My favourite tags on MathOverflow are big-lists, big-picture, soft-question,
reference-request and the like.

Often, answers to such tagged questions contain sound reading advice, style: “road-map to important result/theory X”.

Two more K to go, so let’s spend some more money.

Category theory

One of the problems with my master course on algebraic geometry is that the students are categorical virgins.

They’ve been studying specific categories, functors, natural transformations and more all over their bachelor years, without knowing the terminology.

It then helps to illustrate these concepts with examples. For example that the determinant is a natural transformation, or that $\mathbb{C}[t]$ represents the functor forgetting the ring structure.

The more examples the merrier. I like Riehl’s example that in the category of graphs, the complete graph $K_n$ represents the functor assigning to a graph the set of all its $n$-colourings.

So, I had a look at the MathOverflow question Is Mac Lane still the best place to learn category theory?.

It is always a good idea to support authors offering a free online version of their book.

Abstract and Concrete Categories: The Joy of Cats by J. Adamek,H. Herrlich and G. Strecker. Blurb: “This up-to-date introductory treatment employs the language of category theory to explore the theory of structures. Its unique approach stresses concrete categories, and each categorical notion features several examples that clearly illustrate specific and general cases.”

Free online version : The Joy of Cats

Category Theory for the Sciences by David Spivak. Blurb: “Using databases as an entry to category theory, it begins with sets and functions, then introduces the reader to notions that are fundamental in mathematics: monoids, groups, orders, and graphs — categories in disguise. After explaining the “big three” concepts of category theory — categories, functors, and natural transformations — the book covers other topics, including limits, colimits, functor categories, sheaves, monads, and operads. The book explains category theory by examples and exercises rather than focusing on theorems and proofs. It includes more than 300 exercises, with solutions.”

Free online version: Category theory for scientists

Category Theory in Context by Emily Riehl. Blurb: “Suitable for advanced undergraduates and graduate students in mathematics, the text provides tools for understanding and attacking difficult problems in algebra, number theory, algebraic geometry, and algebraic topology. Drawing upon a broad range of mathematical examples from the categorical perspective, the author illustrates how the concepts and constructions of category theory arise from and illuminate more basic mathematical ideas. ”

Free online version: Category theory in context

Now, for the heavier stuff.

If I want to study Jacob Lurie’s books “Higher Topoi Theory”, “Derived AG”, what prerequisites should I have?

Simplicial Objects in Algebraic Topology by Peter May. Blurb: “Since it was first published in 1967, Simplicial Objects in Algebraic Topology has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of topological spaces. ”

Free online version: Simplicial Objects in Algebraic Topology (h/t David Roberts via the comments)

A Concise Course in Algebraic Topology by Peter May. Blurb: “J. Peter May’s approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. ”

Free online version: A Concise Course in Algebraic Topology

Or in Lurie’s words: “To read Higher Topos Theory, you’ll need familiarity with ordinary category theory and with the homotopy theory of simplicial sets (Peter May’s book “Simplicial Objects in Algebraic Topology” is a good place to learn the latter). Other topics (such as classical topos theory) will be helpful for motivation.”

He also has a suggestion for the classic topos theory stuff:

“”Sheaves in Geometry and Logic” by Moerdijk and MacLane is a pretty good read (as is Uncle John, but I’ve never seen topos theory in there).”

I’ve had this book on permanent loan from our library over the past two years, so it’s about time to have my own copy.

Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Mac Lane and Moerdijk. Blurb: “Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds. Sheaves also appear in logic as carriers for models of set theory. This text presents topos theory as it has developed from the study of sheaves. Beginning with several examples, it explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.”

Higher Topos Theory by Jacob Lurie. Blurb: “Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory’s new language. The result is a powerful theory with applications in many areas of mathematics.”

Free online version: Higher topos theory

Although it is unlikely that I can use this left-over money from a grant to pre-order a book, let’s try

Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic ‘bridges’ by Olivia Caramello. Blurb: “According to Grothendieck, the notion of topos is “the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures”. It is what he had “conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an “essence” which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things”. ”

And, as I also teach a course on the history of mathematics, let’s include:

Tool and Object: A History and Philosophy of Category Theory by Ralph Krömer. Blurb: “This book describes the history of category theory whereby illuminating its symbiotic relationship to algebraic topology, homological algebra, algebraic geometry and mathematical logic and elaboratively develops the connections with the epistemological significance.”

A couple of days ago, there was yet another article by Philippe Douroux on Grothendieck’s Lasserre writings “Inestimables mathématiques, avez-vous donc un prix?” in the French newspaper Liberation.

Not that there is much news to report.

I’ve posted on this before: Grothendieck’s gribouillis, Grothendieck’s gribouillis (2), and more recently Where are Grothendieck’s writings?

In that last post I claimed that the five metallic cases containing Grothendieck’s Lasserre notes were in a white building behind the police station of the sixth arrondissement of Paris.

I was wrong.

There’s a detail in Douroux’ articles I forgot to follow-up before.

Here’s the correct location:

What went wrong?

Here’s my ‘translation’ of part of chapter 46 of Douroux’ book “Alexandre Grothendieck, sur les traces du dernier genie des mathematiques”:

“On November 13th 2015, while the terrorist-attacks on the Bataclan and elsewhere were going on, a Mercedes break with on board Alexandre Jr. Grothendieck and Jean-Bernard, a librarian specialised in ancient writings, was approaching Paris from Lasserre. On board: 5 metallic cases, 2 red ones, 1 green and 2 blues.

At about 2 into the night they arrived at the ‘commissariat du Police’ of the 6th arrondissement. Jean-Bernard pushed open a heavy blue carriage porch, crossed the courtyard opened a second door and then a third one and delivered the cases.”

It all seemed to fit together: the ‘commissariat’ has a courtyard (but then, so do most buildings in the neighborhood) and has a blue carriage porch:

What went wrong?

I should have trusted Google-translate instead.

It translates the original text “…il garait sa voiture pres du commissariat…” more correctly into “…he parked his car near the police station…”. ‘Near’ as apposed to ‘at’…

We should have looked for a location close to the police station.

And, I should have looked up “Jean-Bernard, a librarian specialised in ancient writings”.

Who is Jean-Bernard?

In Douroux’ latest article there’s this sentence:

“Dès lors, on comprend mieux le travail de Jean-Bernard Gillot, libraire à Paris et expert en livres anciens et manuscrits scientifiques pour lequel les cinq malles contenant les écrits de Lasserre représentent l’affaire d’une vie.”

I’m not even going to make an attempt at translation, you know which tool to use if needed. Suffice it to say that the mysterious Jean-Bernard is no other than Jean-Bernard Gillot.

In 2005, Jean-Bernard Gillot took over the Librairie Alain Brieux, specialising in ancient scientific books and objects. Here’s a brief history of this antiques shop.

Relevant to our quest is that it is located 48, rue Jacob in Paris, just around the corner of the Police Station of the 6th arrondissement.

And, there is a beautiful heavy blue carriage porch, leading to an interior courtyard…

A quick look at the vast amount of scientific objects (such as these Napier’s bones) indicates that there must be adequate and ample storage space in the buildings behind the shop.

This is where the five metallic cases containing the Lasserre writings are at this moment.

What’s next?

We’re lightyears removed from Maltsiniotis’ optimistic vision, broadcast at the Grothendieck conference in Montpellier last year, that the BNF would acquire the totality of the writings and make them available to the mathematical community at large.

Apart from Maltsiniotis’ cursory inventory of (part of) the 93.000 pages, nobody knows what’s inside these five boxes, making it impossible to put a price tag on them.

Perhaps, the family should grant some bloggers access to the cases, in return for a series of (live)posts on what they’ll find inside…?!

Please allow for a couple of end-of-semester bluesy ramblings. I just finished grading the final test of the last of five courses I lectured this semester.

Most of them went, I believe, rather well.

As always, it was fun to teach an introductory group theory course to second year physics students.

Personally, I did enjoy our Lie theory course the most, given for a mixed public of both mathematics and physics students. We did the spin-group $SU(2)$ and its connection with $SO_3(\mathbb{R})$ in gruesome detail, introduced the other classical groups, and proved complete reducibility of representations. The funnier part was applying this to the $U(1) \times SU(2) \times SU(3)$-representation of the standard model and its extension to the $SU(5)$ GUT.

Ok, but with a sad undertone, was the second year course on representations of finite groups. Sad, because it was the last time I’m allowed to teach it. My younger colleagues decided there’s no place for RT on the new curriculum.

Soit.

The final lecture is often an eye-opener, or at least, I hope it is/was.

Here’s the idea: someone whispers in your ear that there might be a simple group of order $60$. Armed with only the Sylow-theorems and what we did in this course we will determine all its conjugacy classes, its full character table, and finish proving that this mysterious group is none other than $A_5$.

Right now I’m just a tad disappointed only a handful of students came close to solving the same problem for order $168$ this afternoon.

Clearly, I gave them ample extra information: the group only has elements of order $1,2,3,4$ and $7$ and the centralizer of one order $2$ element is the dihedral group of order $8$. They had to determine the number of distinct irreducible representations, that is, the number of conjugacy classes. Try it yourself (Solution at the end of this post).

For months I felt completely deflated on Tuesday nights, for I had to teach the remaining two courses on that day.

There’s this first year Linear Algebra course. After teaching for over 30 years it was a first timer for me, and probably for the better. I guess 15 years ago I would have been arrogant enough to insist that the only way to teach linear algebra properly was to do representations of quivers…

Now, I realise that linear algebra is perhaps the only algebra course the majority of math-students will need in their further career, so it is best to tune its contents to the desires of the other colleagues: inproducts, determinants as volumes, Markov-processes and the like.

There are thousands of linear algebra textbooks, the one feature they all seem to lack is conciseness. What kept me going throughout this course was trying to come up with the shortest proofs ever for standard results. No doubt, next year the course will grow on me.

Then, there was a master course on algebraic geometry (which was supposed to be on scheme theory, moduli problems such as the classification of fat points (as in the car crash post, etale topology and the like) which had a bumpy start because class was less prepared on varieties and morphisms than I had hoped for.

Still, judging on the quality of the papers students are beginning to hand in (today I received one doing serious stuff with stacks) we managed to cover a lot of material in the end.

I’m determined to teach that first course on algebraic geometry myself next year.

Which brought me wondering about the ideal content of such a course.

Half a decade ago I wrote a couple of posts such as Mumford’s treasure map, Grothendieck’s functor of points, Manin’s geometric axis and the like, which are still quite readable.

In the functor of points-post I referred to a comment thread Algebraic geometry without prime ideals at the Secret Blogging Seminar.

As I had to oversee a test this afternoon, I printed out all comments (a full 29 pages!) and had a good time reading them. At the time I favoured the POV advocated by David Ben-Zvi and Jim Borger (functor of points instead of locally ringed schemes).

Clearly they are right, but then so was I when I thought the ‘right’ way to teach linear algebra was via quiver-representations…

We’ll see what I’ll try out next year.

You may have wondered about the title of this post. It’s derived from a paper Raf Bocklandt (of the Korteweg-de Vries Institute in Amsterdam) arXived some days ago: Reflections in a cup of coffee, which is an extended version of a Brouwer-lecture he gave. Raf has this to say about the Brouwer fixed-point theorem:

“The theorem is usually explained in worldly terms by looking at a cup of coffee. In this setting it states that no matter how you stir your cup, there will always be a point in the liquid that did not change position and if you try to move that part by further stirring you will inevitably move some other part back into its original position. Legend even has it that Brouwer came up with the idea while stirring in a real cup, but whether this is true we’ll never know. What is true however is that Brouwers refections on the topic had a profound impact on mathematics and would lead to lots of new developments in geometry.”

I wish you all a pleasant end of 2016 and a much better 2017.

—

As to the 168-solution: Sylow says there are 8 7-Sylows giving 48 elements of order 7. The centralizer of each of them must be $C_7$ (given the restriction on the order of elements) so two conjugacy classes of them. Similarly each conjugacy class of an order 3 element must contain 56 elements. There is one conjugacy class of an order 2 element having 21 elements (because the centralizer is $D_4$) giving also a conjugacy class of an order 4 element consisting of 42 elements. Together with the identity these add up to 168 so there are 6 irreducible representations.

Geometers will tell you there are two ways to introduce affine schemes.

You can use structure sheaves. That is, compute all prime ideals of your ring and turn them into a space. Then, put a sheaf of rings on this space by localisation. You’ll get your ring back taking global sections.

Or, you might try the ‘functor of points’. That is, you take any other ring. Compute all ring-morphisms from yours to that one. You’ll recover your ring from Yoneda’s lemma.

And here’s the funny part.

Scheme-theorists claim there’s no differences between these two approaches. They are ‘equivalent’, as they prefer to say.

Do you believe them?

Let’s look at an example.

Take the ring of all polynomials with integer coefficients, $\mathbb{Z}[x]$.

Do you know all its prime ideals?

Sure, you’ll say.

There’s zero because it’s a domain. Then there are the ‘curves’. These are all prime numbers and all irreducible polynomials because it’s a UFD.

And then there are the ‘points’. They depend on a prime number $p$ and an irreducible polynomial which does not factor over $p$.

Not exactly rocket science, is it?

Okay, now let’s take them all together into a space.

Can you picture the intersection points of different curves? Let’s keep it simple. Take the curve given by a prime number $p$ and the one given by an irreducible polynomial $F(x)$. How do they intersect?

Easy! They are the factors of $F(x)$ modulo $p$.

Right, but can you picture this pattern for all prime numbers at once?

That depends on $F(x)$. David Mumford sketched the situation for $x^2+1$.

If $-1$ is a square modulo $p$, then $F(x)$ splits in two factors giving two points, such as along $5$. If not, $F(x)$ remains irreducible over $p$ and gives a thicker point like over $3$ or $7$. Except for the ‘odd’ case over $2$ where $F(x)$ is a square. Gauss knew already the situation for every prime.

But, what about arbitrary polynomials?

That’s a lot more difficult. Chebotarev knew how to get their Galois group from the factors at all primes.

So, you’ll need to solve deep problems in number theory before you can picture this space. The structure of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ to name one.

I’m afraid nobody understands the space of all prime ideals of $\mathbb{Z}[x]$ completely, let alone its structure sheaf.

What about the other approach? Let’s try to understand the functor of points of $\mathbb{Z}[x]$.

Take any ring $R$. We need to figure out all ring-maps $\mathbb{Z}[x] \rightarrow R$. But, we know such a map once we know the image of $x$. That is, there are as many ring-maps as there are elements in the set $R$.

Forgetting all about addition and multiplication on $R$. It is just the forgetful functor from rings to sets.

And they claim this is equivalent to solving deep problems in number theory?

Forgetting can’t be that hard, can it?

The Boston subway is a complex system, spreading out from a focus at Park Street.

On March 3rd, the Boylston shuttle went into service, 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…

Never heard of this tragic incident?

Time to read up on A.J. Deutsch’s classic ‘A subway named Moebius’ from 1950. A 12 page pdf of this short story is available via the Rio Rancho Math Camp.

The ‘explanation’ given in the story is that the Moebius strip has a singularity. Before you yell that this is impossible, have a look at this or that.

A ‘non spatial network’ where ‘an exclusion principle operates’, Deutsch’s story says.

Here’s another take.

The train took the exceptional fiber branch, instead of remaining on the desingularisation?

Whatever really happened, it’s a fun read, mathematics clashing with bureaucracy.

In 1996 Gustavo Mosquera directed the film ‘Moebius’, set in Buenos Aires, loosely based on Deutsch’s story.

Here’s the full version (90 min.), with subtitles. Have fun!

MOEBIUS dirigido por Gustavo Mosquera from Universidad del Cine on Vimeo.

Soon, we will be teaching computational geometry courses to football commentators.

If a player is going to be substituted we’ll hear sentences like: “no surprise he’s being replaced, his Voronoi cell has been shrinking since the beginning of the second half!”

David Sumpter, the author of Soccermatics: Mathematical Adventures in the Beautiful Game, wrote a nice article over at Medium The geometry of attacking football.

As an example, he took an attack of Barcelona against Panathinaikos.

and explained the passing possibilities in terms of the Delaunay triangulation between the Barca-players (the corresponding Voronoi cell decomposition is in the header picture).

He concludes: “It is not only their skill on the ball, but also their geometrically accurate positioning that allows them to make the pass.”

Jaime Sampaoi produced a short video of changing Voronoi cells from kick-off by the blue team, with the red team putting pressure until a faulty pass is given, leading to a red-attack and a goal. All in 29 seconds.

I’d love to turn this feature on when watching an actual game.

Oh, and please different cell-colours for the two teams.

And, a remote control to highlight the Voronoi cell of a particular player.

Please?

Today, Samuel Dehority, Xavier Gonzalez, Neekon Vafa and Roger Van Peski arXived their paper Moonshine for all finite groups.

Originally, Moonshine was thought to be connected to the Monster group. McKay and Thompson observed that the first coefficients of the normalized elliptic modular invariant

\[
J(\tau) = q^{-1} + 196884 q + 21493760 q^2 + 864229970 q^3 + \ldots
\]

could be written as sums of dimensions of the first few irreducible representations of the monster group:

\[
1=1,~\quad 196884=196883+1,~\quad 21493760=1+196883+21296876,~\quad … \]

Soon it transpired that there ought to be an infinite dimensional graded vectorspace, the moonshine module

\[
V^{\sharp} = \bigoplus_{n=-1}^{\infty}~V^{\sharp}_n \]

with every component $V^{\sharp}_n$ being a representation of the monster group $\mathbb{M}$ of which the dimension coincides with the coefficient of $q^n$ in $J(\tau)$.

It only got better, for any conjugacy class $[ g ]$ of the monster, if you took the character series

\[
T_g(\tau) = \sum_{n=-1}^{\infty} Tr(g | V^{\sharp}_n) q^n \]

you get a function invariant under the action of the subgroup

\[
\Gamma_0(n) = \{ \begin{bmatrix} a & b \\ c & d \end{bmatrix}~:~c = 0~mod~n \} \]

acting via transformations $\tau \mapsto \frac{a \tau + b}{c \tau + d}$ on the upper half plane where $n$ is the order of $g$ (or, for the experts, almost).

Soon, further instances of ‘moonshine’ were discovered for other simple groups, the unifying feature being that one associates to a group $G$ a graded representation $V$ such that the character series of this representation for an element $g \in G$ is an invariant modular function with respect to the subgroup $\Gamma_0(n)$ of the modular group, with $n$ being the order of $g$.

Today, this group of people proved that there is ‘moonshine’ for any finite group whatsoever.

They changed the definition of moonshine slightly to introduce the notion of moonshine of depth $d$ which meant that they want the dimension sequence of their graded module to be equal to $J(\tau)$ under the action of the normalized $d$-th Hecke operator, which means equal to

\[
\sum_{ac=d,0 \leq b < c} J(\frac{a \tau + b}{c}) \]
as they are interested in the asymptotic behaviour of the components $V_n$ with respect to the regular representation of $G$.

What baffled me was their much weaker observation (remark 2) saying that you get ‘moonshine’ in the form described above, that is, a graded representation $V$ such that for every $g \in G$ you get a character series which is invariant under $\Gamma_0(n)$ with $n=ord(g)$ (and no smaller divisor of $n$), for every finite group $G$.

And, more importantly, you can explain this to any student taking a first course in group theory as all you need is Cayley’s theorem stating that any finite group is a subgroup of some symmetric group $S_n$.

Here’s the idea: take the original monster-moonshine module $V^{\sharp}$ but forget all about the action of $\mathbb{M}$ (that is, consider it as a plain vectorspace) and consider the graded representation

\[
V = (V^{\sharp})^{\otimes n} \]

with the natural action of $S_n$ on the tensor product.

Now, embed a la Cayley $G$ into $S_n$ then you know that the order of $g \in G$ is the least common multiple of the cycle lengths of the permutation it it send to. Now, it is fairly trivial to see that the character series of $V$ with respect to $g$ (having cycle lengths $(k_1,k_2,\dots,k_l)$, including cycles of length one) is equal to the product

\[
J(k_1 \tau) J(k_2 \tau) \dots J(k_l \tau) \]

which is invariant under $\Gamma_0(n)$ with $n = lcm(k_i)$ (but no $\Gamma_0(m)$ with $m$ a proper divisor of $n$).

For example, for $G=S_4$ we have as character series of $(V^{\sharp})^{\otimes 4}$

\[
(1)(2)(3)(4) \mapsto J(\tau)^4 \]

\[
(12)(3)(4) \mapsto J(2 \tau) J(\tau)^2 \]

\[
(12)(34) \mapsto J(2 \tau)^2 \]

\[
(123)(4) \mapsto J(3 \tau) J(\tau) \]

\[
(1234) \mapsto J(4 \tau) \]

Clearly, the main results of the paper are much more subtle, but I’m already happy with this version of ‘moonshine for everyone’!

In several of his talks on #IUTeich, Mochizuki argues that usual scheme theory over $\mathbb{Z}$ is not suited to tackle problems such as the ABC-conjecture.

The idea appears to be that ABC involves both the additive and multiplicative nature of integers, making rings into ‘2-dimensional objects’ (and clearly we use both ‘dimensions’ in the theory of schemes).

So, perhaps we should try to ‘dismantle’ scheme theory, and replace it with something like geometry over the field with one element $\mathbb{F}_1$.

The usual $\mathbb{F}_1$ mantra being: ‘forget all about the additive structure and only retain the multiplicative monoid’.

So perhaps there is yet another geometry out there, forgetting about the multiplicative structure, and retaining just the addition…

This made me wonder.

In the forgetting can’t be that hard, can it?-post we have seen that the forgetful functor

\[
F_{+,\times}~:~\mathbf{rings} \rightarrow \mathbf{sets} \]

(that is, forgetting both multiplicative and additive information of the ring) is representable by the polynomial ring $\mathbb{Z}[x]$.

So, what about our ‘dismantling functors’ in which we selectively forget just one of these structures:

\[
F_+~:~\mathbf{rings} \rightarrow \mathbf{monoids} \quad \text{and} \quad F_{\times}~:~\mathbf{rings} \rightarrow \mathbf{abelian~groups} \]

Are these functors representable too?

Clearly, ring maps from $\mathbb{Z}[x]$ to our ring $R$ give us again the elements of $R$. But now, we want to encode the way two of these elements add (or multiply).

This can be done by adding extra structure to the ring $\mathbb{Z}[x]$, namely a comultiplication $\Delta$ and a counit $\epsilon$

\[
\Delta~:~\mathbb{Z}[x] \rightarrow \mathbb{Z}[x] \otimes \mathbb{Z}[x] \quad \text{and} \quad \epsilon~:~\mathbb{Z}[x] \rightarrow \mathbb{Z} \]

The idea of the comultiplication being that if we have two elements $r,s \in R$ with corresponding ring maps $f_r~:~\mathbb{Z}[x] \rightarrow R \quad x \mapsto r$ and $f_s~:~\mathbb{Z}[x] \rightarrow R \quad x \mapsto s$, composing their tensorproduct with the comultiplication

\[
f_v~:~\mathbb{Z}[x] \rightarrow^{\Delta} \mathbb{Z}[x] \otimes \mathbb{Z}[x] \rightarrow^{f_r \otimes f_s} R
\]

determines another element $v \in R$ which we can take either the product $v=r.s$ or sum $v=r+s$, depending on the comultiplication map $\Delta$.

The role of the counit is merely sending $x$ to the identity element of the operation.

Thus, if we want to represent the functor forgetting the addition, and retaining the multiplication we have to put on $\mathbb{Z}[x]$ the structure of a biring

\[
\Delta(x) = x \otimes x \quad \text{and} \quad \epsilon(x) = 1 \]

(making $x$ into a ‘group-like’ element for Hopf-ists).

The functor $F_{\times}$ forgetting the multiplication but retaining the addition is represented by the Hopf-ring $\mathbb{Z}[x]$, this time with

\[
\Delta(x) = x \otimes 1 + 1 \otimes x \quad \text{and} \quad \epsilon(x) = 0 \]

(that is, this time $x$ becomes a ‘primitive’ element).

Perhaps this adds another feather of weight to the proposal in which one defines algebras over the field with one element $\mathbb{F}_1$ to be birings over $\mathbb{Z}$, with the co-ring structure playing the role of descent data from $\mathbb{Z}$ to $\mathbb{F}_1$.

As, for example, in my note The coordinate biring of $\mathbf{Spec}(\mathbb{Z})/\mathbb{F}_1$.

Since mid May the Montpellier part of Grothendieck’s gribouillis are online and for everyone available at the Archives Grothendieck.

The story is well-known.

End of June 1990, Grothendieck phoned Jean Malgoire warning him to come asap if he wanted to safeguard the better part of G’s mathematical archive, for he was making a bonfire…

A second handover apparently took place on July 28th 1995.

Malgoire kept these notes (in huge Pampers boxes!) until 2010 when he got cold feet as a result of Grothendieck’s letter. He then donated the boxes to the University of Montpellier in 2012.

After Grothendieck’s death in 2014, Montpellier started a project to scan each and every page and put them online, with the backing of Grothendieck’s children (and generous financial support from the local authorities).

So here we are now, and… nobody seems to care.

I’m aware only of this post on MathOverflow by someone who wants to LaTex some of the material on motives.

Perhaps this is due to the less than optimal presentation of the material, or more likely, Grothendieck’s terrible handwriting. Perhaps the University of Montpellier should partner up with the (older generation of) French pharmacists.

But then, there’s this artistic gem in the archive: cote 154 systemes the pseudo-droites written in 1983-84.

It is written on ancient matrix-plotter page. Here’s a typical example

Which mathematical department wouldn’t want to acquire a framed version of one of these original pages?

That’s the point I wanted to make early may in this G+-post, hoping to raise money to safeguard the Lasserre part of Grothendieck’s gribouillis.

When in need for a header picture, I’ll use a page of Grothendieck’s gallery No 154 from now on.

Here’s evidence that Grothendieck was working on GUTS! (literally).

At the moment I’m re-reading Siobhan Roberts’ biography of John Horton Conway, Genius at play – the curious mind of John Horton Conway.

In fact, I’m also re-reading Alexander Masters’ biography of Simon Norton, The genius in my basement – the biography of a happy man.

If you’re in for a suggestion, try to read these two books at about the same time. I believe it is beneficial to both stories.

Whatever. Sooner rather than later the topic of Conway’s game of life pops up.

Conway’s present pose is to yell whenever possible ‘I hate life!’. Problem seems to be that in book-indices in which his name is mentioned (and he makes a habit of checking them all) it is for his invention of the game of Life, and not for his greatest achievement (ihoo), the discovery of the surreal numbers.

If you have an hour to spare (btw. enjoyable), here are Siobhan Roberts and John Conway, giving a talk at Google: “On His LOVE/HATE Relationship with LIFE”

By synchronicity I encounter the game of life now wherever I look.

Today it materialised in following up on an old post by Richard Green on G+ on Gaussian primes.

As you know the Gaussian integers $\mathbb{Z}[i]$ have unique factorization and its irreducible elements are called Gaussian primes.

The units of $\mathbb{Z}[i]$ are $\{ \pm 1,\pm i \}$, so Gaussian primes appear in $4$- or $8$-tuples having the same distance from the origin, depending on whether a prime number $p$ remains prime in $\mathbb{Z}[i]$ or splits.

Here’s a nice picture of Gaussian primes, taken from Oliver Knill’s paper Some experiments in number theory

Note that the natural order of prime numbers is changed in the process (look at the orbits of $3$ and $5$ (or $13$ and $17$).

Because the lattice of Gaussian integers is rectangular we can look at the locations of all Gaussian primes as the living cell in the starting position on which to apply the rules of Life.

Here’s what happens after one move (left) and after three moves (right):

Knill has a page where you can watch life on Gaussian primes in action.

Even though the first generations drastically reduce the number of life spots, you will see that there remains enough action, at least close enough to the origin.

Knill has this conjecture:

When applying the game of life cellular automaton to the Gaussian primes, there is motion arbitrary far away from the origin.

What’s the point?

Well, this conjecture is equivalent to the twin prime conjecture for the Gaussian integers $\mathbb{Z}[i]$, which is formulated as

“there are infinitely pairs of Gaussian primes whose Euclidian distance is $\sqrt{2}$.”

Next week, the brand new séminaire « Lectures grothendieckiennes » will kick off on Tuesday October 24th at 18hr (h/t Isar Stubbe).

There will be one talk a month, on a tuesday evening from 18hr-20hr. Among the lecturers are the ‘usual suspects’:

Pierre Cartier (October 24th) will discuss the state of functional analysis before Grothendieck entered the scene in 1948 and effectively ‘killed the subject’ (said Dieudonné).

Alain Connes (November 7th) will talk on the origins of Grothendieck’s introduction of toposes.

In fact, toposes will likely be a recurrent topic of the seminar.

Laurant Laforgue‘s title will be ‘La notion de vérité selon Grothendieck'(January 9th) and on March 6th there will be a lecture by Olivia Caramello.

Also, Colin McLarty will speak about them on May 3rd: “Nonetheless one should learn the language of topos: Grothendieck on building houses”.

The closing lecture will be delivered by Georges Maltsiniotis on June 5th 2018.

Further Grothendieck news, there’s the exhibition of a sculpture by Nina Douglas, the wife of Michael Douglas, at the Simons Center for Geometry and Physics (h/t Jason Starr).

It depicts Grothendieck as shepherd. The lambs in front of him have Riemann surfaces inserted into them and on the staff is Grothendieck’s ‘Hexenkuche’ (his proof of the Riemann-Roch theorem).

Getting mathematics into Nature (the journal) is next to impossible. Ask David Mumford and John Tate about it.

Last month, John Duncan, Michael Mertens and Ken Ono managed to do just that.

Inevitably, they had to suffer through a photoshoot and give their university’s PR-people some soundbites.

CAPTION

In the simplest terms, an elliptic curve is a doughnut shape with carefully placed points, explain Emory University mathematicians Ken Ono, left, and John Duncan, right. “The whole game in the math of elliptic curves is determining whether the doughnut has sprinkles and, if so, where exactly the sprinkles are placed,” Duncan says.

CAPTION

“Imagine you are holding a doughnut in the dark,” Emory University mathematician Ken Ono says. “You wouldn’t even be able to decide whether it has any sprinkles. But the information in our O’Nan moonshine allows us to ‘see’ our mathematical doughnuts clearly by giving us a wealth of information about the points on elliptic curves.”

(Photos by Stephen Nowland, Emory University. See here and here.)

Some may find this kind of sad, or a bad example of over-popularisation.

I think they do a pretty good job of getting the notion of rational points on elliptic curves across.

That’s what the arithmetic of elliptic curves is all about, finding structure in patterns of sprinkles on special doughnuts. And hey, you can get rich and famous if you’re good at it.

Their Nature-paper Pariah moonshine is a must-read for anyone aspiring to write a math-book aiming at a larger audience.

It is an introduction to and a summary of the results they arXived last February O’Nan moonshine and arithmetic.

Update (October 21st)

John Duncan send me this comment via email:

“Strictly speaking the article was published in Nature Communications (https://www.nature.com/ncomms/). We were also rejected by Nature. But Nature forwarded our submission to Nature Communications, and we had a great experience. Specifically, the review period was very fast (compared to most math journals), and the editors offered very good advice.

My understanding is that Nature Communications is interested in publishing more pure mathematics. If someone reading this has a great mathematical story to tell, I (humbly) recommend to them this option. Perhaps the work of Mumford–Tate would be more agreeably received here.

By the way, our Nature Communications article is open access, available at https://www.nature.com/articles/s41467-017-00660-y.”

Earlier today, John Duncan (of moonshine fame) emailed he was unable to post a comment to the previous post:
“I went to post a comment but somehow couldn’t convince the website to cooperate.”

There’s little point in maintaining a self-hosted blog if people cannot comment on it. If you tried, you got this scary message:

Catchable fatal error: Object of class WP_Error could not be converted to string in /wp-includes/formatting.php on line 1031

The days I meddled with wordpress core php-files are long gone, and a quick Google search didn’t come up with anything helpful.

In despair, there’s always the database to consider.

Here’s a screenshot of this blog’s database in phpMyAdmin:

No surprise you cannot comment here, there isn’t even a wp_comments table in the database! (though surprisingly, there’s a table wp_commentmeta…)

Two weeks ago I moved this blog to a new iMac. Perhaps the database got corrupted in the process, or the quick export option of phpMyAdmin doesn’t include comments (unlikely), or whatever.

Here’s what I did to get things working again. It may solve your problem if you don’t have a backup of another wordpress-blog with a functional wp_comments table.

1. Set up a new WordPress blog in the usual way, including a new database, let’s call it ‘newblog’.

2. In phpMyAdmin drop all tables in newblog except for wp_comments.

3. Export your blog’s database, say ‘oldblog’, via the ‘quick export’ option in phpMyAdmin to get a file oldblog.sql.

4. If this file is small you can use phpMyAdmin to import it into newblog. If not you need to do it with this terminal-command

mysql -h localhost -u root – p newblog

and have the patience for this to finish.

5. Change in your wp-config file the oldblog database to newblog.

Happy commenting!

Once in every six months there’s a flurry of online excitement about Mochizuki’s alleged proof of the abc-conjecture.

It seems to be that time of the year again.

The twitter-account of the ever optimistic @math_jin is probably the best source for (positive) news about IUT/ABC. He now announces the latest version of Yamashita’s ‘summary’ of Mochizuki’s proof:

山下剛さんのIUTサーベイが更新されました。
Go Yamashita
A proof of the abc conjecture after Mochizuki.
preprint. last updated on 18/Nov/2017.https://t.co/XtnMEO3zoQ#IUTABC

— math_jin (@math_jin) November 18, 2017

Another informed source is Ed Frenkel. He sometimes uses his twitter-account @edfrenkel to broadcast Ivan Fesenko‘s enthusiasm.

Big news on Mochizuki's groundbreaking IUT: Over 1000 comments on his 4 papers have been addressed & the final versions sent back to the journal for approval. Hopefully, will be published soon.
Here's Ivan Fesenko's interview about IUT on the AMS website.https://t.co/6GLk3Xh0lm

— Edward Frenkel (@edfrenkel) November 17, 2017

Googling further, I stumbled upon an older (newspaper) article on the subject: das grosse ABC by Marlene Weiss, for which she got silver at the 2017 science journalism awards.

In case you prefer an English translation: The big ABC.

Here’s her opening paragraph:

“In a children’s story written by the Swiss author Peter Bichsel, a lonely man decides to invent his own language. He calls the table “carpet”, the chair “alarm clock”, the bed “picture”. At first he is enthusiastic about his idea and always thinks of new words, his sentences sound original and funny. But after a while, he begins to forget the old words.”

The article is less optimistic than other recent popular accounts of Mochizuki’s story, including:

Monumental proof to torment mathematicians for years to come in Nature by Davide Castelvecchi.

Hope Rekindled for Perplexing Proof in Quanta-magazine by Kevin Hartnett.

Baffling ABC maths proof now has impenetrable 300-page ‘summary’ in the New Scientist by Timothy Revell.

Marlene Weiss fears a sad ending:

“Table is called “carpet”, chair is called “alarm clock”, bed is called “picture”. In the story by Peter Bichsel, the lonely man ends up having so much trouble communicating with other people that he speaks only to himself. It is a very sad story.”

Perhaps things will turn out for the better, and we’ll hear about it sometime.

In six months, I’d say…

A few weeks more of (heavy) teaching ahead, and then I finally hope to start on a project, slumbering for way too long: to write a book for a broader audience.

Prepping for this I try to read most of the popular math-books hitting the market.

The latest two explore how the internet changed the way we discuss, learn and do mathematics. Think Math-Blogs, MathOverflow and Polymath.

‘Gina says’, Adventures in the Blogosphere String War

The ‘string wars’ started with the publication of the books by Peter Woit:

Not even wrong: the failure of string theory and the search for unity in physical law

and Lee Smolin:

The trouble with physics: the rise of string theory, the fall of a science, and what comes next.

In the summer of 2006, Gil Kalai got himself an extra gmail acount, invented the fictitious ‘Gina’ and started commenting (some would argue trolling) on blogs such as Peter Woit’s own Not Even Wring, John Baez and Co.’s the n-Category Cafe and Clifford Johnson’s Asymptotia.

Gil then copy-pasted Gina’s comments, and the replies they provoked, into a leaflet and put it on his own blog in June 2009: “Gina says”, Adventures in the Blogosphere String War.

Back then, it was fun to waste an afternoon re-reading all of this, and I wrote about it here:

Now here’s an idea (June 2009)

Gina says, continued (August 2009)

With only minor editing, and including some drawings by Gil’s daughter, these leaflets have now resurfaced as a book…?!

After more than 10 years I had hoped that Gil would have taken this test-case to say some smart things about the math-blogging scene and its potential to attract more people to mathematics, or whatever.

In 2009 I wrote:

“Having read the first 20 odd pages in full and skimmed the rest, two remarks : (1) it shouldn’t be too difficult to borrow this idea and make a much better book out of it and (2) it raises the question about copyrights on blog-comments…”

Closing the gap: the quest to understand prime numbers

I can hear you sigh, but no, this is not yet another prime number book.

In May 2013, Yitang Zhang startled the mathematical world by proving that there are infinitely many prime pairs, each no more than 70.000.000 apart.

Perhaps a small step towards the twin prime conjecture but it was the first time someone put a bound on this prime gap.

Vicky Neal‘s book tells the story of closing this gap. In less than a year the bound of 70.000.000 was brought down to 246.

If you’ve read all popular prime books, there are a handful of places in the book where you might sigh: ‘oh no, not that story again’, but by far the larger part of the book explains exciting results on prime number progressions, not found anywhere else.

Want to know about sieve methods?

Which results made Tim Gowers or Terry Tao famous?

What is Szemeredi’s theorem or the Hardy-Littlewood circle method?

Ever heard about the Elliot-Halberstam or the Erdos-Turan conjecture? The work by Tao on Erdos discrepancy problem or that of James Maynard (and Tao) on closing the prime gap?

Closing the gap is the book to read about all of this.

But it is much more.

It tells about the origins and successes of the Polymath project, and details the progress made by Polymath8 on closing the gap, it gives an insight into how mathematics is done, what role conferences, talks and research institutes a la Oberwolfach play, and more.

Looking for a gift for that niece of yours interested in maths? Look no further. Closing the gap is a great book!

Consider subscribing to the feed of the mathematics section of Quantamagazine.

The articles there are invariably of high quality and quite informative.

Their latest is Secret Link Uncovered Between Pure Math and Physics by Kevin Hartnett.

It features the work by number-theorist Minhyong Kim of Oxford University.

In it, Minhyong Kim comes out of the closet, revealing that many of his results on rational points of algebraic curves were inspired by analogies he sees between number theory and physics.

So far he refrained from mentioning this inspiration in papers because “Number theorists are a pretty tough-minded group of people,” he said.

Yesterday, Peter Woit had a post on this on his blog ‘Not Even Wrong’, stuffed with interesting links to recent talks and papers by Minhyong Kim.

Minhyong Kim’s ideas grew out the topic of arithmetic topology, that is, the analogy between number rings and $3$-dimensional compact manifolds and between their prime ideals and embedded knots.

In this analogy, which is based on the similarity between finite connected covers of manifolds on the one hand and connected etale extensions of rings on the other, the prime spectrum of $\mathbb{Z}$ should correspond (due to Minkowski’s result on discriminants and Perelman’s proof of the Poincare-conjecture) to the $3$-sphere $S^3$.

I’ve written more about this analogy here:

– Mazur’s knotty dictionary.

– What is the knot associated to a prime?

– Who dreamed up the knots=primes analogy?

– The birthday of the primes=knots analogy.

And probably I’ll mention it later this month when I give a couple of talks at the $\mathbb{F}_1$-seminar in Ghent.

Last night, on our way to the fireworks in Antwerp, we walked by this definition of prime numbers:

“The numbers, only divisible by $1$ and itself are: $2,3$ and every number before or after a multiple of $6$, without their squares or products.” (Peter Wynen)

True enough.

And a lot more user-friendly than: the generators of the multiplicative monoid of all natural numbers which are $\pm 1$ modulo $6$ are the prime numbers, except for $2$ and $3$.

I wish you a 2018 full of math (and artistic) pleasures.

From Wikipedia on 24:

“$24$ is the only number whose divisors, namely $1, 2, 3, 4, 6, 8, 12, 24$, are exactly those numbers $n$ for which every invertible element of the commutative ring $\mathbb{Z}/n\mathbb{Z}$ is a square root of $1$. It follows that the multiplicative group $(\mathbb{Z}/24\mathbb{Z})^* = \{ \pm 1, \pm 5, \pm 7, \pm 11 \}$ is isomorphic to the additive group $(\mathbb{Z}/2\mathbb{Z})^3$. This fact plays a role in monstrous moonshine.”

Where did that come from?

In the original “Monstrous Moonshine” paper by John Conway and Simon Norton, section 3 starts with:

“It is a curious fact that the divisors $h$ of $24$ are precisely those numbers $h$ for which $x.y \equiv 1~(mod~h)$ implies $x \equiv y~(mod~h)$.”

and a bit further they even call this fact:

“our ‘defining property of $24$'”.

The proof is pretty straightforward.

We want all $h$ such that every unit in $\mathbb{Z}/h \mathbb{Z}$ has order two.

By the Chinese remainder theorem we only have to check this for prime powers dividing $h$.

$5$ is a unit of order $4$ in $\mathbb{Z}/16 \mathbb{Z}$.

$2$ is a unit of order $6$ in $\mathbb{Z}/ 9 \mathbb{Z}$.

A generator of the cyclic group $(\mathbb{Z}/p\mathbb{Z})^*$ is a unit of order $p-1 > 2$ in $\mathbb{Z}/p \mathbb{Z}$, for any prime number $p \geq 5$.

This only leaves those $h$ dividing $2^3.3=24$.

But, what does it have to do with monstrous moonshine?

Moonshine assigns to elements of the Monster group $\mathbb{M}$ a specific subgroup of $SL_2(\mathbb{Q})$ containing a cofinite congruence subgroup

\[
\Gamma_0(N) = \{ \begin{bmatrix} a & b \\ cN & d \end{bmatrix}~|~a,b,c,d \in \mathbb{Z}, ad-Nbc = 1 \} \]

for some natural number $N = h.n$ where $n$ is the order of the monster-element, $h^2$ divides $N$ and … $h$ is a divisor of $24$.

To begin to understand how the defining property of $24$ is relevant in this, take any strictly positive rational number $M$ and any pair of coprime natural numbers $g fixes $M \frac{g}{h}$ if we have that
\[
\alpha_{M\frac{g}{h}} \Gamma_0(N) \alpha_{M\frac{g}{h}}^{-1} \subset SL_2(\mathbb{Z}) \]

For those in the know, $M \frac{g}{h}$ stands for the $2$-dimensional integral lattice
\[
\mathbb{Z} (M \vec{e}_1 + \frac{g}{h} \vec{e}_2) \oplus \mathbb{Z} \vec{e}_2 \]
and the condition tells that $\Gamma_0(N)$ preserves this lattice under base-change (right-multiplication).

In “Understanding groups like $\Gamma_0(N)$” Conway describes the groups appearing in monstrous moonshine as preserving specific finite sets of these lattices.

For this, it is crucial to determine all $M\frac{g}{h}$ fixed by $\Gamma_0(N)$.

\[
\alpha_{M\frac{g}{h}}.\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}.\alpha_{M\frac{g}{h}}^{-1} = \begin{bmatrix} 1 & M \\ 0 & 1 \end{bmatrix} \]

so we must have that $M$ is a natural number, or that $M\frac{g}{h}$ is a number-like lattice, in Conway-speak.

\[
\alpha_{M\frac{g}{h}}.\begin{bmatrix} 1 & 0 \\ N & 1 \end{bmatrix}.\alpha_{M\frac{g}{h}}^{-1} = \begin{bmatrix} 1 + \frac{Ng}{Mh} & – \frac{Ng^2}{Mh^2} \\ \frac{N}{M} & 1 – \frac{Ng}{Mh} \end{bmatrix} \]

so $M$ divides $N$, $Mh$ divides $Ng$ and $Mh^2$ divides $Ng^2$. As $g$ and $h$ are coprime it follows that $Mh^2$ must divide $N$.

Now, for an arbitrary element of $\Gamma_0(N)$ we have

\[
\alpha_{M\frac{g}{h}}.\begin{bmatrix} a & b \\ cN & d \end{bmatrix}.\alpha_{M\frac{g}{h}}^{-1} = \begin{bmatrix} a + c \frac{Ng}{Mh} & Mb – c \frac{Ng^2}{Mh^2} – (a-d) \frac{g}{h} \\ c \frac{N}{M} & d – c \frac{Ng}{Mh} \end{bmatrix} \]
and using our divisibility requirements it follows that this matrix belongs to $SL_2(\mathbb{Z})$ if $a-d$ is divisible by $h$, that is if $a \equiv d~(mod~h)$.

We know that $ad-Nbc=1$ and that $h$ divides $N$, so $a.d \equiv 1~(mod~h)$, which implies $a \equiv d~(mod~h)$ if $h$ satisfies the defining property of $24$, that is, if $h$ divides $24$.

Concluding, $\Gamma_0(N)$ preserves exactly those lattices $M\frac{g}{h}$ for which
\[
1~|~M~|~\frac{N}{h^2}~\quad~\text{and}~\quad~h~|~24 \]

A first step towards figuring out the Moonshine Picture.

Conway’s Big Picture consists of all pairs of rational numbers $M,\frac{g}{h}$ with $M > 0$ and $0 \leq \frac{g}{h} last time that $M,\frac{g}{h}$ stands for the lattice
\[
\mathbb{Z} (M \vec{e}_1 + \frac{g}{h} \vec{e}_2) \oplus \mathbb{Z} \vec{e}_2 \subset \mathbb{Q}^2 \]
and we associate to it the rational $2 \times 2$ matrix
\[
\alpha_{M,\frac{g}{h}} = \begin{bmatrix} M & \frac{g}{h} \\ 0 & 1 \end{bmatrix} \]

If $M$ is a natural number we write $M \frac{g}{h}$ and call the corresponding lattice number-like, if $g=0$ we drop the zero and write $M$.

The Big Picture carries a wealth of structures. Today, we will see that it can be factored as the product of Bruhat-Tits buildings for $GL_2(\mathbb{Q}_p)$, over all prime numbers $p$.

Here’s the factor-building for $p=2$, which is a $3$-valent tree:

To see this, define the distance between lattices to be
\[
d(M,\frac{g}{h}~|~N,\frac{i}{j}) = log~Det(q(\alpha_{M,\frac{g}{h}}.\alpha_{N,\frac{i}{j}}^{-1})) \]
where $q$ is the smallest strictly positive rational number such that $q(\alpha_{M,\frac{g}{h}}.\alpha_{N,\frac{i}{j}}^{-1}) \in GL_2(\mathbb{Z})$.

We turn the Big Picture into a (coloured) graph by drawing an edge (of colour $p$, for $p$ a prime number) between any two lattices distanced by $log(p)$.

\[
\xymatrix{M,\frac{g}{h} \ar@[red]@{-}[rr]|p & & N,\frac{i}{j}} \qquad~\text{iff}~\qquad d(M,\frac{g}{h}~|~N,\frac{i}{j})=log(p) \]

The $p$-coloured subgraph is $p+1$-valent.

The $p$-neighbours of the lattice $1 = \mathbb{Z} \vec{e}_1 \oplus \mathbb{Z} \vec{e}_2$ are precisely these $p+1$ lattices:

\[
p \qquad \text{and} \qquad \frac{1}{p},\frac{k}{p} \qquad \text{for} \qquad 0 \leq k

To check that the $p$-coloured subgraph is indeed the Bruhat-Tits building of $GL_2(\mathbb{Q}_p)$ it remains to see that it is a tree.

For this it is best to introduce $p+1$ operators on lattices

\[
p \ast \qquad \text{and} \qquad \frac{k}{p} \ast \qquad \text{for} \qquad 0 \leq k This follows from the fact that the operator
\[
(\frac{k_n}{p} \ast) \circ \dots \circ (\frac{k_2}{p} \ast) \circ (\frac{k_1}{p} \ast) \]
is given by left-multiplication with the matrix
\[
\begin{bmatrix} \frac{1}{p^n} & \frac{k_1}{p^n}+\frac{k_2}{p^{n-1}}+\dots+\frac{k_n}{p} \\ 0 & 1 \end{bmatrix} \]
which determines the order in which the $k_i$ occur.

A lattice at distance $n log(p)$ from $1$ can be uniquely written as
\[
(\frac{k_{n-l}}{p} \ast) \circ \dots \circ (\frac{k_{l+1}}{p} \ast) \circ (p^l \ast) 1 \]
which gives us the unique path to it from $1$.

The Big Picture itself is then the product of these Bruhat-Tits trees over all prime numbers $p$. Decomposing the distance from $M,\frac{g}{h}$ to $1$ as
\[
d(M,\frac{g}{h}~|~1) = n_1 log(p_1) + \dots + n_k log(p_k) \]
will then allow us to find minimal paths from $1$ to $M,\frac{g}{h}$.

But we should be careful in drawing $2$-dimensional cells (or higher dimensional ones) in this ‘product’ of trees as the operators
\[
\frac{k}{p} \ast \qquad \text{and} \qquad \frac{l}{q} \ast \]
for different primes $p$ and $q$ do not commute, in general. The composition
\[
(\frac{k}{p} \ast) \circ (\frac{l}{q} \ast) \qquad \text{with matrix} \qquad \begin{bmatrix} \frac{1}{pq} & \frac{kq+l}{pq} \\ 0 & 1 \end{bmatrix} \]
has as numerator in the upper-right corner $0 \leq kq + l For the primes $2$ and $3$, relevant in the description of the Moonshine Picture, the commutation relations are

\[
(\frac{0}{2} \ast) \circ (\frac{0}{3} \ast) = (\frac{0}{3} \ast) \circ (\frac{0}{2} \ast), \quad
(\frac{0}{2} \ast) \circ (\frac{1}{3} \ast) = (\frac{0}{3} \ast) \circ (\frac{1}{2} \ast),
\quad
(\frac{0}{2} \ast) \circ (\frac{2}{3} \ast) = (\frac{1}{3} \ast) \circ (\frac{0}{2} \ast) \]

\[
(\frac{1}{2} \ast) \circ (\frac{0}{3} \ast) = (\frac{1}{3} \ast) \circ (\frac{1}{2} \ast), \quad
(\frac{1}{2} \ast) \circ (\frac{1}{3} \ast) = (\frac{2}{3} \ast) \circ (\frac{0}{2} \ast),
\quad
(\frac{1}{2} \ast) \circ (\frac{2}{3} \ast) = (\frac{2}{3} \ast) \circ (\frac{1}{2} \ast) \]

My one and only resolution for 2018: ban vampires from my life!

Here’s the story.

In the 1920’s, Montparnasse was at the heart of the intellectual and artistic life in Paris because studios and cafés were inexpensive.

Artists including Picasso, Matisse, Zadkine, Modigliani, Dali, Chagall, Miro, and the Romanian sculptor Constantin Brancusi all lived there.

You’ll find many photographs of Picasso in the company of others (here center, with Modigliani and Salmon), but … not with Brancusi.

From A Life of Picasso: The Triumphant Years, 1917-1932 (Vol 3) by John Richardson:

“Brancusi disapproved of one of of Picasso’s fundamental characteristics—one that was all too familiar to the latter’s fellow artists and friends—his habit of making off not so much with their ideas as with their energy. “Picasso is a cannibal,” Brancusi said. He had a point. After a pleasurable day in Picasso’s company, those present were apt to end up suffering from collective nervous exhaustion. Picasso had made off with their energy and would go off to his studio and spend all night living off it. Brancusi hailed from vampire country and knew about such things, and he was not going to have his energy or the fruits of his energy appropriated by Picasso.”

I learned this story via Austin Kleon who made this video about it: