## A Bicategory of Decorated Cospans

8 July, 2017

My students are trying to piece together general theory of networks, inspired by many examples. A good general theory should clarify and unify these examples. What some people call network theory, I’d just call ‘applied graph invariant theory’: they come up with a way to calculate numbers from graphs, they calculate these numbers for graphs that show up in nature, and then they try to draw conclusions about this. That’s fine as far as it goes, but there’s a lot more to network theory!

There are many kinds of networks. You can usually create big networks of a given kind by sticking together smaller networks of this kind. The networks usually do something, and the behavior of the whole is usually determined by the behavior of the parts and how the parts are stuck together.

So, we should think of networks of a given kind as morphisms in a category, or more generally elements of an algebra of some operad, and define a map sending each such network to its behavior. Then we can study this map mathematically!

All these insights (and many more) are made precise in Fong’s theory of ‘decorated cospans’:

• Brendan Fong, The Algebra of Open and Interconnected Systems, Ph.D. thesis, University of Oxford, 2016. (Blog article here.)

Kenny Courser is starting to look at the next thing: how one network can turn into another. For example, a network might change over time, or we might want to simplify a complicated network somehow. If a network is morphism, a process where one network turns into another could be a ‘2-morphism’: that is, a morphism between morphisms. Just as categories have objects and morphisms, bicategories have objects, morphisms and 2-morphisms.

So, Kenny is looking at bicategories. As a first step, Kenny took Brendan’s setup and souped it up to define ‘decorated cospan bicategories’:

• Kenny Courser, Decorated cospan bicategories, to appear in Theory and Applications of Categories.

In this paper, he showed that these bicategories are often ‘symmetric monoidal’. This means that you can not only stick networks together end to end, you can also set them side by side or cross one over the other—and similarly for processes that turn one network into another! A symmetric monoidal bicategory is a somewhat fearsome structure, so Kenny used some clever machinery developed by Mike Shulman to get the job done:

• Mike Shulman, Constructing symmetric monoidal bicategories.

I would love to talk about the details, but they’re a bit technical so I think I’d better talk about something more basic. Namely: what’s a decorated cospan category and what’s a decorated cospan bicategory?

First: what’s a decorated cospan? A cospan in some category $C$ is a diagram like this:

where the objects and morphisms are all in $C.$ For example, if $C$ is the category of sets, we’ve got two sets $X$ and $Y$ mapped to a set $\Gamma.$

In a ‘decorated’ cospan, the object $\Gamma$ is equipped or, as we like to say, ‘decorated’ with extra structure. For example:

Here the set $\Gamma$ consists of 3 points—but it’s decorated with a graph whose edges are labelled by numbers! You could use this to describe an electrical circuit made of resistors. The set $X$ would then be the set of ‘input terminals’, and $Y$ the set of ‘output terminals’.

In this example, and indeed in many others, there’s no serious difference between inputs and outputs. We could reflect the picture, switching the roles of $X$ and $Y,$ and the inputs would become outputs and vice versa. One reason for distinguishing them is that we can then attach the outputs of one circuit to the inputs of another and build a larger circuit. If we think of our circuit as a morphism from the input set $X$ to the output set $Y,$ this process of attaching circuits to form larger ones can be seen as composing morphisms in a category.

In other words, if we get the math set up right, we can compose a decorated cospan from $X$ to $Y$ and a decorated cospan from $Y$ to $Z$ and get a decorated cospan from $X$ to $Z.$ So with luck, we get a category with objects of $C$ as objects, and decorated cospans between these guys as morphisms!

For example, we can compose this:

and this:

to get this:

What did I mean by saying ‘with luck’? Well, there’s not really any luck involved, but we need some assumptions for all this to work. Before we even get to the decorations, we need to be able to compose cospans. We can do this whenever our cospans live in a category with pushouts. In category theory, a pushout is how we glue two things together.

So, suppose our category $C$ has pushouts. IF we then have two cospans in $C,$ one from $X$ to $Y$ and one from $Y$ to $Z:$

we can take a pushout:

and get a cospan from $X$ to $Z:$

All this is fine and dandy, but there’s a slight catch: the pushout is only defined up to isomorphism, so we can’t expect this process of composing cospans to be associative: it will only be associative up to isomorphism.

What does that mean? What’s an isomorphism of cospans?

I’m glad you asked. A map of cospans is a diagram like this:

where the two triangles commmute. You can see two cospans in this picture; the morphism $f$ provides the map from one to the other. If $f$ is an isomorphism, then this is an isomorphism of cospans.

To get around this problem, we can work with a category where the morphisms aren’t cospans, but isomorphism classes of cospans. That’s what Brendan did, and it’s fine for many purposes.

But back around 1972, when Bénabou was first inventing bicategories, he noticed that you could also create a bicategory with

• objects of $C$ as objects,
• spans in $C$ as morphisms, and
• maps of spans in $C$ as 2-morphisms.

Bicategories are perfectly happy for composition of 1-morphisms to be associative only up to isomorphism, so this solves the problem in a somewhat nicer way. (Taking equivalence classes of things when you don’t absolutely need to is regarded with some disdain in category theory, because it often means you’re throwing out information—and when you throw out information, you often regret it later.)

So, if you’re interested in decorated cospan categories, and you’re willing to work with bicategories, you should consider thinking about decorated cospan bicategories. And now, thanks to Kenny Courser’s work, you can!

He showed how the decorations work in the bicategorical approach: for example, he proved that whenever $C$ has finite colimits and

$F : (C,+) \to (\mathrm{Set}, \times)$

is a lax symmetric monoidal functor, you get a symmetric monoidal bicategory where a morphism is a cospan in $C:$

with the object $\Gamma$ decorated by an element of $F(\Gamma).$

Proving this took some virtuosic work in category theory. The key turns out to be this glorious diagram:

For the explanation, check out Proposition 4.1 in his paper.

I’ll talk more about applications of cospan bicategories when I blog about some other papers Kenny Courser and Daniel Cicala are writing.

## The Geometric McKay Correspondence (Part 2)

2 July, 2017

Last time I sketched how the $E_8$ Dynkin diagram arises from the icosahedron. This time I’m fill in some details. I won’t fill in all the details, because I don’t know how! Working them out is the goal of this series, and I’d like to enlist your help.

(In fact, I’m running this series of posts both here and at the n-Category Café. So far I’m getting many more comments over there. So, to keep the conversation in one place, I’ll disable comments here and urge you to comment over there.)

Remember the basic idea. We start with the rotational symmetry group of the isosahedron and take its double cover, getting a 120-element group $\Gamma$ called the binary icosahedral group. Since this is naturally a subgroup of $\mathrm{SU}(2)$ it acts on $\mathbb{C}^2,$ and we can form the quotient space

$S = \mathbb{C}^2/\Gamma$

This is a smooth manifold except at the origin—by which I mean the point coming from $0 \in \mathbb{C}^2.$ Luckily we can ‘resolve’ this singularity! This implies that we can find a smooth manifold $\widetilde{S}$ and a smooth map

$\pi \colon \widetilde{S} \to S$

that’s one-to-one and onto except at the origin. There may be various ways to do this, but there’s one best way, the ‘minimal’ resolution, and that’s what I’ll be talking about.

The origin is where all the fun happens. The map $\pi$ sends 8 spheres to the origin in $\mathbb{C}^2/\Gamma,$ one for each dot in the $\mathrm{E}_8$ Dynkin diagram:

Two of these spheres intersect in a point if their dots are connected by an edge; otherwise they’re disjoint.

This is wonderful! So, the question is just how do we really see it? For starters, how do we get our hands on this manifold $\widetilde{S}$ and this map $\pi \colon \widetilde{S} \to S?$

For this we need some algebraic geometry. Indeed, the whole subject of ‘resolving singularities’ is part of algebraic geometry! However, since I still remember my ignorant youth, I want to avoid flinging around the vocabulary of this subject until we actually need it. So, experts will have to pardon my baby-talk. Nonexperts can repay me in cash, chocolate, bitcoins or beer.

What’s $\widetilde{S}$ like? First I’ll come out and tell you, and then I’ll start explaining what the heck I just said.

Theorem. $\widetilde{S}$ is the space of all $\Gamma$-invariant ideals $I \subseteq \mathbb{C}[x,y]$ such that $\mathbb{C}[x,y]/I$ is isomorphic, as a representation of $\Gamma,$ to the regular representation of $\Gamma.$

If you want a proof, this is Corollary 12.8 in Kirillov’s Quiver Representations and Quiver Varieties. It’s on page 245, so you’ll need to start by reading lots of other stuff. It’s a great book! But it’s not completely self-contained: for example, right before Corollary 12.8 he brings in a crucial fact without proof: “it can be shown that in dimension 2, if a crepant resolution exists, it is minimal”.

I will not try to prove the theorem; instead I will start explaining what it means.

Suppose you have a bunch of points $p_1, \dots, p_n \in \mathbb{C}^2.$ We can look at all the polynomials on $\mathbb{C}^2$ that vanish at these points. What is this collection of polynomials like?

Let’s use $x$ and $y$ as names for the standard coordinates on $\mathbb{C}^2,$ so polynomials on $\mathbb{C}^2$ are just polynomials in these variables. Let’s call the ring of all such polynomials $\mathbb{C}[x,y].$ And let’s use $I$ to stand for the collection of such polynomials that vanish at our points $p_1, \dots, p_n.$

Here are two obvious facts about $I$:

A. If $f \in I$ and $g \in I$ then $f + g \in I.$

B. If $f \in I$ and $g \in \mathbb{C}[x,y]$ then $fg \in I.$

We summarize these by saying $I$ is an ideal, and this is why we called it $I.$ (So clever!)

Here’s a slightly less obvious fact about $I$:

C. If the points $p_1, \dots, p_n$ are all distinct, then $\mathbb{C}[x,y]/I$ has dimension n.

The point is that the value of a function $f \in \mathbb{C}[x,y]$ at a point $p_i$ doesn’t change if we add an element of $I$ to $f,$ so this value defines a linear functional on $\mathbb{C}[x,y]/I.$ Guys like this form a basis of linear functionals on $\mathbb{C}[x,y]/I,$ so it’s n-dimensional.

All this should make you interested in the set of ideals $I$ with $\mathrm{dim}(\mathbb{C}[x,y]/I) = n.$ This set is called the Hilbert scheme $\mathrm{Hilb}^n(\mathbb{C}^2).$

Why is it called a scheme? Well, Hilbert had a bunch of crazy schemes and this was one. Just kidding: actually Hilbert schemes were invented by Grothendieck in 1961. I don’t know why he named them after Hilbert. The kind of Hilbert scheme I’m using is a very basic one, more precisely called the ‘punctual’ Hilbert scheme.

The Hilbert scheme $\mathrm{Hilb}^n(\mathbb{C}^2)$ is a whole lot like the set of unordered n-tuples of distinct points in $\mathbb{C}^2.$ Indeed, we’ve seen that every such n-tuple gives a point in the Hilbert scheme. But there are also other points in the Hilbert scheme! And this is where the fun starts!

Imagine n particles moving in $\mathbb{C}^2,$ with their motion described by polynomial functions of time. As long as these particles don’t collide, they define a curve in the Hilbert scheme. But it still works when they collide! When they collide, this curve will hit a point in the Hilbert scheme that doesn’t come from an unordered n-tuple of distinct points in $\mathbb{C}^2.$ This point describes a ‘type of collision’.

More precisely: n-tuples of distinct points in $\mathbb{C}^2$ give an open dense set in the Hilbert scheme, but there are other points in the Hilbert scheme which can be reached as limits of those in this open dense set! The topology here is very subtle, so let’s look at an example.

Let’s look at the Hilbert scheme $\mathrm{Hilb}^2(\mathbb{C}^2).$ Given two distinct points $p_1, p_2 \in \mathbb{C}^2,$ we get an ideal

$\{ f \in \mathbb{C}[x,y] \, : \; f(p_1) = f(p_2) = 0 \}$

This ideal is a point in our Hilbert scheme, since $\mathrm{dim}(\mathbb{C}[x,y]/I) = 2 .$

But there are other points in our Hilbert scheme! For example, if we take any point $p \in \mathbb{C}^2$ and any vector $v \in \mathbb{C}^2,$ there’s an ideal consisting of polynomials that vanish at $p$ and whose directional derivative in the $v$ direction also vanishes at $p$:

$\displaystyle{ I = \{ f \in \mathbb{C}[x,y] \, : \; f(p) = \lim_{t \to 0} \frac{f(p+t v) - f(p)}{t} = 0 \} }$

It’s pretty easy to check that this is an ideal and that $\mathrm{dim}(\mathbb{C}[x,y]/I) = 2 .$ We can think of this ideal as describing two particles in $\mathbb{C}^2$ that have collided at $p$ with relative velocity some multiple of $v$.

For example you could have one particle sitting at $p$ while another particle smacks into it while moving with velocity $v;$ as they collide the corresponding curve in the Hilbert scheme would hit $I.$

This would also work if the velocity were any multiple of $v,$ since we also have

$\displaystyle{ I = \{ f \in \mathbb{C}[x,y] \, : \; f(p) = \lim_{t \to 0} \frac{f(p+ c t v) - f(p)}{t} = 0 \} }$

for any constant $c \ne 0.$ And note, this constant can be complex. I’m trying to appeal to your inner physicist, but we’re really doing algebraic geometry over the complex numbers, so we can do weird stuff like multiply velocities by complex numbers.

Or, both particles could be moving and collide at $p$ while their relative velocity was some complex multiple of $v.$ As they collide, the corresponding point in the Hilbert scheme would still hit $I.$

But here’s the cool part: such ‘2-particle collisions with specified position and relative velocity’ give all the points in the Hilbert scheme $\mathrm{Hilb}^2(\mathbb{C}^2),$ except of course for those points coming from 2 particles with distinct positions.

What happens when we go to the next Hilbert scheme, $\mathrm{Hilb}^3(\mathbb{C}^2)?$ This Hilbert scheme has an open dense set corresponding to triples of particles with distinct positions. It has other points coming from situations where two particles collide with some specified position and relative velocity while a third ‘bystander’ particle sits somewhere else. But it also has points coming from triple collisions. And these are more fancy! Not only velocities but accelerations play a role!

I could delve into this further, but for now I’ll just point you here:

• John Baez, The Hilbert scheme for 3 points on a surface, MathOverflow, June 7, 2017.

The main thing to keep in mind is this. As n increases, there are more and more ways we can dream up ideals $I$ with $\mathrm{dim}(\mathbb{C}[x,y]/I) = n.$ But all these ideals consist of functions that vanish at n or fewer points and also obey other equations saying that various linear combinations of their first, second, and higher derivatives vanish. We can think of these ideals as ways for n particles to collide, with conditions on their positions, velocities, accelerations, etc. The total number of conditions needs to be n.

Now let’s revisit that description of the wonderful space we’re seeking to understand, $\widetilde{S}$:

Theorem. $\widetilde{S}$ is the space of all $\Gamma$-invariant ideals $I \subseteq \mathbb{C}[x,y]$ such that $\mathbb{C}[x,y]/I$ is isomorphic, as a representation of $\Gamma,$ to the regular representation of $\Gamma.$

Since $\Gamma$ has 120 elements, its regular representation—the obvious representation of this group on the space of complex functions on this group—is 120-dimensional. So, points in $\widetilde{S}$ are ideals $I$ with $\mathrm{dim}(\mathbb{C}[x,y]/I) = 120 .$ So, they’re points in the Hilbert scheme $\mathrm{Hilb}^{120}(\mathbb{C}^2).$

But they’re not just any old points in this Hilbert scheme! The binary icosahedral group $\Gamma$ acts on $\mathbb{C}^2$ and thus anything associated with it. In particular, it acts on the Hilbert scheme $\mathrm{Hilb}^{120}(\mathbb{C}^2).$ A point in this Hilbert scheme can lie in $\widetilde{S}$ only if it’s invariant under the action of $\Gamma.$ And given this, it’s in $\widetilde{S}$ if and only if $\mathbb{C}[x,y]/I$ is isomorphic to the regular representation of $\Gamma.$

Given all this, there’s an easy way to get your hands on a point $I \in \widetilde{S}.$ Just take any nonzero element of $\mathbb{C}^2$ and act on it by $\Gamma.$ You’ll get 120 distinct points in $\mathbb{C}^2$ — I promise. Do you see why? Then let $I$ be the set of polynomials that vanish on all these points.

In fact, we saw last time that your 120 points will be the vertices of a 600-cell centered at the origin of $\mathbb{C}^2$:

By this construction we get enough points to form an open dense subset of $\widetilde{S}.$ These are the points that aren’t mapped to the origin by

$\pi \colon \widetilde{S} \to S$

Alas, it’s the other points in $\widetilde{S}$ that I’m really interested in. As I hope you see, these are certain ‘limits’ of 600-cells that have ‘shrunk to the origin’… or in other words, highly symmetrical ways for 120 points in $\mathbb{C}^2$ to collide at the origin, with some highly symmetrical conditions on their velocities, accelerations, etc.

That’s what I need to understand.

## The Theory of Devices

20 June, 2017

I’m visiting the University of Genoa and talking to two category theorists: Marco Grandis and Giuseppe Rosolini. Grandis works on algebraic topology and higher categories, while Rosolini works on the categorical semantics of programming languages.

Yesterday, Marco Grandis showed me a fascinating paper by his thesis advisor:

• Gabriele Darbo, Aspetti algebrico-categoriali della teoria dei dispotivi, Symposia Mathematica IV (1970), Istituto Nazionale di Alta Matematica, 303–336.

It’s closely connected to Brendan Fong’s thesis, but also different—and, of course, much older. According to Grandis, Darbo was the first person to work on category theory in Italy. He’s better known for other things, like ‘Darbo’s fixed point theorem’, but this piece of work is elegant, and, it seems to me, strangely ahead of its time.

The paper’s title translates as ‘Algebraic-categorical aspects of the theory of devices’, and its main concept is that of a ‘universe of devices’: a collection of devices of some kind that can be hooked up using wires to form more devices of this kind. Nowadays we might study this concept using operads—but operads didn’t exist in 1970, and Darbo did quite fine without them.

The key is the category $\mathrm{FinCorel},$ which has finite sets as objects and ‘corelations’ as morphisms. I explained corelations here:

Corelations in network theory, 2 February 2016.

Briefly, a corelation from a finite set $X$ to a finite set $Y$ is a partition of the disjoint union of $X$ and $Y.$ We can get such a partition from a bunch of wires connecting points of $X$ and $Y.$ The idea is that two points lie in the same part of the partition iff they’re connected, directly or indirectly, by a path of wires. So, if we have some wires like this:

they determine a corelation like this:

There’s an obvious way to compose corelations, giving a category $\mathrm{FinCorel}.$

Gabriele Darbo doesn’t call them ‘corelations’: he calls them ‘trasduttori’. A literal translation might be ‘transducers’. But he’s definitely talking about corelations, and like Fong he thinks they are basic for studying ways to connect systems.

Darbo wants a ‘universe of devices’ to assign to each finite set $X$ a set $D(X)$ of devices having $X$ as their set of ‘terminals’. Given a device in $D(X)$ and a corelation $f \colon X \to Y,$ thought of as a bunch of wires, he wants to be able to attach these wires to the terminals in $X$ and get a new device with $Y$ as its set of terminals. Thus he wants a map $D(f): D(X) \to D(Y).$ If you draw some pictures, you’ll see this should give a functor

$D : \mathrm{FinCorel} \to \mathrm{Set}$

Moreover, if we have device with a set $X$ of terminals and a device with a set $Y$ of terminals, we should be able to set them side by side and get a device whose set of terminals form the set $X + Y$, meaning the disjoint union of $X$ and $Y.$ So, Darbo wants to have maps

$\delta_{X,Y} : D(X) \times D(Y) \to D(X + Y)$

If you draw some more pictures you can convince yourself that $\delta$ should be a lax symmetric monoidal functor… if you’re one of the lucky few who knows what that means. If you’re not, you can look it up in many places, such as Section 1.2 here:

• Brendan Fong, The Algebra of Open and Interconnected Systems, Ph.D. thesis, University of Oxford, 2016. (Blog article here.)

Darbo does not mention lax symmetric monoidal functors, perhaps because such concepts were first introduced by Mac Lane only in 1968. But as far as I can tell, Darbo’s definition is almost equivalent to this:

Definition. A universe of devices is a lax symmetric monoidal functor $D \colon \mathrm{FinCorel} \to \mathrm{Set}.$

One difference is that Darbo wants there to be exactly one device with no terminals. Thus, he assumes $D(\emptyset)$ is a one-element set, say $1$, while the definition above would only demand the existence of a map $\delta \colon 1 \to D(\emptyset)$ obeying a couple of axioms. That gives a particular ‘favorite’ device with no terminals. I believe we get Darbo’s definition from the above one if we further assume $\delta$ is the identity map. This makes sense if we take the attitude that ‘a device is determined by its observable behavior’, but not otherwise. This attitude is called ‘black-boxing’.

Darbo does various things in his paper, but the most exciting to me is his example connected to linear electrical circuits. He defines, for any pair of objects $V$ and $I$ in an abelian category $C,$ a particular universe of devices. He calls this the universe of linear devices having $V$ as the object of potentials and $I$ as the object of currents.

If you don’t like abelian categories, think of $C$ as the category of finite-dimensional real vector spaces, and let $V = I = \mathbb{R}.$ Electric potential and electric current are described by real numbers so this makes sense.

The basic idea will be familiar to Fong fans. In an electrical circuit made of purely conductive wires, when two wires merge into one we add the currents to get the current on the wire going out. When one wire splits into two we duplicate the potential to get the potentials on the wires going out. Working this out further, any corelation $f : X \to Y$ between finite set determines two linear relations, one

$f_* : I^X \rightharpoonup I^Y$

relating the currents on the wires coming in to the currents on the wires going out, and one

$f^* : V^Y \rightharpoonup V^X$

relating the potentials on the wires going out to the potentials on the wires coming in. Here $I^X$ is the direct sum of $X$ copies of $I,$ and so on; the funky arrow indicates that we have a linear relation rather than a linear map. Note that $f_*$ goes forward while $f^*$ goes backward; this is mainly just conventional, since you can turn linear relations around, but we’ll see it’s sort of nice.

If we let $\mathrm{Rel}(A,B)$ be the set of linear relations between two objects $A, B \in C,$ we can use the above technology to get a universe of devices where

$D(X) = \mathrm{Rel}(V^X, I^X)$

In other words, a device of this kind is simply a linear relation between the potentials and currents at its terminals!

How does $D$ get to be a functor $D : \mathrm{FinCorel} \to \mathrm{FinSet}$? That’s pretty easy. We’ve defined it on objects (that is, finite sets) by the above formula. So, suppose we have a morphism (that is, a corelation) $f \colon X \to Y.$ How do we define $D(f) : D(X) \to D(Y)?$

To answer this question, we need a function

$D(f) : \mathrm{Rel}(V^X, I^X) \to \mathrm{Rel}(V^Y, I^Y)$

Luckily, we’ve got linear relations

$f_* : I^X \rightharpoonup I^Y$

and

$f^* : V^Y \rightharpoonup V^X$

So, given any linear relation $R \in \mathrm{Rel}(V^X, I^X),$ we just define

$D(f)(R) = f_* \circ R \circ f^*$

Voilà!

People who have read Fong’s thesis, or my paper with Blake Pollard on reaction networks:

• John Baez and Blake Pollard, A compositional framework for reaction networks.

will find many of Darbo’s ideas eerily similar. In particular, the formula

$D(f)(R) = f_* \circ R \circ f^*$

appears in Lemma 16 of my paper with Blake, where we are defining a category of open dynamical systems. We prove that $D$ is a lax symmetric monoidal functor, which is just what Darbo proved—though in a different context, since our $R$ is not linear like his, and for a different purpose, since he’s trying to show $D$ is a ‘universe of devices’, while we’re trying to construct the category of open dynamical systems as a ‘decorated cospan category’.

In short: if this work of Darbo had become more widely known, the development of network theory could have been sped up by three decades! But there was less interest in a general theory of networks at the time, lax monoidal functors were brand new, operads unknown… and, sadly, few mathematicians read Italian.

Darbo has other papers, and so do his students. We should read them and learn from them! Here are a few open-access ones:

• Franco Parodi, Costruzione di un universo di dispositivi non lineari su una coppia di gruppi abeliani , Rendiconti del Seminario Matematico della Università di Padova 58 (1977), 45–54.

• Franco Parodi, Categoria degli universi di dispositivi e categoria delle T-algebre, Rendiconti del Seminario Matematico della Università di Padova 62 (1980), 1–15.

• Stefano Testa, Su un universo di dispositivi monotoni, Rendiconti del Seminario Matematico della Università di Padova 65 (1981), 53–57.

At some point I will scan in G. Darbo’s paper and make it available here.

## The Geometric McKay Correspondence (Part 1)

19 June, 2017

The ‘geometric McKay correspondence’, actually discovered by Patrick du Val in 1934, is a wonderful relation between the Platonic solids and the ADE Dynkin diagrams. In particular, it sets up a connection between two of my favorite things, the icosahedron:

and the $\mathrm{E}_8$ Dynkin diagram:

When I recently gave a talk on this topic, I realized I didn’t understand it as well as I’d like. Since then I’ve been making progress with the help of this book:

• Alexander Kirillov Jr., Quiver Representations and Quiver Varieties, AMS, Providence, Rhode Island, 2016.

I now think I glimpse a way forward to a very concrete and vivid understanding of the relation between the icosahedron and E8. It’s really just a matter of taking the ideas in this book and working them out concretely in this case. But it takes some thought, at least for me. I’d like to enlist your help.

The rotational symmetry group of the icosahedron is a subgroup of $\mathrm{SO}(3)$ with 60 elements, so its double cover up in $\mathrm{SU}(2)$ has 120. This double cover is called the binary icosahedral group, but I’ll call it $\Gamma$ for short.

This group $\Gamma$ is the star of the show, the link between the icosahedron and E8. To visualize this group, it’s good to think of $\mathrm{SU}(2)$ as the unit quaternions. This lets us think of the elements of $\Gamma$ as 120 points in the unit sphere in 4 dimensions. They are in fact the vertices of a 4-dimensional regular polytope, which looks like this:

It’s called the 600-cell.

Since $\Gamma$ is a subgroup of $\mathrm{SU}(2)$ it acts on $\mathbb{C}^2,$ and we can form the quotient space

$S = \mathbb{C}^2/\Gamma$

This is a smooth manifold except at the origin—that is, the point coming from $0 \in \mathbb{C}^2.$ There’s a singularity at the origin, and this where $\mathrm{E}_8$ is hiding! The reason is that there’s a smooth manifold $\widetilde{S}$ and a map

$\pi : \widetilde{S} \to S$

that’s one-to-one and onto except at the origin. It maps 8 spheres to the origin! There’s one of these spheres for each dot here:

Two of these spheres intersect in a point if their dots are connected by an edge; otherwise they’re disjoint.

The challenge is to find a nice concrete description of $\widetilde{S},$ the map $\pi : \widetilde{S} \to S,$ and these 8 spheres.

But first it’s good to get a mental image of $S.$ Each point in this space is a $\Gamma$ orbit in $\mathbb{C}^2,$ meaning a set like this:

$\{g x : \; g \in \Gamma \}$

for some $x \in \mathbb{C}^2.$ For $x = 0$ this set is a single point, and that’s what I’ve been calling the ‘origin’. In all other cases it’s 120 points, the vertices of a 600-cell in $\mathbb{C}^2.$ This 600-cell is centered at the point $0 \in \mathbb{C}^2,$ but it can be big or small, depending on the magnitude of $x.$

So, as we take a journey starting at the origin in $S,$ we see a point explode into a 600-cell, which grows and perhaps also rotates as we go. The origin, the singularity in $S,$ is a bit like the Big Bang.

Unfortunately not every 600-cell centered at the origin is of the form I’ve shown:

$\{g x : \; g \in \Gamma \}$

It’s easiest to see this by thinking of points in 4d space as quaternions rather than elements of $\mathbb{C}^2.$ Then the points $g \in \Gamma$ are unit quaternions forming the vertices of a 600-cell, and multiplying $g$ on the right by $x$ dilates this 600-cell and also rotates it… but we don’t get arbitrary rotations this way. To get an arbitrarily rotated 600-cell we’d have to use both a left and right multiplication, and consider

$\{x g y : \; g \in \Gamma \}$

for a pair of quaternions $x, y.$

Luckily, there’s a simpler picture of the space $S.$ It’s the space of all regular icosahedra centered at the origin in 3d space!

To see this, we start by switching to the quaternion description, which says

$S = \mathbb{H}/\Gamma$

Specifying a point $x \in \mathbb{H}$ amounts to specifying the magnitude $\|x\|$ together with $x/\|x\|,$ which is a unit quaternion, or equivalently an element of $\mathrm{SU}(2).$ So, specifying a point in

$\{g x : \; g \in \Gamma \} \in \mathbb{H}/\Gamma$

amounts to specifying the magnitude $\|x\|$ together with a point in $\mathrm{SU}(2)/\Gamma$. But $\mathrm{SU}(2)$ modulo the binary icosahedral group $\Gamma$ is the same as $\mathrm{SO(3)}$ modulo the icosahedral group (the rotational symmetry group of an icosahedron). Furthermore, $\mathrm{SO(3)}$ modulo the icosahedral group is just the space of unit-sized icosahedra centered at the origin of $\mathbb{R}^3.$

So, specifying a point

$\{g x : \; g \in \Gamma \} \in \mathbb{H}/\Gamma$

amounts to specifying a nonnegative number $\|x\|$ together with a unit-sized icosahedron centered at the origin of $\mathbb{R}^3.$ But this is the same as specifying an icosahedron of arbitrary size centered at the origin of $\mathbb{R}^3.$ There’s just one subtlety: we allow the size of this icosahedron to be zero, but then the way it’s rotated no longer matters.

So, $S$ is the space of icosahedra centered at the origin, with the ‘icosahedron of zero size’ being a singularity in this space. When we pass to the smooth manifold $\widetilde{S},$ we replace this singularity with 8 spheres, intersecting in a pattern described by the $\mathrm{E}_8$ Dynkin diagram.

Points on these spheres are limiting cases of icosahedra centered at the origin. We can approach these points by letting an icosahedron centered at the origin shrink to zero size in a clever way, perhaps spinning about wildly as it does.

I don’t understand this last paragraph nearly as well as I’d like! I’m quite sure it’s true, and I know a lot of relevant information, but I don’t see it. There should be a vivid picture of how this works, not just an abstract argument. Next time I’ll start trying to assemble the material that I think needs to go into building this vivid picture.

## The Mathematics of Open Reaction Networks

8 June, 2017

Next week, Blake Pollard and I will talk about our work on reaction networks. We’ll do this at Dynamics, Thermodynamics and Information Processing in Chemical Networks, a workshop at the University of Luxembourg organized by Massimiliano Esposito and Matteo Polettini. We’ll do it on Tuesday, 13 June 2017, from 11:00 to 13:00, in room BSC 3.03 of the Bâtiment des Sciences. If you’re around, please stop by and say hi!

Here are the slides for my talk:

Abstract. To describe systems composed of interacting parts, scientists and engineers draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Feynman diagrams and the like. In principle all these different diagrams fit into a common framework: the mathematics of monoidal categories. This has been known for some time. However, the details are more challenging, and ultimately more rewarding, than this basic insight. Here we explain how various applications of reaction networks and Petri nets fit into this framework.

If you see typos or other problems please let me know now!

I hope to blog a bit about the workshop… it promises to be very interesting.

## The Dodecahedron, the Icosahedron and E8

16 May, 2017

Here you can see the slides of a talk I’m giving:

The dodecahedron, the icosahedron and E8, Annual General Meeting of the Hong Kong Mathematical Society, Hong Kong University of Science and Technology.

It’ll take place on 10:50 am Saturday May 20th in Lecture Theatre G. You can see the program for the whole meeting here.

The slides are in the form of webpages, and you can see references and some other information tucked away at the bottom of each page.

In preparing this talk I learned more about the geometric McKay correspondence, which is a correspondence between the simply-laced Dynkin diagrams (also known as ADE Dynkin diagrams) and the finite subgroups of $\mathrm{SU}(2).$

There are different ways to get your hands on this correspondence, but the geometric way is to resolve the singularity in $\mathbb{C}^2/\Gamma$ where $\Gamma \subset \mathrm{SU}(2)$ is such a finite subgroup. The variety $\mathbb{C}^2/\Gamma$ has a singularity at the origin–or more precisely, the point coming from the origin in $\mathbb{C}^2.$ To make singularities go away, we ‘resolve’ them. And when you take the ‘minimal resolution’ of this variety (a concept I explain here), you get a smooth variety $S$ with a map

$\pi \colon S \to \mathbb{C}^2/\Gamma$

which is one-to-one except at the origin. The points that map to the origin lie on a bunch of Riemann spheres. There’s one of these spheres for each dot in some Dynkin diagram—and two of these spheres intersect iff their two dots are connected by an edge!

In particular, if $\Gamma$ is the double cover of the rotational symmetry group of the dodecahedron, the Dynkin diagram we get this way is $E_8$:

The basic reason $\mathrm{E}_8$ is connected to the icosahedron is that the icosahedral group is generated by rotations of orders 2, 3 and 5 while the $\mathrm{E}_8$ Dynkin diagram has ‘legs’ of length 2, 3, and 5 if you count right:

In general, whenever you have a triple of natural numbers $a,b,c$ obeying

$\displaystyle{ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} > 1}$

you get a finite subgroup of $\mathrm{SU}(2)$ that contains rotations of orders $a,b,c,$ and a simply-laced Dynkin diagram with legs of length $a,b,c.$ The three most exciting cases are:

$(a,b,c) = (2,3,3)$: the tetrahedron, and $E_6,$

$(a,b,c) = (2,3,4)$: the octahedron, and $E_7,$

$(a,b,c) = (2,3,5)$: the icosahedron, and $E_8.$

But the puzzle is this: why does resolving the singular variety $\mathbb{C}^2/\Gamma$ gives a smooth variety with a bunch of copies of the Riemann sphere $\mathbb{C}\mathrm{P}^1$ sitting over the singular point at the origin, with these copies intersecting in a pattern given by a Dynkin diagram?

It turns out the best explanation is in here:

• Klaus Lamotke, Regular Solids and Isolated Singularities, Vieweg & Sohn, Braunschweig, 1986.

In a nutshell, you need to start by blowing up $\mathbb{C}^2$ at the origin, getting a space $X$ containing a copy of $\mathbb{C}\mathrm{P}^1$ on which $\Gamma$ acts. The space $X/\Gamma$ has further singularities coming from the rotations of orders $a, b$ and $c$ in $\Gamma$. When you resolve these, you get more copies of $\mathbb{C}\mathrm{P}^1,$ which intersect in the pattern given by a Dynkin diagram with legs of length $a,b$ and $c.$

I would like to understand this better, and more vividly. I want a really clear understanding of the minimal resolution $S.$ For this I should keep rereading Lamotke’s book, and doing more calculations.

I do, however, have a nice vivid picture of the singular space $\mathbb{C}^2/\Gamma.$ For that, read my talk! I’m hoping this will lead, someday, to an equally appealing picture of its minimal resolution.

## Periodic Patterns in Peptide Masses

6 April, 2017

Gheorghe Craciun is a mathematician at the University of Wisconsin who recently proved the Global Attractor Conjecture, which since 1974 was the most famous conjecture in mathematical chemistry. This week he visited U. C. Riverside and gave a talk on this subject. But he also told me about something else—something quite remarkable.

### The mystery

A peptide is basically a small protein: a chain of made of fewer than 50 amino acids. If you plot the number of peptides of different masses found in various organisms, you see peculiar oscillations:

These oscillations have a frequency of about 14 daltons, where a ‘dalton’ is roughly the mass of a hydrogen atom—or more precisely, 1/12 the mass of a carbon atom.

Biologists had noticed these oscillations in databases of peptide masses. But they didn’t understand them.

Can you figure out what causes these oscillations?

It’s a math puzzle, actually.

Next I’ll give you the answer, so stop looking if you want to think about it first.

### The solution

Almost all peptides are made of 20 different amino acids, which have different masses, which are almost integers. So, to a reasonably good approximation, the puzzle amounts to this: if you have 20 natural numbers $m_1, ... , m_{20},$ how many ways can you write any natural number $N$ as a finite ordered sum of these numbers? Call it $F(N)$ and graph it. It oscillates! Why?

(We count ordered sums because the amino acids are stuck together in a linear way to form a protein.)

There’s a well-known way to write down a formula for $F(N)$. It obeys a linear recurrence:

$F(N) = F(N - m_1) + \cdots + F(N - m_{20})$

and we can solve this using the ansatz

$F(N) = x^N$

Then the recurrence relation will hold if

$x^N = x^{N - m_1} + x^{N - m_2} + \dots + x^{N - m_{20}}$

for all $N.$ But this is fairly easy to achieve! If $m_{20}$ is the biggest mass, we just need this polynomial equation to hold:

$x^{m_{20}} = x^{m_{20} - m_1} + x^{m_{20} - m_2} + \dots + 1$

There will be a bunch of solutions, about $m_{20}$ of them. (If there are repeated roots things get a bit more subtle, but let’s not worry about.) To get the actual formula for $F(N)$ we need to find the right linear combination of functions $x^N$ where $x$ ranges over all the roots. That takes some work. Craciun and his collaborator Shane Hubler did that work.

But we can get a pretty good understanding with a lot less work. In particular, the root $x$ with the largest magnitude will make $x^N$ grow the fastest.

If you haven’t thought about this sort of recurrence relation it’s good to look at the simplest case, where we just have two masses $m_1 = 1, m_2 = 2.$ Then the numbers $F(N)$ are the Fibonacci numbers. I hope you know this: the $N$th Fibonacci number is the number of ways to write $N$ as the sum of an ordered list of 1’s and 2’s!

1

1+1,   2

1+1+1,   1+2,   2+1

1+1+1+1,   1+1+2,   1+2+1,   2+1+1,   2+2

If I drew edges between these sums in the right way, forming a ‘family tree’, you’d see the connection to Fibonacci’s original rabbit puzzle.

In this example the recurrence gives the polynomial equation

$x^2 = x + 1$

and the root with largest magnitude is the golden ratio:

$\Phi = 1.6180339...$

The other root is

$1 - \Phi = -0.6180339...$

With a little more work you get an explicit formula for the Fibonacci numbers in terms of the golden ratio:

$\displaystyle{ F(N) = \frac{1}{\sqrt{5}} \left( \Phi^{N+1} - (1-\Phi)^{N+1} \right) }$

But right now I’m more interested in the qualitative aspects! In this example both roots are real. The example from biology is different.

Puzzle 1. For which lists of natural numbers $m_1 < \cdots < m_k$ are all the roots of

$x^{m_k} = x^{m_k - m_1} + x^{m_k - m_2} + \cdots + 1$

real?

I don’t know the answer. But apparently this kind of polynomial equation always one root with the largest possible magnitude, which is real and has multiplicity one. I think it turns out that $F(N)$ is asymptotically proportional to $x^N$ where $x$ is this root.

But in the case that’s relevant to biology, there’s also a pair of roots with the second largest magnitude, which are not real: they’re complex conjugates of each other. And these give rise to the oscillations!

For the masses of the 20 amino acids most common in life, the roots look like this:

The aqua root at right has the largest magnitude and gives the dominant contribution to the exponential growth of $F(N).$ The red roots have the second largest magnitude. These give the main oscillations in $F(N),$ which have period 14.28.

For the full story, read this:

• Shane Hubler and Gheorghe Craciun, Periodic patterns in distributions of peptide masses, BioSystems 109 (2012), 179–185.

Most of the pictures here are from this paper.

My main question is this:

Puzzle 2. Suppose we take many lists of natural numbers $m_1 < \cdots < m_k$ and draw all the roots of the equations

$x^{m_k} = x^{m_k - m_1} + x^{m_k - m_2} + \cdots + 1$

What pattern do we get in the complex plane?

I suspect that this picture is an approximation to the answer you’d get to Puzzle 2:

If you stare carefully at this picture, you’ll see some patterns, and I’m guessing those are hints of something very beautiful.

Earlier on this blog we looked at roots of polynomials whose coefficients are all 1 or -1:

The pattern is very nice, and it repays deep mathematical study. Here it is, drawn by Sam Derbyshire:

But now we’re looking at polynomials where the leading coefficient is 1 and all the rest are -1 or 0. How does that change things? A lot, it seems!

By the way, the 20 amino acids we commonly see in biology have masses ranging between 57 and 186. It’s not really true that all their masses are different. Here are their masses:

57, 71, 87, 97, 99, 101, 103, 113, 113, 114, 115, 128, 128, 129, 131, 137, 147, 156, 163, 186

I pretended that none of the masses $m_i$ are equal in Puzzle 2, and I left out the fact that only about 1/9th of the coefficients of our polynomial are nonzero. This may affect the picture you get!