all its neighbours, and repeating this with the new lattices obtained.

In other words, Kneser’s neighbourhood graph, with vertices the unimodular lattices (of fixed dimension and signature) and edges between them whenever the lattices are neighbours, is connected.



Martin Kneser (1928-2004) – Photo Credit

Last time, we’ve constructed the Niemeier lattice $(A_1^{24})^+$ from the binary Golay code $\mathcal{C}_{24}$
\[
L = (A_1^{24})^+ = \mathcal{C}_{24} \underset{\mathbb{F}_2}{\times} (A_1^{24})^* = \{ \tfrac{1}{\sqrt{2}} \vec{v} ~|~\vec{v} \in \mathbb{Z}^{\oplus 24},~v=\vec{v}~mod~2 \in \mathcal{C}_{24} \} \]
With hindsight, we know that $(A_1^{24})^+$ is the unique neighbour of the Leech lattice in the Kneser neighbourhood graph of the positive definite, even unimodular $24$-dimensional lattices, aka the Niemeier lattices.

Let’s try to construct the Leech lattice $\Lambda$ from $L=(A_1^{24})^+$ by Kneser’s neighbour-finding trick.



Sublattices of $L$ of index two are in one-to-one correspondence with non-zero elements in $L/2L$. Take $l \in L – 2L$ and $m \in L$ such that the inner product $l.m$ is odd, then
\[
L_l = \{ x \in L~|~l.x~\text{is even} \} \]
is an index two sublattice because $L = L_l \sqcup (L_l+m)$. By definition $l.x$ is even for all $x \in L_l$ and therefore $\frac{l}{2} \in L_l^*$. We have this situation
\[
L_l \subsetneq L = L^* \subsetneq L_l^* \]
and $L_l^*/L_l \simeq \mathbb{F}_2 \oplus \mathbb{F}_2$, with the non-zero elements represented by $\{ \frac{l}{2}, m, \frac{l}{2}+m \}$. That is,
\[
L_l^* = L_l \sqcup (L_l+m) \sqcup (L_l+\frac{l}{2}) \sqcup (L_l+(\frac{l}{2}+m)) \]
This gives us three lattices
\[
\begin{cases}
M_1 &= L_l \sqcup (L_l+m) = L \\
M_2 &= L_l \sqcup (L_l+\frac{l}{2}) \\
M_3 &= L_l \sqcup (L_l+(\frac{l}{2}+m))
\end{cases}
\]
and all three of them are unimodular because
\[
L_l \subsetneq M_i \subseteq M_i^* \subsetneq L_l^* \]
and $L_l$ is of index $4$ in $L_l^*$.

Now, let’s assume the norm of $l$, that is, $l.l \in 4 \mathbb{Z}$. Then, either the norm of $\frac{l}{2}$ is odd (but then the norm of $\frac{l}{2}+m$ must be even), or the norm of $\frac{l}{2}$ is even, in which case the norm of $\frac{l}{2}+m$ is odd.

That is, either $M_2$ or $M_3$ is an even unimodular lattice, the other one being an odd unimodular lattice.

Let’s take for $l$ and $m$ the vectors $\lambda = \frac{1}{\sqrt{2}} (1,1,\dots,1) \in L – 2L$ and $\mu = \sqrt{2}(1,0,\dots,0) \in L$, then
\[
\lambda.\lambda = \frac{1}{2}\times 24 = 12 \quad \text{and} \quad \mu.\lambda = 1 \]
Because $\frac{\lambda}{2}.\frac{\lambda}{2} = \frac{12}{4}=3$ is odd, we have that
\[
\Lambda = L_{\lambda} \sqcup (L_{\lambda} + (\frac{\lambda}{2} + \mu)) \]
is an even unimodular lattice, which is the Leech lattice, and
\[
\Lambda_{odd} = L_{\lambda} \sqcup (L_{\lambda} + \frac{\lambda}{2}) \]
is an odd unimodular lattice, called the odd Leech lattice.



John Leech (1926-1992) – Photo Credit

Let’s check that these are indeed the Leech lattices, meaning that they do not contain roots (vectors of norm two).

The only roots in $L = (A_1^{24})^+$ are the $48$ roots of $A_1^{24}$ and they are of the form $\pm \sqrt{2} [ 1, 0^{23} ]$, but none of them lies in $L_{\lambda}$ as their inproduct with $\lambda$ is one. So, all non-zero vectors in $L_{\lambda}$ have norm $\geq 4$.

As for the other part of $\Lambda$ and $\Lambda_{odd}$
\[
(L_{\lambda} + \frac{\lambda}{2}) \sqcup (L_{\lambda} + \mu + \frac{\lambda}{2}) = (L_{\lambda} \sqcup (L_{\lambda}+\mu))+\frac{\lambda}{2} = L + \frac{\lambda}{2} \]
From the description of $L=(A_1^{24})^+$ it follows that every coordinate of a vector in $L + \frac{\lambda}{2}$ is of the form
\[
\frac{1}{\sqrt{2}}(v+\frac{1}{2}) \quad \text{or} \quad \frac{1}{\sqrt{2}}(v+\frac{3}{2}) \]
with $v \in 2 \mathbb{Z}$, with the second case instances forming a codeword in $\mathcal{C}_{24}$. In either case, the square of each of the $24$ coordinates is $\geq \frac{1}{8}$, so the norm of such a vector must be $\geq 3$, showing that there are no roots in this region either.

If one takes for $l$ a vector of the form $\frac{1}{\sqrt{2}} v = \frac{1}{\sqrt{2}}[1^a,0^{24-a}]$ where $a=8,12$ or $16$ and $v \in \mathcal{C}_{24}$, takes $m=\mu$ as before, and repeats the construction, one gets the other Niemeier-neighbours of $(A_1^{24})^+$, that is, the lattices $(A_2^{12})^+$, $(A_3^8)^+$ and $(D_4^6)^+$.

For $a=12$ one needs a slightly different argument, see section 0.2 of Richard Borcherds’ Ph.D. thesis.

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