Freeman Dyson is a famous physicist who has also dabbled in number theory quite productively. If some random dude said the Riemann Hypothesis was connected to quasicrystals, I’d probably dismiss him as a crank. But when Dyson says this, it’s a lot more interesting. So I’ve been trying to understand his remarks on this. And it’s been productive, in that I’ve learned some interesting things, and I now feel closer to seeing why the Riemann Hypothesis is a natural and important conjecture.

But still, I could use a lot of help: I don’t have much time for number theory, and a few pointers from experts could keep me from going down dead ends.

Those of you who were using the internet around 1990 may remember newsgroups like sci.math, sci.physics, sci.math.research and sci.physics.research. That’s how internet-savvy mathematicians and physicists communicated back then. And if you were around then, you may remember Matt McIrvin, who wrote consistently intelligent and good-natured posts about physics.

I lost track of him for a while, but met him again on Google+, where has been using the open-source math software called Sage to play around with ideas from number theory.

Sage comes equipped with a table of nontrivial zeros of the Riemann zeta function computed by Andrew Odlyzko. Remember, this function is given by

$$\zeta (s)=\sum _{n=1}^{\mathrm{\infty }}{n}^{-s}$$

when $\mathrm{Re}(s)>1$, but it can be analytically continued over the whole complex plane except for a pole at 1, and then it turns out that $\zeta (z)=0$ for a lot of points on a line:

$$z=\frac{1}{2}+ik$$

where $k$ is a real number. Here are the first few:

$$k_1\approx 14.1347$$

$$k_2\approx 21.0220$$

$$k_3\approx 25.0109$$

The Riemann Hypothesis claims that all the zeros of the zeta function lie on this line, except for the so-called trivial zeros at $-2,-4,-6,-8$ and so on. Riemann only checked this for the first three cases… but by now people have checked it for the first 10,000,000,000,000 cases. So it seems to be true, but nobody can prove it.

The nontrivial zeros of the Riemann zeta function are really interesting. There’s no simple formula for them, but they encode information about prime numbers. Riemann was the first to notice this… but Matt ran into it on his own.

He took the first ten thousand positive numbers $k_j$ that make

$$\zeta (\frac{1}{2}+i{k}_j)=0,$$

and he added up a bunch of functions like this:

$$\exp (i{k}_{j}x)$$

one for each $j=1,\dots ,\mathrm{10,000}$.

The result is a function of $x$, and he graphed its absolute value. The graph has lots of sharp spikes!

And where are these spikes? He zoomed in on the first few:

He wrote:

A closeup of those first spikes. I wonder what those numbers are, exactly? Probably they’re in the literature somewhere.

I looked around and soon found that those spikes should be here:

$$\ln (2),\ln (3),\ln (4),\ln (5),\ln (7),\ln (8),\ln (9),\ln (11),\ln (13),\ln (16),...$$

See the pattern?

Dyson’s remarks

In math jargon, what Matt did is take the Fourier transform of a sum of Dirac deltas supported at the imaginary parts of the nontrivial Riemann zeta zeros. The answer seemed to be another sum of Dirac deltas, times different numbers: the different spikes in the pictures above seem to have different heights. It’s unusual to take the Fourier transform of such a spiky function and get another spiky function. And according to Freeman Dyson, this is the defining feature of a quasicrystal!

When I was looking around for clues, one of the first things I ran into was a lecture by Dyson. He never actually delivered this lecture—it was cancelled at the last minute for some reason—but it was printed here:

• Freeman Dyson, Frogs and birds, Notices of the American Mathematical Society 56 (2009), 212–223.

It was about two styles of doing mathematics, hence the curious title. He said:

The proof of the Riemann Hypothesis is a worthy goal, and it is not for us to ask whether we can reach it. I will give you some hints describing how it might be achieved. Here I will be giving voice to the mathematician that I was fifty years ago before I became a physicist. I will talk first about the Riemann Hypothesis and then about quasicrystals.

There were until recently two supreme unsolved problems in the world of pure mathematics, the proof of Fermat’s Last Theorem and the proof of the Riemann Hypothesis. Twelve years ago, my Princeton colleague Andrew Wiles polished off Fermat’s Last Theorem, and only the Riemann Hypothesis remains. Wiles’ proof of the Fermat Theorem was not just a technical stunt. It required the discovery and exploration of a new field of mathematical ideas, far wider and more consequential than the Fermat Theorem itself. It is likely that any proof of the Riemann Hypothesis will likewise lead to a deeper understanding of many diverse areas of mathematics and perhaps of physics too. Riemann’s zeta-function, and other zeta-functions similar to it, appear ubiquitously in number theory, in the theory of dynamical systems, in geometry, in function theory, and in physics. The zeta-function stands at a junction where paths lead in many directions. A proof of the hypothesis will illuminate all the connections. Like every serious student of pure mathematics, when I was young I had dreams of proving the Riemann Hypothesis. I had some vague ideas that I thought might lead to a proof. In recent years, after the discovery of quasicrystals, my ideas became a little less vague. I offer them here for the consideration of any young mathematician who has ambitions to win a Fields Medal.

Quasicrystals can exist in spaces of one, two, or three dimensions. From the point of view of physics, the three-dimensional quasicrystals are the most interesting, since they inhabit our three-dimensional world and can be studied experimentally. From the point of view of a mathematician, one-dimensional quasicrystals are much more interesting than two-dimensional or three-dimensional quasicrystals because they exist in far greater variety. The mathematical definition of a quasicrystal is as follows. A quasicrystal is a distribution of discrete point masses whose Fourier transform is a distribution of discrete point frequencies. Or to say it more briefly, a quasicrystal is a pure point distribution that has a pure point spectrum. This definition includes as a special case the ordinary crystals, which are periodic distributions with periodic spectra.

Excluding the ordinary crystals, quasicrystals in three dimensions come in very limited variety, all of them associated with the icosahedral group. The two-dimensional quasicrystals are more numerous, roughly one distinct type associated with each regular polygon in a plane. The two-dimensional quasicrystal with pentagonal symmetry is the famous Penrose tiling of the plane.

Finally, the one-dimensional quasicrystals have a far richer structure since they are not tied to any rotational symmetries. So far as I know, no complete enumeration of one-dimensional quasicrystals exists. It is known that a unique quasicrystal exists corresponding to every Pisot–Vijayaraghavan number or PV number. A PV number is a real algebraic integer, a root of a polynomial equation with integer coefficients, such that all the other roots have absolute value less than one [1]. The set of all PV numbers is infinite and has a remarkable topological structure. The set of all one-dimensional quasicrystals has a structure at least as rich as the set of all PV numbers and probably much richer. We do not know for sure, but it is likely that a huge universe of one-dimensional quasicrystals not associated with PV numbers is waiting to be discovered.

Here comes the connection of the one-dimensional quasicrystals with the Riemann Hypothesis. If the Riemann Hypothesis is true, then the zeros of the zeta-function form a one-dimensional quasicrystal according to the definition. They constitute a distribution of point masses on a straight line, and their Fourier transform is likewise a distribution of point masses, one at each of the logarithms of ordinary prime numbers and prime-power numbers. My friend Andrew Odlyzko has published a beautiful computer calculation of the Fourier transform of the zeta-function zeros [2]. The calculation shows precisely the expected structure of the Fourier transform, with a sharp discontinuity at every logarithm of a prime or prime-power number and nowhere else.

My suggestion is the following. Let us pretend that we do not know that the Riemann Hypothesis is true. Let us tackle the problem from the other end. Let us try to obtain a complete enumeration and classification of one-dimensional quasicrystals. That is to say, we enumerate and classify all point distributions that have a discrete point spectrum […] We shall then find the well-known quasicrystals associated with PV numbers, and also a whole universe of other quasicrystals, known and unknown. Among the multitude of other quasicrystals we search for one corresponding to the Riemann zeta-function and one corresponding to each of the other zeta-functions that resemble the Riemann zeta-function. Suppose that we find one of the quasicrystals in our enumeration with properties that identify it with the zeros of the Riemann zeta-function. Then we have proved the Riemann Hypothesis and we can wait for the telephone call announcing the award of the Fields Medal.

These are of course idle dreams. The problem of classifying one-dimensional quasicrystals is horrendously difficult, probably at least as difficult as the problems that Andrew Wiles took seven years to explore. But if we take a Baconian point of view, the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible. The classification of quasicrystals is a worthy goal, and might even turn out to be achievable.

[1] M. J. Bertin et al., Pisot and Salem Numbers, Birkhäuser, Boston, 1992.

[2] A. M. Odlyzko, Primes, quantum chaos and computers, in Number Theory: Proceedings of a Symposium, 4 May 1989, Washington, DC, USA (National Research Council, 1990), pp. 35–46.

Questions

I wanted to understand this better, so I asked around on Mathoverflow. I got a lot of help. For starters, I got pointed to some critical remarks by Nick S., who said:

Well his definition of quasicrystals is not the one used by the quasicrystal community (we actually don’t have a formal mathematical definition yet)…. His statement about icosahedral group is false, actually most 3-dimensional models don’t have any symmetry group. Same issue for the 2 dimensional quasicrystals, most of them are not related to polygons in the plane. […] I really have no idea what he mean by “it is well known that a unique quasicrystal exists corresponding to every PV number”. The existence is true, the uniqueness is far for true… Unless I’m making a terrible mistake, there are constructions which produce pure point diffractive sets from PV numbers, and they produce uncountably many… In many situations, but not always, one can probably get that most of them are “equivalent” in some sense, but not all of them… And the big issue is that any equivalence in this sense, unless one adds very strong extra conditions, allows for .. small translations of the points… And there are of course uncountably many models which are not associated to PV numbers. Another issue is that the zeroes of the RZF are not a Delone set, so anything done so far by the quasicrystal community is not relevant to the problem… And last, I really don’t see how one can go around the following issue: Let $\Lambda$ be the set of zeroes. Let $\Lambda \prime$ be the set obtained by moving all the zeroes, such that the $n$th zero is moved by at most $1/n$. Then diffraction cannot differentiate between $\Lambda$ and $\mathrm{Lambda}\prime$.

I don’t know if these remarks are true, so if any of you know, please tell me… preferably with references!

I don’t care too much if Dyson is using ‘quasicrystal’ in a nonstandard sense. He at least seems to be hinting at a fairly precise definition, perhaps “a countable sum of Dirac deltas on ${ℝ}^n$ that defines a tempered distribution whose Fourier transform is a countable linear combination of Dirac deltas”. The phrase ‘defines a tempered distribution’ just means the Dirac deltas don’t bunch up too fast, so the Fourier transform is well-defined. Allowing the original sum to also be a more general linear combination of Dirac deltas might be be nice, too: then the Fourier transform of a quasicrystal would be another quasicrystal!

Anyway, what I’d like to learn is what’s known about such entities. In what sense, if any, does any Pisot–Vijayaraghavan number give a unique quasicrystal in 1 dimension? Do de Bruijn’s quasiperiodic tilings, like this one drawn by Greg Egan’s software, give quasicrystals in Dyson’s sense?

Is it really true that all quasicrystals in 3 dimensions are related to the icosahedral group? What’s the theorem there? And what about the 4-dimensional pattern built from the E₈ lattice—is that a quasicrystal in Dyson’s sense? Is it hard to find higher-dimensional quasicrystals, or easy?

The Guinand–Weil explicit formula

But now I’d like to come to my actual point, which concerns this remark:

If the Riemann Hypothesis is true, then the zeros of the zeta-function form a one-dimensional quasicrystal according to the definition. They constitute a distribution of point masses on a straight line, and their Fourier transform is likewise a distribution of point masses, one at each of the logarithms of ordinary prime numbers and prime-power numbers.

Is this actually true? In other words, does the Riemann Hypothesis actually imply this? And if so, why?

At first I thought it was true. That would make a very nice bumper sticker explaining the ‘meaning’ of the Riemann Hypothesis… something like

RIEMANN ZETA ZEROS ARE FOURIER DUAL TO LOGS OF PRIME POWERS!

This is the first version of the Riemann Hypothesis I’ve seen that makes me really want it to be true. You can see it discussed in this excellent book draft here, with lots of pretty pictures:

• Barry Mazur and Richard Stein, Primes.

But it seems the truth is a bit more complicated. The truth is called the explicit formula of Guinand and Weil and it involves terms, not only for the nontrivial zeros of the zeta function, but also for the trivial zeros, and the pole. And in fact it’s best to think of this formula not in terms of the original Riemann zeta function, but the ‘corrected’ version that takes the ‘prime at infinity’ into account using the gamma function, namely:

$$\Lambda (s)={\pi }^{-s/2}\Gamma (s/2)\zeta (s)$$

This ‘corrected’ version has the all-important symmetry:

$$\Lambda (s)=\Lambda (1-s).$$

that lets you see the zeros of this function, and thus all the nontrivial zeros of the original Riemann zeta function, lie in the strip $0\le \mathrm{Re}(s)\le 1$.

So, I still have hope for getting a conceptually clear statement of the Riemann Hypothesis that’s exactly correct! However, so far, I can’t seem to say something correct without it looking rather messy. For example, on Mathoverflow Brad Rogers stated a version of the Guinand–Weil explicit formula that looks about like this:

For a compactly supported smooth function $g:ℝ\to ℂ$ with Fourier transform $\hat{g}$,

$$\sum _k\hat{g}(k/2\pi )={\int }_{ℝ}[g(x)+g(-x)]e^{-x/2}d(e^x-\psi (e^x))+{\int }_{ℝ}\frac{\Omega (\xi )}{2\pi }\hat{g}(\xi /2\pi )d\xi$$

Here the sum is over $k$ such that $1/2+ik$ is a non-trivial zero of the Riemann zeta function. $\psi$ is the Chebyshev prime counting function:

$$\psi (x)=\sum _{p^k\le x}\ln (p)$$

and

$$\Omega (\xi )=\frac{1}{2}\frac{\Gamma \prime }{\Gamma }(1/4+i\xi /2)+\frac{1}{2}\frac{\Gamma \prime }{\Gamma }(1/4-i\xi /2)-\log \pi .$$

Alas, this formula doesn’t look very ‘conceptual’, though I think I’m beginning to understand it, and Marc Palm gave a nice sketch of a proof.

In this formula, $k$ can be complex if the Riemann Hypothesis is false. But if it’s true, $k$ is always real, and the left-hand side of the big equation

$$\sum _k\hat{g}(k/2\pi )$$

is really just the test function $g$ integrated against a sum of complex exponentials, one for each nontrivial zero of the zeta function. (I should warn you that these zeros come in complex conjugate pairs, so for each positive real $k$ we get a corresponding negative $k$).

The right-hand side of the big equation contains a ‘nice’ term that’s a sum over prime powers, but also some ‘corrections’ that seem to make Dyson’s claim fail to be literally true. For example, besides a linear combination of Dirac deltas at logarithms of prime powers, there’s the correction term proportional to the function $\Omega$. Owen Maresh has plotted this function:

So, I’m thinking that the slow rise at right here:

might not be an artifact of numerical approximations, but an actual real thing: this function $\Omega$.

So: what’s the really neat way to write an ‘explicit formula’ relating prime powers and Riemann zeta zeros… which simplifies in some way iff the Riemann Hypothesis holds? And is there a way to modify Freeman Dyson’s claim here, that makes it correct while still maintaining a connection to quasicrystals?

If the Riemann Hypothesis is true, then the zeros of the zeta-function form a one-dimensional quasicrystal according to the definition. They constitute a distribution of point masses on a straight line, and their Fourier transform is likewise a distribution of point masses, one at each of the logarithms of ordinary prime numbers and prime-power numbers.

Applications of category theory are described by Julie Rehmeyer in ScienceNews under the banner

One of the most abstract fields in math finds application in the ‘real’ world.

Now, how about applications in the real world?

Here it is at last: the Selected Papers Network. Given that social networks already exist, all we need for truly open scientific communication is a convention on a consistent set of tags and IDs for discussing papers. Christopher Lee, in bioinformatics at UCLA, has developed software that makes this work. Try it out!

What’s cool about this system is that it’s federated. Instead of locking up your comments within its own website—the “walled garden” strategy followed by many other service—it explicitly shares these data in a way that people not on the Selected Papers Network can easily see. Any other service can see and use them too.

To learn about the problems that make us want this system, read this:

• The Selected Papers Network (Part 1)

To learn how the system works, and to try it out, go here:

• The Selected Papers Network (Part 2)

I just returned from a month at Hong Kong University, visiting James Fullwood, an algebraic geometer who likes to think about the mathematics of string theory. There, I gave a colloquium on G₂ and the rolling ball, a paper John Baez and I wrote that is due to appear in Transactions of the AMS. This project began over a decade ago in conversations between John and Jim Dolan, later continued between Jim and me. Though Jim opted not to be a coauthor, his insights were crucial.

I would like to tell you about this paper, but I’ll warm up with a puzzle—one you’ve seen in several guises if you’ve read John Baez’s posts about this paper, but well-worth revisiting.

Puzzle: Roll a ball of unit radius on a fixed ball of radius $R$, being careful not to let your ball slip or twist as you roll it. Suppose you roll along a great circle from the North Pole to the South Pole and back to the start at the North. How many ${360}^{\circ }$ turns did the ball make as it rolled?

Here’s a variant of this puzzle you can work out very concretely: for $R=1$, roll one coin around another of the same kind, without slipping, and count the number of times the rolling coin turns.

Below the fold, I’ll give you the answer, and I’ll also tell you about something amazing that happens when $R=3$, bringing in the exceptional Lie group ${\mathrm{G}}_2$ and a funky 8-dimensional number system called the split octonions.

When Cartan and Killing classified the simple Lie groups, they found something unexpected. Besides the perfectly respectable infinite families, $\mathrm{SO}(n)$, $\mathrm{SU}(n)$ and $\mathrm{Sp}(n)$, there were five exceptions. In order of increasing dimension, these exceptional groups are prosaically called ${\mathrm{G}}_2$, ${\mathrm{F}}_4$, ${\mathrm{E}}_6$, ${\mathrm{E}}_7$ and ${\mathrm{E}}_8$. While the first three kinds of groups are all symmetry groups of vector spaces equipped with some kind of bilinear form, the exceptions do not, at first glance, have a reason to exist. I like to imagine Cartan and Killing reacting like physicist I.I. Rabi to the discovery of the muon: “Who ordered that?” Since their discovery, finding simple models for the exceptional Lie groups has been an important program in mathematics. Here, I’ll tell you about one such model for ${\mathrm{G}}_2$, essentially due to Cartan: ${\mathrm{G}}_2$ is almost the symmetry group of one ball rolling without slipping or twisting on another ball, provided the ratio of radii is 1:3 or 3:1.

When Jim Dolan and I started talking about this problem, we set out specifically to explain that funny ratio. We took our cue from Bor and Montgomery, who in their excellent paper G₂ and the rolling distribution, write:

Open problem. Find a geometric or dynamical interpretation for the “3” of the 3:1 ratio.

Why only 1:3 or 3:1? You can see a hint of “threeness” in the Dynkin diagram for ${\mathrm{G}}_2$:

And that’s probably responsible for the 1:3 ratio, but by the end of the post, I’ll have shown you another explanation, far more geometric in nature. The key is going to be to relate the rolling ball picture to another description of ${\mathrm{G}}_2$, also due to Cartan: ${\mathrm{G}}_2$ is the automorphism group of the ‘split octonions’, $𝕆\prime$.

I’ll explain all of these ideas, so don’t worry if you’re unfamiliar with them. But first, an aside to those in the know: throughout this post, I’ll focus on the adjoint, split real form of ${\mathrm{G}}_2$, which is the only form for which all of these ideas make sense, though a lot of them continue to work for the complex form as well, as long as you replace the split octonions with their complexification. This is in keeping with a well-known theme in Lie theory: split real forms and complex forms behave in almost identical fashion.

The rolling ball

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Now let’s get down to business, by working out a mathematical framework to describe a ball rolling on another ball, without slipping or twisting. To start off, I’ll let the fixed ball have radius $R$, and the rolling ball have radius 1. Only later will we see what happens when we set $R=3$.

First, a configuration of this system corresponds to a point in $S^2\times \mathrm{SO}(3)$, as follows: for $(x,g)\in S^2\times \mathrm{SO}(3)$, $\mathrm{Rx}$ is the point of contact where the rolling ball touches the fixed ball, and $g$ tells us the present orientation of the rolling ball, rotating it from your favorite starting orientation. To show you what I mean, I can’t do better than a picture from the paper of Bor and Montgomery (though note that what I call $g$ they call $\phi$, and they let their rolling ball have arbitrary radius $r$):

That’s how we describe configurations of the rolling ball system. What does it mean for the ball to roll without slipping or twisting? To describe that, we will turn to ‘incidence geometry’. This kind of geometry sounds very bare bones: in its simplest incarnation, an incidence geometry has points, lines, and an incidence relation (a point lies on a line, or is incident on the line). Yet this minimal structure will suffice to capture what we mean by rolling without slipping or twisting.

The rolling ball system defines an incidence geometry where:

• Points are configurations of the rolling ball—elements of $S^2\times \mathrm{SO}(3)$.
• Lines are given by rolling without slipping or twisting along great circles.

A line, then, is some one-dimensional submanifold of $S^2\times \mathrm{SO}(3)$. Which one-dimensional submanifolds are lines? Well, that’s the point of the puzzle at the start of this post. If you didn’t think about it before, go back and give it a shot now!

Here’s the answer: each time we roll a ball of unit radius around a ball of radius $R$, the rolling ball turns $R+1$ times! And that’s how we roll. For instance, if begin rolling from the North Pole and sweep out a central angle $\theta$ as we do so, we rotate our ball by an angle $(R+1)\theta$ about the axis going into the picture:

Now that we have a incidence geometry, we can talk about its symmetries. These symmetries are the diffeomorphisms that preserve the lines:

$$\mathrm{Aut}(S^2\times \mathrm{SO}(3))=\{f\in \mathrm{Diff}(S^2\times \mathrm{SO}(3)):f\text{takes lines to lines}\}.$$

Since the lines depend on $R$, so does this group. Alas, this group is not ${\mathrm{G}}_2$, even when $R=3$.

Remember, I said that ${\mathrm{G}}_2$ was almost the symmetry group of the rolling ball, and we have run head first into that “almost”: as shown by Bor and Montgomery, ${\mathrm{G}}_2$ does not act on $S^2\times \mathrm{SO}(3)$. Instead, it acts on its double cover:

$$S^2\times \mathrm{SU}(2)\stackrel{p}{⟶}{S}^2\times \mathrm{SO}(3).$$

And, if we stretch our imaginations a bit, we can think of this as a variant of the rolling ball system, which we call the ‘rolling spinor’.

The rolling spinor…

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What’s a rolling spinor? It’s like a ball, but thanks to the “double” in the double cover $\mathrm{SU}(2)\to \mathrm{SO}(3)$, a ${360}^{\circ }$ rotation does not act like the identity. Instead, we need to rotate by ${720}^{\circ }$ degrees to get back where we started.

Spinors show up physically and mathematically. Famously, electrons are spinors, but a far more concrete example is given by the belt trick: put a ${360}^{\circ }$ twist in a belt, and you cannot undo the twist just by translating the ends around, but for ${720}^{\circ }$ twist, you can!

I can make more sense of this using quaternions. If I identify $\mathrm{SU}(2)$ with the group of unit quaternions:

$$\mathrm{SU}(2)=\{q\in ℍ:\mid q\mid =1\}$$

and identify ${ℝ}^3$ with the subspace of imaginary quaternions:

$${ℝ}^3=\mathrm{Im}(ℍ)=\{xi+yj+zk:x,y,z\in ℝ\}$$

than it’s easy for me to describe the covering map, $\mathrm{SU}(2)\to \mathrm{SO}(3)$. Each unit quaternion $q\in \mathrm{SU}(2)$ goes to the rotation of ${ℝ}^3$ given by conjugation:

$$x\mapsto qx{q}^{-1},x\in {ℝ}^3.$$

The kernel of $\mathrm{SU}(2)\to \mathrm{SO}(3)$ is $\{\pm 1\}$. If I think of $+1$ as giving a rotation by 0°, and $-1$ as giving a rotation by 360°, then the idea that a 720° rotation is the identity becomes $(-1{)}^2=1$, which is indeed true. If this idea sounds funny, or you want an overview of quaternions, read this talk I gave at Fullerton College.

The rolling spinor system comes with an incidence geometry where:

• Points are elements of $S^2\times \mathrm{SU}(2)$.
• Lines are preimages of lines in $S^2\times \mathrm{SO}(3)$.

When $R=3$, ${\mathrm{G}}_2$ acts on $S^2\times \mathrm{SU}(2)$ as symmetries preserving lines. To understand this action, it helps to introduce yet another variant of the rolling ball system: the rolling spinor on a projective plane.

…on a projective plane

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This system is double-covered by the rolling spinor:

$$S^2\times \mathrm{SU}(2)\stackrel{q}{⟶}ℝ{\mathrm{P}}^2\times \mathrm{SU}(2),$$

where the map $q$ identifies antipodal points of $S^2$. We write $(\pm x,q)\in ℝ{\mathrm{P}}^2\times \mathrm{SU}(2)$ for the equivalence class containing $(x,q)$ and $(-x,q)$. There is an incidence geometry where:

• Points are elements of $ℝ{\mathrm{P}}^2\times \mathrm{SU}(2)$.
• Lines are images of lines in $S^2\times \mathrm{SU}(2)$.

Unlike the original rolling ball system, when $R=3$, ${\mathrm{G}}_2$ acts as symmetries of this incidence geometry. To see why, it’s time to introduce the split octonions.

Split octonions

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The split octonions $𝕆\prime$ are an eight dimensional real composition algebra. There are two such algebras over the reals, the other being their more famous cousin, the octonions. We can define the split octonions as pairs of quaternions:

$$𝕆\prime ={ℍ}^2$$

with the following multiplication:

$$(a,b)(c,d)=(\mathrm{ac}+\overline{d}b,\overline{a}d+cb).$$

Here, the barred quaternions $\overline{a}$ and $\overline{d}$ denote the conjugate, in $ℍ$, of $a$ and $d$, obtained by flipping the sign of the imaginary part, just as we would do for a complex number. With this multiplication, one can check that the quadratic form:

$$Q(a,b)=\mid a{\mid }^2-\mid b{\mid }^2,(a,b)\in {ℍ}^2$$

makes $𝕆\prime$ into a composition algebra, with the property:

$$Q(xy)=Q(x)Q(y),x,y\in 𝕆\prime .$$

Now we can define ${\mathrm{G}}_2=\mathrm{Aut}(𝕆\prime )$. And, as promised, this description of ${\mathrm{G}}_2$ can be related to the rolling ball descriptions I have been sketching. But first, it helps to note that we don’t really need all of $𝕆\prime$: since ${\mathrm{G}}_2$ fixes $1\in 𝕆\prime$, we really only need the subspace of imaginary split octonions. This is the subspace orthogonal to 1 with respect to $Q$. In terms of pairs of quaternions, this is the subspace where the first quaternion is purely imaginary while the second can be arbitrary:

$$𝕀=\mathrm{Im}(ℍ)\oplus ℍ$$ ${\mathrm{G}}_2$ acts on the imaginary split octonions as well, and in fact it is the smallest irreducible representation of ${\mathrm{G}}_2$. We’re close now. Because ${\mathrm{G}}_2$ preserves $Q$, ${\mathrm{G}}_2$ acts on the space of 1d null subspaces of $𝕀$: $$\mathrm{PC}=\{\mathrm{span}(x):x\in 𝕀-\{0\},Q(x)=0.\}$$

This space of null lines is almost the configuration space of the spinor rolling on the projective plane, $ℝP^2\times \mathrm{SU}(2)$. Choosing a representative of each line in $\mathrm{Im}(ℍ)\oplus ℍ$, we can normalize its first component to have length 1, forcing the second component to also have unit length. Doing this, we see that:

$$\frac{S^2\times \mathrm{SU}(2)}{(x,q)\sim (-x,-q)}\cong \mathrm{PC}.$$

This ought to remind you of the spinor rolling on a projective plane, which has configuration space:

$$ℝ{\mathrm{P}}^2\times \mathrm{SU}(2)\cong \frac{S^2\times \mathrm{SU}(2)}{(x,q)\sim (-x,q)}.$$

In the first case, we mod out both components by their sign, and in the second, we just mod out the first. And in fact, these spaces are diffeomorphic:

$$\begin{array}{ccc}ℝ{\mathrm{P}}^2\times \mathrm{SU}(2)& \to & \mathrm{PC}\\ (\pm x,q)& \mapsto & \pm (x,xq)\\ \end{array}$$

Note that this is not true for the ordinary rolling ball. We needed to think about a spinor rolling on a projective plane.

Under the diffeomorphism above, lines in $ℝ{\mathrm{P}}^2\times \mathrm{SU}(2)$ incidence geometry become submanifolds of $\mathrm{PC}$. If and only if $R=3$, these submanifolds ‘straighten out’: they are given by projectivizing 2-dimensional null subspaces of the imaginary split octonions $𝕀$.

However, not all 2d null subspaces of $𝕀$ yield lines in $\mathrm{PC}$. When $R=3$, the incidence geometry of $ℝ{\mathrm{P}}^2\times \mathrm{SU}(2)$ coincides with the incidence geometry on $\mathrm{PC}$ with:

• Points are 1d null subspaces of $𝕀$.
• Lines are 2d null subspaces of $𝕀$ on which the product also vanishes.

In fact, if we call a subspace of $𝕀$ a null subalgebra if the product of any two elements vanishes, we can give a very snappy description of the incidence geometry on $\mathrm{PC}$, thanks to the helpful fact that a 1d subspace $\mathrm{span}(x)$ of $𝕀$ is null if and only if $x^2=0$:

• Points are 1d null subalgebras of $𝕀$.
• Lines are 2d null subalgebras of $𝕀$.

Now we have what we wanted: the geometry of a spinor rolling on a projective plane, when $R=3$, is the geometry of null subalgebras of the imaginary split octonions, $𝕀$. Hence, ${\mathrm{G}}_2$ acts as symmetries of the spinor rolling on the projective plane!

Coda

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We’ve seen how a spinor rolling on a projective plane three times as big acquires ${\mathrm{G}}_2$ symmetry, but we still haven’t seen a completely satisfying explanation of that 1:3 ratio. It’s hiding somewhere in the proofs of the theorems I described above.

Yet there is an intuitive way to see the 1:3 ratio. Let’s think of our spinor as a ball that comes back to itself after turning twice, and let’s think of our projective plane as a ball with antipodal points identified. Rolling from a point on our fixed ball to its antipode, we’ve come back to where we’ve started in our projective plane. For what ratio of radii will the rolling spinor also be back where it started? That is, for what ratio of radii does the rolling ball turn an even number of times as it goes from a point to its antipode? At minimum, we need to turn twice. So:

• For what ratio of radii does the rolling ball turn twice as it goes from a point to its antipode? Or, put another way, for what ratio of radii does the rolling ball turn four times as it rolls once around the fixed ball?

By the solution to our puzzle, a ball of unit radius turns $R+1$ times as it rolls once around a fixed ball of radius $R$. To turn four times, we need $R=3$.

In another instalment of my occasional series on ‘Things you didn’t realise were enriched categories (unless you’re an expert!)’ I want to talk about torsors and how they can be considered enriched categories where the enriching category is a group, considered as a discrete monoidal category.

I’ll start off by telling you what a torsor is and then explain how it can be thought of as having something like a ‘group-valued distance’ and how this relates to enriched category theory.

What is a torsor?

“Torsor” is a word that used to strike a dread fear into my heart. It is a word that was seemingly only used by high-brow mathematicians who wished to intimidate more lowly souls. That was until I read Dan Freed’s paper Higher Algebraic Structures and Quantization which made me realise that torsor is just a fancy name for a simple concept. For a group $G$ a torsor is something that looks like $G$ but isn’t actually a group. That might sound like an overly cryptic description but hopefully it will make things clear in a minute.

On my desk I have a coffee cup — we are in a café, after all.

The rim $R$ of my coffee cup is a circle: it looks like the group of unit complex numbers $𝕋$, but I don’t know how to multiply two points on the coffee cup rim, $R$, so the coffee-cup rim $R$ is not canonically a group. However, the group of unit complex number $𝕋$ acts on the rim $R$: if you give me a unit complex number $e^{i\theta }$, I can rotate my coffee cup through the angle $\theta$.

This action is transitive in that I can move any point on the rim to any other point by a rotation and the action is free in that precisely one rotation will move a specified point to another specified point. A set with a $G$ action which is free and transitive is known as a principal homogeneous $G$-set, or simply a $G$-torsor. So the rim of my coffee cup is a $𝕋$-torsor.

If I pick any point $p$ on the rim $R$ then I get an bijection between the points of $R$ and the points in the group $𝕋$ by identifying the point $p$ with the identity in $𝕋$. So there are as many ways to identify the rim with the circle group $𝕋$ as there are points in the rim but none of them is canonical.

Just as an affine space can be thought of as a vector space without an origin, so a torsor can be thought of as a group without an identity: a torsor is to a group as an affine space is to a vector space.

John has written a nice piece explaining how torsors, particularly in physics, are more prevalent than you might first think, because our gut reaction is to think in terms of groups. We see a line, so we want to put an origin on it and identify it with $ℝ$ rather than accept that it doesn’t naturally have an origin, and should be thought of as an $ℝ$-torsor.

Another view of torsors

Torsors can be thought of in a different fashion and thinking in this fashion this will lead us enriched categories. Between each pair of points in a torsor there is something like a ‘group-valued distance’. More precisely, associated to each ordered pair $(t_1,t_2)$ of elements of a $G$-torsor $T$ there is a unique element of the group $G$ which sends $t_1$ to $t_2$, we can write this as $g(t_2,t_1)$, or as $g_T(t_2,t_1)$ if we want to emphasise the torsor $T$. Symbolically, this is the unique element of the group $G$ that satisfies $g(t_2,t_1)\cdot t_1=t_2$. This is the ‘distance’ from $t_1$ to $t_2$.

[I have chosen to write $g(t_2,t_1)$ rather than the more usual category theoretic convention $g(t_1,t_2)$, because the actions are on the left and the chosen conventions make the formulas look a bit nicer.]

Of course, in the coffee cup example, the ‘distance’ from one point to another is the rotation required to move the first point to the second.

Because we have a $G$-action on $T$ we know that $(gh)\cdot t_1=g\cdot (h\cdot t_1)$ and this implies $g(t_3,t_2)g(t_2,t_1)=g(t_3,t_1)$ for all $t_1,t_2,t_3$.; for the same reason we also have $g(t_1,t_1)=e$. Moreover, we can recover the $G$-action from this data because given $g\cdot t_1$ is the unique element of $T$ such that $g(g\cdot t_1,t_1)=g$.

In summary, we can think of a $G$-torsor as a set $T$ such that

1. for each pair $t_1,t_2\in T$ there is a group element $g(t_2,t_1)\in G$;

2. for each triple $t_1,t_2,t_3\in T$ there is an equality $g(t_3,t_2)g(t_2,t_1)=g(t_3,t_1)$;

3. for each element $t_1\in T$ there is the equality $g(t_1,t_1)=e$;

4. for each $t_1\in T$, $g\in G$ there is a unique $t_2$ such that $g(t_2,t_1)=g$.

You might have noticed that the first three of the above conditions form precisely the definition of a certain kind of enriched category, namely a category enriched over the monoidal category ${𝒱}_G$ which is the group $G$ considered as a discrete monoidal category. This means that ${𝒱}_G$ has the elements of $G$ as its objects, has only identity morphisms and has the group multiplication as the monoidal multiplication: $g\otimes h≔gh$.

We have shown that a $G$-torsor is a ${𝒱}_G$-category satisfying an extra condition. A reasonable question to ask at this point is “What’s the good in that?” One answer is that it gives us another layer of intuition; it allows us to make analogies with categories, with metric spaces, with posets, amongst other things, and it allows us to use the tools from enriched category theory.

We should now ask what a ${𝒱}_G$-functor $\varphi :T\to S$ between ${𝒱}_G$-torsors is. This is a function $\varphi :T\to S$ such that

$$g(t_1,t_2)=g(\varphi (t_1),\varphi (t_2)).$$

This is quite a rigid condition, saying that the right notion of map is some kind of isometry. Unsurprisingly, if $T$ and $S$ correspond to $G$-torsors then this is precisely the condition that $\varphi$ is an equivariant map:

$$\varphi (g\cdot t)=g\cdot \varphi (t)\text{for all}g\in G,t\in T.$$

New torsors from old

When we look at torsors over abelian groups there are ways of combining torsors. We will see below how these relate to standard enriched category theory constructions.

Hom torsor: If $A$ is an abelian group and both $T$ and $S$ are $A$-torsors then we can form the set ${\text{Hom}}_A(T,S)$ of equivariant maps from $T$ to $S$. I learnt many years ago from Freed’s paper that this set of maps ${\text{Hom}}_A(T,S)$ is itself an $A$-torsor. We can define the action of $A$ on an equivariant map $\varphi$ by $(a\cdot \varphi )(t):=a\cdot (\varphi (t))$. You can check that $a\cdot \varphi$ is an equivariant map, but you will see that it is necessary that $A$ is abelian for this to work.

Tensor torsor: If $A$ is an abelian group and both $T$ and $S$ are $A$-torsors then we can also define the tensor product $T{\otimes }_{A}S$ as $T\times S/\sim$ where $\sim$ is the relation defined by

$$(a\cdot t,s)\sim (t,a\cdot s)\text{for all}a\in A,t\in T,s\in S.$$

We find that $T{\otimes }_{A}S$ is also an $A$-torsor if we define the action by $a\cdot (t,s)≔(a\cdot t,s)$. Again, if you check, you will see that you need $A$ to be abelian for this to be well defined.

Properties of ${𝒱}_G$

Things get more interesting with enriched categories when we enrich over categories which are closed, braided and bicomplete. Let’s consider each of these conditions for ${𝒱}_G$.

Closedness: A monoidal category $𝒱$ is left-closed if for each object $v$ the functor $v\otimes -:𝒱\to 𝒱$ has a right adjoint $[v,-{]}_{\text{left}}:𝒱\to 𝒱$. Unwrapping this definition for ${𝒱}_G$ we find that for each $g\in G$ we need a function of sets $[g,-{]}_{\text{left}}:G\to G$ such that for every $h\in G$ we have

$$gh=x\text{if and only if}h=[g,x{]}_{\text{left}}.$$

Clearly, we can take $[g,x{]}_{\text{left}}:=g^{-1}x$. So for any group $G$ the monoidal category ${𝒱}_G$ is left-closed.

Similarly, a monoidal category $𝒱$ is right-closed if for each object $v$ the functor $-\otimes v:𝒱\to 𝒱$ has a right adjoint $[v,-{]}_{\text{right}}:𝒱\to 𝒱$. For ${𝒱}_G$, we can take $[g,x{]}_{\text{right}}:=x{g}^{-1}$. So for any group $G$ the monoidal category ${𝒱}_G$ is both left- and right-closed.

Braidedness: For a monoidal category to be braided we need isomorphisms $v\otimes w\cong w\otimes v$ for all objects $v$ and $w$. For the category ${𝒱}_G$ that would mean we need $gh=hg$ for all $g,h\in G$, in other words, we need that $G$ is abelian. In that case we have that ${𝒱}_G$ is in fact symmetric.

Bicompleteness: A category is bicomplete if it has all limits and colimits. We have no hope of any non-trivial group $G$ having a bicomplete category ${𝒱}_G$. This is because this category is discrete: to form a product $g\prod h$ we would need projections to $g$ and $h$ but, because the only morphisms are identities, that would mean $g\prod h=g$ and $g\prod h=h$. You can check that the self products are actually defined though: $g\prod g=g$. This looks like it might spoil our fun, but it will transpire that the only limits that we need will be the self-products.

Summary: In summary then, if $G$ is abelian then ${𝒱}_G$ is a closed symmetric monoidal category; if $G$ is non-abelian then ${𝒱}_G$ is a closed monoidal category which is not braided.

As the definition of the tensor product of $𝒱$-categories requires that $𝒱$ is braided, we will restrict ourselves to the case that $G$ is an abelian group, and we will emphasize this fact by renaming it $A$.

Functor category and tensor product

We can now look at how the hom torsor and tensor torsor from above arise in enriched category theory. We will work with an abelian group $A$ so that ${𝒱}_A$ is closed symmetric monoidal.

Functor categories and hom torsors: Suppose $𝒱$ is a closed symmetric monoidal category and that $𝒞$ and $𝒟$ are $𝒱$-categories then (provided that $𝒱$ has sufficiently many limits) there is $𝒱$-category $[[𝒞,𝒟]]$ of $𝒱$-morphisms where the objects are the $𝒱$-functors and the hom object $[[𝒞,𝒟]](\varphi ,\theta )$ is given by the equalizer of the following diagram.

$$\prod _{c\in 𝒞}𝒟(\varphi (c),\theta (c))\rightrightarrows \prod _{c\prime ,c″\in 𝒞}[𝒞(c\prime ,c″),𝒟(\varphi (c\prime ),\theta (c″))]$$

We don’t need to worry here about what the two maps are as, in the case of interest when $𝒱={𝒱}_A$, they are both going to be equalities.

If $T$ and $S$ are $A$-torsors, thought of as ${𝒱}_A$-categories, then we try to construct the functor category $[[T,S]]$. This has ${𝒱}_A$-functors, i.e. equivariant maps as morphisms. The hom object $g(\varphi ,\theta )$, between equivariant maps $\varphi ,\theta :T\to S$ is given by the equalizer of the above diagram, but as mentioned, the maps are equalities, so the equalizer is just the left-hand-side term, so

$$g_{[[T,S]]}(\varphi ,\theta )=\prod _{t\in T}{g}_S(\varphi (t),\theta (t)).$$

As mentioned in the previous section, ${𝒱}_A$ does not have many limits, but fortunately it does have this product as all the terms in the product are the same: a calculation shows that

$$g_S(\varphi (t_1),\theta (t_1))=g_S(\varphi (t_2),\theta (t_2))\text{for all}{t}_1,t_2\in T.$$

Hence we find that the “distance” between two functors is the distance between the two images of any point:

$$g_{[[T,S]]}(\varphi ,\theta )=g_S(\varphi (t),\theta (t))\text{for all}t\in T.$$

This means that the functor ${𝒱}_A$-category $[[T,S]]$ exists and is actually a torsor. You can easily check that the $A$-action is exactly that of the hom torsor, so

$$[[T,S]]={\text{Hom}}_A(T,S).$$

Tensor product categories and tensor torsors: If $𝒱$ is a braided monoidal category then you can define the tensor product $𝒞\otimes 𝒟$ of two $𝒱$-categories $𝒞$ and $𝒟$. An object of $𝒞\otimes 𝒟$ is a pair

$$(c,d)\in \text{ob}𝒞\times \text{ob}𝒟$$

and the hom objects are defined by

$$(𝒞\otimes 𝒟)((c,d),(c\prime ,d\prime ))≔𝒞(c,c\prime )\otimes 𝒟(d,d\prime ).$$

The braiding of $𝒱$ is needed to define the composition morphisms.

If $T$ and $S$ are $A$-torsors, thought of as ${𝒱}_A$-categories, then we can construct the tensor product ${𝒱}_A$-category $T\otimes S$. You might expect that this is actually a torsor equal to the tensor torsor $T{\otimes }_{A}S$ but that’s not quite true!

In the torsor $T{\otimes }_{A}S$ we have made the identification $(acdott,s)=(t,a\cdot s)$ however in the ${𝒱}_A$-category $T\otimes S$ these two points are distinct despite having `no distance’ between them: you can check that $g_{T\otimes S}((a\cdot t,s),(t,a\cdot s))=e$.

This means that the ${𝒱}_A$-category $T\otimes S$ is not actually a torsor but it is equivalent to the torsor $T{\otimes }_{A}S$: the quotient map on objects gives an equivalence of ${𝒱}_A$-categories $T\otimes S\to T{\otimes }_{A}S$. You can think of $T\otimes S$ as a fattened-up version of $T{\otimes }_{A}S$; it is like using a stack instead of a quotient space. This is what you might expect from a categorical approach, in that you don’t violently quotient out but rather encode the equivalence relation via isomorphisms.

Final words

I’ll just finish by mentioning the Yoneda map. For any ${𝒱}_A$-category $T$, the presheaf ${𝒱}_A$-category $\hat{T}≔[[T^{\text{op}},{𝒱}_A]]$ is a torsor.

If $T$ is actually a torsor then the Yoneda map $T\to \hat{T}$ is an isomorphism, which is not something you would expect from the Yoneda map!

If $T$ is not a torsor then the Yoneda map $T\to \hat{T}$ is a torsorization of $T$. For instance, in the tensor example above we have $T{\otimes }_{A}S\equiv \widehat{T\otimes S}$.

I’m happy to announce that Emily Riehl has joined us as a host at the Café. Emily is based at Harvard, and you can deduce some of her interests from the many guest posts she has written for us before:

• Associativity data in an $(\mathrm{\infty },1)$-category

• Understanding the homotopy coherent nerve

• Algebraic model structures

• Cellularity in algebraic model structures

• Parametrized mates and multivariable adjunctions

• Bounded gaps between primes

Welcome, Emily!

Guest post by Bruce Bartlett

On Monday, David Corfield and Kobi Kremnitzer gave philosophy talks in a snazzy new building at Oxford:

• Kobi Kremnitzer, What is geometry?, 2-4pm.
• David Corfield, What might philosophy make of homotopy type theory?, 4.30-6.30pm.

The talks shared homotopy type theory as a common theme. The name “Per Martin-Löf” was mentioned a lot, which was good for me since I had always thought Martin and Löf were two separate people:

Notes are available above, but I will try to give some brief impressions.

Kobi Kremnitzer, What is geometry?

1. Introduction

He started by answering the question “Why am I giving this talk?”, and explained that he followed the pragmatic approach to philosophy of mathematics. I think then he said his approach was somehow similar to that of Wittgenstein and Carnap (but he could have been saying the exact opposite :-)), and that for him, there is no Metaphysics in the joint carving. I’m afraid this totally went over my head, but I did imagine some Oxford dons pleasantly carving a roast chicken, which started to make me hungry!

He stressed that for him mathematics is not in the business of theorem proving only, but mathematicians create new systems, new languages, and that in the correct language a problem becomes trivial… the approach of Grothendieck.

2. Categorical language Since it was a philosophy talk, he motivated categories by starting with Kripke semantics, going via posets, and then he went from posets to categories by literally “raising the bar”!

3. Crashcourse in algebraic geometry Algebraic geometry is the study of solutions to polynomial equations, like a parabola:

$$X=\{y-x^2=0\}.$$

I haven’t specified what “y” and “x” actually are, and that’s the point. We can interpret them in any ring. Hence the Grothendieck view is to think of an affine variety $X$ defined by a bunch of polynomial equations $f_1,\dots ,f_n$ as being a presheaf on the (opposite) category of rings,

$$X:\mathrm{Rings}\to \mathrm{Set},$$

defined by

$$X(R)=\{(a_1,\dots ,a_n)\in R^n:f_i(a_1,\dots ,a_n)=0\mathrm{for}\mathrm{all}i\in I\}.$$

This leads us to define the category of algebraic spaces as being nothing but the category of presheaves on ${\mathrm{Ring}}^{\mathrm{op}}$.

This category of algebraic spaces has lots of nice properties. Inside it live subcategories of objects having nice properties, such as schemes and sheaves. But Kobi stressed that it is very handy to understand them as living in this general universe of algebraic spaces.

4. What is algebra? There was a crucial idea lurking above – that a ring is a gismo $R$ which allows you to take any polynomial $f\in \mathbb{Z}[x_1,\dots ,x_n]$ and evaluate it on elements of $R$.

So – to go from algebraic to differential geometry, we could replace the concept of a “ring” with the concept of a “$C^{\mathrm{\infty }}$-ring” – this is a gismo $R$ which allows you to take any smooth function $f\in C^{\mathrm{\infty }}({ℝ}^n,ℝ)$ and evaluate it on elements of $R$!

For instance, the space of smooth functions on a manifold is a $C^{\mathrm{\infty }}$-ring… but so is $ℂ[ϵ]/{ϵ}^2$ ! So by this slight change of view, we have accomplished Leibniz’s dream – calculus and infinitesimals in the same universe.

5. Final comments He spoke a bit about: derived geometry, set-theoretic foundations, noncommutative geometry, synthetic differential geometry, elementary theory of the category of sets – have a look at the last few pages of his notes.

He stressed that ordinary set-theoretic foundations pulverizes spaces into “atomic dust” where the elements have no “cohesion” with each other… we have to put this back in by hand using topology. As a foundation, homotopy type theory will have this cohesiveness natively built in, and that is attractive to a geometer.

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David Corfield, What might philosophy make of homotopy type theory?

David began by defining three “camps” in the philosophy of mathematics:

• Antiformalism (eg. Heidegger, Wittgenstein)
• Proformalism (eg. Russell, Carnap, Quine)
• Historical / dialectical philosophy (eg. Cassirer, Collingwood, Lautman, Polanyi, Lakatos, Shapere, MacIntyre, Friedman)

This last camp has the longest list of names, so you guessed it, David placed himself there! He quoted from Domski and Dixon, which I couldn’t understand, but I could understand this quote from his book:

Straight away, from simple inductive considerations, it should strike us as implausible that mathematicians dealing with number, function and space have produced nothing of philosophical significance in the past seventy years in view of their record over the previous three centuries. Implausible, that is, unless by some extraordinary event in the history of philosophy a way had been found to filter, so to speak, the findings of mathematicians working in core areas, so that even the transformations brought about by the development of category theory, which surfaced explicitly in 1940s algebraic topology, or the rise of non-commutative geometry over the past seventy years, are not deemed to merit philosophical attention.

He had rather sobering news for philosophers keen on getting involved with homotopy type theory:

If one is not on-board already, that’s leaving it rather late to assist with Homotopy Type Theory.

He mentioned the n-category exposition on homotopy type theory by Mike Shulman (starting here) quite a lot, and also Urs Schreiber’s explanation of how homotopy type theory natively produces the “right answer” in a differential topology problem in string theory (flux quantization condition).

He also explained how homotopy type theory gets around certain problems, such as Have you left off beating your wife? and The present king of France is bald.

He also spoke about polarity, inference, modal logic and lots of other stuff - but my understanding was severely cramped.

Finally he ended on a positive note, there is plenty of places for philosophers to get involved:

• For philosophers of the historical / dialectical persuasion, homotopy type theory is just the kind of development that needs to be written up. There is plenty to learn from the development – the role of logician-philosopher Martin-Löf, constructive type theory, category theory, homotopic mathematics, influence of physics, computer science.

• For proformalist philosophers, there are also plenty of opportunities. Look closely at intensionality, modality, polarity, the dependent type theory-language relation, and philosophy of physics.

He ended by linking to the upcoming book by Mike Shulman and co., and the nLab.

Guest post by Emily Riehl

Whether we grow up to become category theorists or applied mathematicians, one thing that I suspect unites us all is that we were once enchanted by prime numbers. It comes as no surprise then that a seminar given yesterday afternoon at Harvard by Yitang Zhang of the University of New Hampshire reporting on his new paper “Bounded gaps between primes” attracted a diverse audience. I don’t believe the paper is publicly available yet, but word on the street is that the referees at the Annals say it all checks out.

What follows is a summary of his presentation. Any errors should be ascribed to the ignorance of the transcriber (a category theorist, not an analytic number theorist) rather than to the author or his talk, which was lovely.

Prime gaps

Let us write $p_1,p_2,\dots$ for the primes in increasing cardinal order. We know of course that this list is countably infinite. A prime gap is an integer $p_{n+1}-p_n$. The Prime Number Theorem tells us that $p_{n+1}-p_n$ is approximately $\log (p_n)$ as $n$ approaches infinity.

The twin primes conjecture, on the other hand asserts that

$$\underset{n\to \mathrm{\infty }}{\mathrm{liminf}}(p_{n+1}-p_n)=2$$

i.e., that there are infinitely many pairs of twin primes for which the prime gap is just two. A generalization, attributed to Alphonse de Polignac, states that for any positive even integer, there are infinitely many prime gaps of that size. This conjecture has been neither proven nor disproven in any case. These conjectures are related to the Hardy-Littlewood conjecture about the distribution of prime constellations.

The strategy

The basic question is whether there exists some constant $C$ so that $p_{n+1}-p_n<C$ infinitely often. Now, for the first time, we know that the answer is yes…when $C=7\times {10}^7$.

Here is the basic proof strategy, supposedly familiar in analytic number theory. A subset $H=\{h_1,\dots ,h_k\}$ of distinct natural numbers is admissible if for all primes $p$ the number of distinct residue classes modulo $p$ occupied by these numbers is less than $p$. (For instance, taking $p=2$, we see that the gaps between the $h_j$ must all be even.) If this condition were not satisfied, then it would not be possible for each element in a collection $\{n+h_1,\dots ,n+h_k\}$ to be prime. Conversely, the Hardy-Littlewood conjecture contains the statement that for every admissible $H$, there are infinitely many $n$ so that every element of the set $\{n+h_1,\dots ,n+h_k\}$ is prime.

Let $\theta (n)$ denote the function that is $\log (n)$ when $n$ is prime and 0 otherwise. Fixing a large integer $x$, let us write $n\sim x$ to mean $x$ ≤ $n<2x$. Suppose we have a positive real valued function $f$—to be specified later—and consider two sums:

$$S_1=\sum _{n\sim x}f(n)$$ $$S_2=\sum _{n\sim x}(\sum _{j=1}^k\theta (n+h_j))f(n)$$

Then if $S_2>(\log 3x)S_1$ for some function $f$ it follows that ${\sum }_{j=1}^k\theta (n+h_j)>\log 3x$ for some $n\sim x$ (for any $x$ sufficiently large) which means that at least two terms in this sum are non-zero, i.e., that there are two indices $i$ and $j$ so that $n+h_i$ and $n+h_j$ are both prime. In this way we can identify bounded prime gaps.

Some details

The trick is to find an appropriate function $f$. Previous work of Daniel Goldston, János Pintz, and Cem Yildirim suggests define $f(n)=\lambda (n{)}^2$ where

$$\lambda (n)=\sum _{d\mid P(n),d<D}\mu (d)\left(\log \right(\frac{D}{d}){)}^{k+\ell }P(n)=\prod _{j=1}^k(n+h_j)$$

where $\ell >0$ and $D$ is a power of $x$.

Now think of the sum $S_2-(\log 3x)S_1$ as a main term plus an error term. Taking $D=x^{\vartheta }$ with $\vartheta <\frac{1}{4}$, the main term is negative, which won’t do. When $\vartheta =\frac{1}{4}+\omega$ the main term is okay but the question remains how to bound the error term.

Zhang’s work

Zhang’s idea is related to work of Enrico Bombieri, John Friedlander, and Henryk Iwaniec. Let $\vartheta =\frac{1}{4}+\omega$ where $\omega =\frac{1}{1168}$ (which is “small but bigger than $ϵ$”). Then define $\lambda (n)$ using the same formula as before but with an additional condition on the index $d$, namely that $d$ divides the product of the primes less that $x^{\omega }$. In other words, we only sum over square-free $d$ with small prime factors.

The point is that when $d$ is not too small (say $d>x^{1/3}$) then $d$ has lots of factors. If $d=p_1\cdots p_b$ and $R<d$ there is some $a$ so that $r=p_1\cdots p_a<R$ and $p_1\cdots p_{a+1}>R$. This gives a factorization $d=rq$ with $R/x^{\omega }<r<R$ which we can use to break the sum over $d$ into two sums (over $r$ and over $q$) which are then handled using techniques whose names I didn’t recognize.

On the size of the bound

You might be wondering where the number 70 million comes from. This is related to the $k$ in the admissible set. (My notes say $k=3.5\times {10}^6$ but maybe it should be $k=3.5\times {10}^7$.) The point is that $k$ needs to be large enough so that the change brought about by the extra condition that $d$ is square free with small prime factors is negligible. But Zhang believes that his techniques have not yet been optimized and that smaller bounds will soon be possible.

There’s going to be a “thematic trimester” in Paris starting next spring:

• Semantics of proofs and certified mathematics, Institut Henri Poincaré, April 22nd - July 11th, 2014, organized by Pierre-Louis Curien, Hugo Herbelin, Paul-André Melliès.

If you like applications of category theory to logic and computer science, there should be a lot for you here!

The basic layout is this:

• Week 1 — Kick-off: Formalisation in mathematics and in computer science
• Week 3 — Workshop 1: Formalization of mathematics in proof assistants, organized by Georges Gonthier and Vladimir Voevodsky.
• Week 6 — Workshop 2: Constructive mathematics and models of type theory, organized by Thierry Coquand and Thomas Streicher.
• Week 8 — Workshop 3: Semantics of proofs and programs, organized by Thomas Ehrhard and Alex Simpson.
• Week 10 — Workshop 4: Abstraction and verification in semantics, organized by Paul-André Melliès and Luke Ong.
• Week 12 — Workshop 5, Certification of high-level and low-level programs organized by Christine Paulin and Zhong Shao.

A lot of people I know will attend parts of this, such as Jean Benabou, Marcelo Fiore, Dan Ghica, André Joyal, Samuel Mimram, and Bas Spitters. And that makes me happy, because Paul-André Melliès has invited me to spend up to a month attending this series of workshops, perhaps in two 2-week stretches. With a little luck I’ll be able to actually do this.

(My wife Lisa Raphals has gotten invited to Erlangen for the spring of 2014, meaning roughly April 1 - June 1. If she and I succeed in getting leaves of absence, I’ll go with her, and then take some trips to nearby places. Since I split my time between the Wild West and the Far East, Paris seems nearby to Erlangen to me. I also have vague invitations to IHES, Prague and Berlin which I might try to take advantage of. And if you have a luxurious villa in northern Italy or the French Riviera, let me know.)

Suppose $X$ is a topological space and $U\subseteq X$ is an open subset, with closed complement $K=X\setminus U$. Then $U$ and $K$ are, of course, topological spaces in their own right, and we have $X=U\bigsqcup K$ as a set. What additional information beyond the topologies of $U$ and $K$ is necessary to enable us to recover the topology of $X$ on their disjoint union?

Recall that the subspace topologies of $U$ and $K$ say that for each open $V\subseteq X$, the intersections $V\cap U$ and $V\cap K$ are open in $U$ and $K$, respectively. Thus, if a subset of $X$ is to be open, it must yield open subsets of $U$ and $K$ when intersected with them. However, this condition is not in general sufficient for a subset of $X$ to be open — it does define a topology on $X$, but it’s the coproduct topology, which may not be the original one.

One way we could start is by asking what sort of structure relating $U$ and $K$ we can deduce from the fact that both are embedded in $X$. For instance, suppose $A\subseteq U$ is open. Then there is some open $V\subseteq X$ such that $V\cap U=A$. But we could also consider $V\cap K$, and ask whether this defines something interesting as a function of $A$.

Of course, it’s not clear that $V\cap K$ is a function of $A$ at all, since it depends on our choice of $V$ such that $V\cap U=A$. Is there a canonical choice of such $V$? Well, yes, there’s one obvious canonical choice: since $U$ is open in $X$, $A$ is also open as a subset of $X$, and we have $A\cap U=A$. However, $A\cap K=\varnothing$, so choosing $V=A$ wouldn’t be very interesting.

The choice $V=A$ is the smallest possible $V$ such that $V\cap U=A$. But there’s also a largest such $V$, namely the union of all such $V$. This set is open in $X$, of course, since open sets are closed under arbitrary unions, and since intersections distribute over arbitrary unions, its intersection with $U$ is still $A$.

Let’s call this set $i_{*}(A)$. In fact, it’s part of a triple of adjoint functors $i_{!}⊣i^{*}⊣i_{*}$ between the posets $O(U)$ and $O(X)$ of open sets in $U$ and $X$, where $i^{*}:O(X)\to O(U)$ is defined by $i^{*}(V)=V\cap U$, and $i_{!}:O(U)\to O(X)$ is defined by $i_{!}(A)=A$. Here $i$ denotes the continuous inclusion $U\hookrightarrow X$.

Now we can consider the intersection $i_{*}(A)\cap K$, which I’ll also denote $j^{*}{i}_{*}(A)$, where $j:K\hookrightarrow X$ is the inclusion. It turns out that this is interesting! Consider the following example, which is easy to visualize:

• $X={ℝ}^2$.
• $U=\{(x,y)\mid x<0\}$, the open left half-plane.
• $K=\{(x,y)\mid x\ge 0\}$, the closed right half-plane.

If an open subset $A\subseteq U$ “doesn’t approach the boundary” between $U$ and $K$, such as the open disc of radius $1$ centered at $(-2,0)$, then it’s fairly easy to see that $i_{*}(A)=A\cup \{(x,y)\mid x>0\}$, and therefore $j^{*}{i}_{*}(A)=\{(x,y)\mid x>0\}$ is the open right half-plane.

On the other hand, consider some open subset $A\subseteq U$ which does approach the boundary, such as

$$A=\{(x,y)\mid x^2+y^2<1\text{and}x<0\}$$

the intersection with $U$ of the open disc of radius $1$ centered at $(0,0)$. A little thought should convince you that in this case, $i_{*}(A)$ is the union of the open right half-plane with the whole open disc of radius $1$ centered at $(0,0)$. Therefore, $j^{*}{i}_{*}(A)$ is the open right half-plane together with the strip $\{(0,y)\mid -1<y<1\}$.

This example suggests that in general, $j^{*}{i}_{*}(A)$ measures how much of the “boundary” between $U$ and $K$ is “adjacent” to $A$. I leave it to some enterprising reader to try to make that precise. Here’s another nice exercise: what can you say about $i^{*}{j}_{*}(B)$ for an open subset $B\subseteq K$?

Let us however go back to our original question of recovering the topology of $X$. Suppose $A\subseteq U$ and $B\subseteq K$ are open such that $A\cup B$ is open in $X$; how does this latter fact manifest as a property of $A$ and $B$? Note first that $(A\cup B)\cap U=A$. Thus, since $i_{*}(A)$ is the largest $V$ such that $V\cap U=A$, we have $A\cup B\subseteq i_{*}(A)$, and therefore $B=j^{*}(A\cup B)\subseteq j^{*}{i}_{*}(A)$. Let me say that again:

$$B\subseteq j^{*}{i}_{*}(A).$$

This is a relationship between $A$ and $B$ which is expressed purely in terms of the topological spaces $U$ and $K$ and the function $j^{*}{i}_{*}:O(U)\to O(K)$, which we have just shown is necessary for $A\cup B$ to be open in $X$.

In fact, it is also sufficient! For suppose this to be true. Since $B$ is open in $K$, there is some open $C\subseteq X$ such that $C\cap K=B$. Given such a $C$, the union $C\cup U$ also has this property, since $U\cap K=\varnothing$. Note that in fact $C\cup U=B\cup U$, and also $B\cup U=j_{*}(B)$, the largest open subset of $X$ whose intersection with $K$ is $B$. (Since $K$, unlike $U$, is not open, there may not be a smallest such, but there is always a largest such.) Now I claim we have

$$A\cup B=j_{*}(B)\cap i_{*}(A)$$

To show this, it suffices to show that the two sides become equal after intersecting with $U$ and with $K$. For the first, we have

$$(j_{*}(B)\cap i_{*}(A))\cap U=j_{*}(B)\cap (i_{*}(A)\cap U)=j_{*}(B)\cap A=A=(A\cup B)\cap U$$

and for the second we have

$$(j_{*}(B)\cap i_{*}(A))\cap K=(j_{*}(B)\cap K)\cap i_{*}(A)=B\cap i_{*}(A)=B=(A\cup B)\cap K$$

using the assumption at the step $B\cap i_{*}(A)=B$.

In conclusion, the topology of $X$ is entirely determined by

• the induced topology of an open subspace $U\subseteq X$,
• the induced topology on its closed complement $K=X\setminus U$, and
• the induced function $j^{*}{i}_{*}:O(U)\to O(K)$.

Specifically, the open subsets of $X$ are those of the form $A\cup B$ — or equivalently, by the above argument, $i_{*}(A)\cap j_{*}(B)$ — where $A\subseteq U$ is open in $U$, $B\subseteq K$ is open in $K$, and $B\subseteq j^{*}{i}_{*}(A)$.

An obvious question to ask now is, suppose given two arbitrary topological spaces $U$ and $K$ and a function $f:O(U)\to O(K)$; what conditions on $f$ ensure that we can define a topology on $X≔U\bigsqcup K$ in this way, which restricts to the given topologies on $U$ and $K$ and induces $f$ as $j^{*}{i}_{*}$? We may start by asking what properties $j^{*}{i}_{*}$ has. Well, it preserves inclusion of open sets (i.e. $A\subseteq A\prime \Rightarrow j^{*}{i}_{*}(A)\subseteq j^{*}{i}_{*}(A\prime )$) and also finite intersections ($j^{*}{i}_{*}(A\cap A\prime )=j^{*}{i}_{*}(A)\cap j^{*}{i}_{*}(A\prime )$), including the empty intersection ($j^{*}{i}_{*}(U)=K$). In other words, it is a finite-limit-preserving functor between posets. Perhaps surprisingly, it turns out that this is also sufficient: any finite-limit-preserving $f:O(U)\to O(K)$ allows us to glue $U$ and $K$ in this way; I’ll leave that as an exercise too.

Okay, that was some fun point-set topology. Now let’s categorify it. Open subsets of $X$ are the same as 0-sheaves on it, i.e. sheaves of truth values, or of subsingleton sets, and the poset $O(X)$ is the (0,1)-topos of 0-sheaves on $X$. So a certain sort of person immediately asks, what about $n$-sheaves for $n>0$?

In other words, suppose we have $X$, $U$, and $K$ as above; what additional data on the toposes $\mathrm{Sh}(U)$ and $\mathrm{Sh}(K)$ of sheaves (of sets, or groupoids, or homotopy types, etc.) allows us to recover the topos $\mathrm{Sh}(X)$? As in the posetal case, we have adjunctions $i_{!}⊣i^{*}⊣i_{*}$ and $j^{*}⊣j_{*}$ relating these toposes, and we may consider the composite $j^{*}{i}_{*}:\mathrm{Sh}(U)\to \mathrm{Sh}(K)$.

The corresponding theorem is then that $\mathrm{Sh}(X)$ is equivalent to the comma category of ${\mathrm{Id}}_{\mathrm{Sh}(K)}$ over $j^{*}{i}_{*}$, i.e. the category of triples $(A,B,\varphi )$ where $A\in Sh(U)$, $B\in \mathrm{Sh}(K)$, and $\varphi :B\to j^{*}{i}^{*}(A)$. This is true for 1-sheaves, $n$-sheaves, $\mathrm{\infty }$-sheaves, etc. Moreover, the condition on a functor $f:\mathrm{Sh}(U)\to \mathrm{Sh}(K)$ ensuring that its comma category is a topos is again precisely that it preserves finite limits. Finally, this all works for arbitrary toposes, not just sheaves on topological spaces. I mentioned in my last post some applications of gluing for non-sheaf toposes (namely, syntactic categories).

One new-looking thing does happen at dimension 1, though, relating to what exactly the equivalence

$$\mathrm{Sh}(X)\simeq ({\mathrm{Id}}_{\mathrm{Sh}(K)}\downarrow j^{*}{i}_{*})$$

looks like. The left-to-right direction is easy: we send $C\in \mathrm{Sh}(X)$ to $(i^{*}C,j^{*}C,\varphi )$ where $\varphi :j^{*}C\to j^{*}{i}_{*}{i}^{*}C$ is $j^{*}$ applied to the unit of the adjunction $i^{*}⊣i_{*}$. But in the other direction, suppose given $(A,B,\varphi )$; how can we reconstruct an object of $\mathrm{Sh}(X)$?

In the case of open subsets, we obtained the corresponding object (an open subset of $X$) as $A\cup B$, but now we no longer have an ambient “set of points” in which to take such a union. However, we also had the equivalent characterization of the open subset of $X$ as $i_{*}(A)\cap j_{*}(B)$, and in the categorified case we do have objects $i_{*}(A)$ and $j_{*}(B)$ of $\mathrm{Sh}(X)$. We might initially try their cartesian product, but this is obviously wrong because it doesn’t incorporate the additional datum $\varphi$. It turns out that the right generalization is actually the pullback of $j_{*}(\varphi )$ and the unit of the adjunction $j^{*}⊣j_{*}$ at $i_{*}(A)$:

$$\begin{array}{ccc}C& \to & j_{*}(B)\\ \downarrow & & {\downarrow }^{j^{*}(\varphi )}\\ i_{*}(A)& \to & j_{*}{j}^{*}{i}_{*}(A)\end{array}$$

In particular, any object $C\in \mathrm{Sh}(X)$ can be recovered from $i^{*}C$ and $j^{*}C$ by this pullback:

$$\begin{array}{ccc}C& \to & j_{*}{j}^{*}C\\ \downarrow & & \downarrow \\ i_{*}{i}^{*}C& \to & j_{*}{j}^{*}{i}_{*}{i}^{*}C\end{array}$$

Now let’s shift perspective a bit, and ask what all this looks like in the internal language of the topos $\mathrm{Sh}(X)$. Inside $\mathrm{Sh}(X)$, the subtoposes $\mathrm{Sh}(U)$ and $\mathrm{Sh}(K)$ are visible through the left-exact idempotent monads $i_{*}{i}^{*}$ and $j_{*}{j}^{*}$, whose corresponding reflective subcategories are equivalent to $\mathrm{Sh}(U)$ and $\mathrm{Sh}(K)$ respectively. In the internal type theory of $\mathrm{Sh}(X)$, $i_{*}{i}^{*}$ and $j_{*}{j}^{*}$ are modalities, which I will denote $I_U$ and $J_U$ respectively. Thus, inside $\mathrm{Sh}(X)$ we can talk about “sheaves on $U$” and “sheaves on $K$” by talking about $I_U$-modal and $J_U$-modal types (or sets).

Moreover, these particular modalities are actually definable in the internal language of $\mathrm{Sh}(X)$. Open subsets $U\subseteq X$ can be identified with subterminal objects of $\mathrm{Sh}(X)$, a.k.a. h-propositions or “truth values” in the internal logic. Thus, $U$ is such a proposition. Now $I_U$ is definable in terms of $U$ by

$$I_U(C)=(U\to C)$$

I’m using type-theorists’ notation here, so $U\to C$ is the exponential $C^U$ in $\mathrm{Sh}(X)$. The other modality $J_U$ is also definable internally, though a bit less simply: it’s the following pushout:

$$\begin{array}{ccc}U\times C& \to & C\\ \downarrow & & \downarrow \\ U& \to & J_U(C)\end{array}.$$

In homotopy-theoretic language, $J_U(C)$ is the join of $C$ and $U$, written $U*C$. And if we identify $\mathrm{Sh}(U)$ and $\mathrm{Sh}(K)$ with their images under $i_{*}$ and $j_{*}$, then the functor $j^{*}{i}_{*}:\mathrm{Sh}(U)\to \mathrm{Sh}(K)$ is just the modality $J_U$ applied to $I_U$-modal types.

Finally, the fact that $\mathrm{Sh}(X)$ is the gluing of $\mathrm{Sh}(U)$ with $\mathrm{Sh}(K)$ means internally that any type $C$ can be recovered from $I_U(C)$, $J_U(C)$, and the induced map $J_U(C)\to J_U(I_U(C))$ as a pullback:

$$\begin{array}{ccc}C& \to & J_U(C)\\ \downarrow & & \downarrow \\ I_U(C)& \to & J_U(I_U(C))\end{array}$$

Now recall that internally, $U$ is a proposition: something which might be true or false. Logically, $I_U(C)=(U\to C)$ has a clear meaning: its elements are ways to construct an element of $C$ under the assumption that $U$ is true.

The logical meaning of $J_U$ is somewhat murkier, but there is one case in which it is crystal clear. Suppose $U$ is decidable, i.e. that it is true internally that “$U$ or not $U$”. If the law of excluded middle holds, then all propositions are decidable — but of course, internally to a topos, the LEM may fail to hold in general. If $U$ is decidable, then we have $U+\neg U=1$, where $\neg U=(U\to 0)$ is its internal complement. It’s a nice exercise to show that under this assumption we have $J_U(C)=(\neg U\to C)$.

In other words, if $U$ is decidable, then the elements of $J_U(C)$ are ways to construct an element of $C$ under the assumption that $U$ is false. In the decidable case, we also have $J_U(I_U(C))=1$, so that $C=I_U(C)\times J_U(C)$ — and this is just the usual way to construct an element of $C$ by case analysis, doing one thing if $U$ is true and another if it is false.

This suggests that we might regard internal gluing as a “generalized sort of case analysis” which applies even to non-decidable propositions. Instead of ordinary case analysis, where we have to do two things:

• assuming $U$, construct an element of $C$; and
• assuming not $U$, construct an element of $C$

in the non-decidable case we have to do three things:

• assuming $U$, construct an element of $C$;
• construct an element of the join $U*C$; and
• check that the two constructions agree in $U*(U\to C)$.

I have no idea whether this sort of generalized case analysis is useful for anything. I kind of suspect it isn’t, since otherwise people would have discovered it, and be using it, and I would have heard about it. But you never know, maybe it has some application. In any case, I find it a neat way to think about gluing.

Let me end with a tantalizing remark (at least, tantalizing to me). People who calculate things in algebraic topology like to work by “localizing” or “completing” their topological spaces at primes, since it makes lots of things simpler. Then they have to try to put this “prime-by-prime” information back together into information about the original space. One important class of tools for this “putting back together” is called fracture theorems. A simple fracture theorem says that if $X$ is a $p$-local space (meaning that all primes other than $p$ are inverted) and some technical conditions hold, then there is a pullback square:

$$\begin{array}{ccc}X& \to & X_p^{\wedge }\\ \downarrow & & \downarrow \\ X_{\mathbb{Q}}& \to & (X_p^{\wedge }{)}_{\mathbb{Q}}\end{array}$$

where $(-{)}_p^{\wedge }$ denotes $p$-completion and $(-{)}_{\mathbb{Q}}$ denotes “rationalization” (inverting all primes). A similar theorem applies to any space $X$ (with technical conditions), yielding a pullback square

$$\begin{array}{ccc}X& \to & \prod _p{X}_{(p)}\\ \downarrow & & \downarrow \\ X_{\mathbb{Q}}& \to & (\prod _p{X}_{(p)}{)}_{\mathbb{Q}}\end{array}$$

where $(-{)}_{(p)}$ denotes localization at $p$.

Clearly, there is a formal resemblance to the pullback square involved in the gluing theorem. At this point I feel like I should be saying something about $\mathrm{Spec}(\mathbb{Z})$. Unfortunately, I don’t know what to say! Maybe some passing expert will enlighten us.

This weekend I’m giving a talk on “The Foundations of Applied Mathematics”. It’s mostly about how the widespread use of diagrams in engineering, biology, and the like may force us to think about math in new ways, at least if we take those diagrams seriously.

I wouldn’t typically emphasize the term ‘foundations of mathematics’ when talking about this. But I’m speaking at a Category-Theoretic Foundations of Mathematics Workshop, so I want to fit in.

Unfortunately, this means I need to think a bit more generally about how applications of mathematics can impinge on foundational issues. I need to do this just so I’m not taken by surprise when people start objecting or asking tough questions.

So, I’d like to hear what you have to say. Preferably before Sunday morning!

Here’s a bit of what I’m saying. Again, this isn’t the important part of my talk, just the introduction… but I’m afraid I’m biting off more than I can chew.

We often picture the flow of information about mathematics a bit like this:

Of course this picture is oversimplified in many ways! For example:

• The details depend enormously on time and place. Let’s focus on post-WWII mathematics just to keep the topic from growing too large.
• There are many branches of science and engineering, and a very complex flow of information among these.
• In academia, only some applications of mathematics are now classified as “applied mathematics”.
• Some branches of physics now communicate more directly to “pure mathematics” than “applied mathematics”.
• Computer science also plays a distinctive role, often communicating directly to “foundations of mathematics”.

But the picture is close enough to true that deviations are interesting.

In particular: can developments in applied mathematics force changes in the foundations of mathematics?

Applied mathematics often provokes new developments in pure mathematics. But can it demand new foundations?

Can applications of math directly affect the foundations of math, or is the conversation always mediated by pure mathematicians?

Computer science is the biggest example of “applied math” that grew directly out of work in logic, where new ideas directly impact foundations:

• Uncomputability, undecidability,…
• Computer-aided proofs: what is a proof?
• Category-theoretic logic
• etcetera

But I want to talk about some other applications of mathematics that seem to call for category-theoretic foundations….

A new portrait gallery opened this week in the James Clark Maxwell Building which houses the Edinburgh University maths department. It consists of seventy portraits of mathematicians selected, as the name suggests, by Michael and Lily Atiyah.

Due to the magic of the internet, you do not have to travel to Edinburgh to see the exhibition as Andrew Ranicki, who helped organise the gallery, has made it available on his website.

The Michael and Lily Atiyah Portrait Gallery

The commentaries on each of the photographs give interesting personal insights of the Atiyahs. I’d definitely recommend that you younger mathematicians out there give them a read.

If Martin-Löf dependent type theory is a formal system which projects down via truncation to (typed) first-order logic, might we not expect there to be a modal type theory which would project down to first-order modal logic? I’ll use this post to play around with the idea.

Normally we take a modal logic to be a system which allows us to study certain ways of qualifying propositions.

It is necessarily the case that P; It is obligatory that P ; It is known that P, etc.

With first-order modal logic, these propositions may have free variables which we can then bind by a quantifier. This allows us to express controversial metaphysical theses such as:

‘Everything is necessarily self-identical’ or ‘For every thing, it is necessarily the case that that thing is identical to itself’ ($\forall x\square (x=x)$).

It still seems that only propositions are modalised. But given all we’ve been hearing at the Café from Mike on type theory, we know that propositions can be considered as just a certain kind of type. Then, if we can modalise proposition-as-types, why not modalise all types? This is what happens in modal type theory.

Despite the enormous amount of attention given over to modal logic by philosophers, I’m unaware of any mention of the idea that modal operators could apply to anything other than propositions. Instead, the majority of the work in modal type theory occurs in computer science. To pick out one interpretation of ‘necessity’ from de Paiva, Goré and Mendler’s useful summary Modalities in constructive logics and type theories

…the constructive $\square A$ singles out those terms of type $A$ that are “closed”, i.e., which are completely built from constructors over which primitive recursion is well-defined.

On the other hand, we’ve seen Urs and Mike working out higher modalities for physics, in the case of cohesive structures.

But in view of the fact that dependent type theory seems to fit well with natural language, is it not possible that we already have a trace of modal types in our everyday conversations? Let’s think. A director might say while casting

He’s a possible Hamlet.

Now, we might try to rewrite every instance of

X is a possible Y

as

It is possibly the case that X is a Y,

but this may do some violence to our inclination to attach the modality directly to a noun phrase. When cooking, some things are necessary ingredients for a dish, others optional. Beef is a necessary ingredient of a beef stew, while ale is only a possible ingredient.

Looking now at the epistemic modality, in my current state of knowledge, I may have France and Germany as known EU members, and China as a known non-EU member. I may also have Norway as a possible EU-member. Indeed, I can tag on ‘known (to me)’ to most noun phrases, and the dual ‘possible’ = ‘not known to be not’.

Thinking of dependent types, I may not know for sure someone’s nationality.

Karen is a possible citizen of Norway, hence, since I think Norway a possible EU country, as far as I know she is a possible EU citizen.

We would then need to see how modalities and dependent types interact. It seems we have

$\diamond ({\sum }_{c:\mathrm{EU}}\mathrm{Citizen}(c))\cong {\sum }_{c:\diamond \mathrm{EU}}\diamond \mathrm{Citizen}(c)$.

Given that dependent sum is what projects down to existential quantification, this might put us in mind of the so-called ‘Barcan formula’. One of the originators of first-order modal logic was the philosopher Ruth Barcan, who gave her name to a postulated entailment and its converse. These postulated entailments run between $\exists \diamond$ and $\diamond \exists$.

I like that idea that intuitions concerning one level of logic or mathematics often arise from something deeper behind the scenes. So the intuitionists had picked up on something when they saw that $P\to \neg \neg P$ was acceptable but not the converse. Behind the scenes, the former is natural, arising from currying the evaluation of an exponential ($B^A\times A\to B$, where $B$ is falsity here).

What then of the Barcan and converse Barcan formulas? Well we had some discussion over at the nForum, and came to see that the converse Barcan was a projection of something natural in the presence of a certain pullback. So

$$\begin{array}{c}\sum _{X}E\\ & \searrow \\ & & \sum _X{\diamond }_{X}E& \to & \diamond \sum _{X}E\\ & & \downarrow & \mathrm{pb}& \downarrow \\ & & X& \to & \diamond X.\end{array}$$

Let’s illustrate this using the dependent type of players of teams:

$$\begin{array}{c}\mathrm{players}\mathrm{of}\mathrm{actual}\mathrm{teams}\\ & \searrow \\ & & \mathrm{poss}.\mathrm{players}\mathrm{of}\mathrm{actual}\mathrm{teams}& \to & \mathrm{poss}.\mathrm{players}\mathrm{of}\mathrm{poss}.\mathrm{teams}\\ & & \downarrow & \mathrm{pb}& \downarrow \\ & & \mathrm{actual}\mathrm{teams}& \to & \mathrm{poss}.\mathrm{teams}\end{array},$$

where that top right entry can be formulated differently

possible players of possible teams = possible (players of teams).

Converse Barcan is a projected shadow of the inclusion of possible players of actual teams within possible (players of teams), i.e., possible players of possible teams. Only when all possible teams are actual teams do we have Barcan, that is, that every possibly player is a possible player of an actual team.

In view of the large philosophical literature on topics such as possible objects, there are opportunities for modal dependent type theory and philosophy to interact. One thing to work out would be how to think about modalities applied to the universe of types itself. Semantically, are we to imagine a collection of possible worlds over which the given types vary? We may want the base here to be a full $\mathrm{\infty }$-groupoid.

Last summer in Barcelona, Joachim Kock floated the idea that there might be a connection between two invariants of graphs: the Tutte polynomial and the magnitude function. Here I’ll explain what these two invariants are, I’ll give you some examples, and I’ll ask for your help in understanding what the magnitude function tells us.

The Tutte polynomial of a graph is old and well-understood. It turns a graph $G$ into a polynomial $T_G$ in two variables, with natural number coefficients:

$$T_G(x,y)\in \mathbb{N}[x,y].$$

The magnitude function of a graph is new and ill-understood. It turns a graph $G$ into a rational function ${\mu }_G$ in one variable, with rational coefficients:

$${\mu }_G(q)\in \mathbb{Q}(q).$$

It can also be expanded as a power series with integer coefficients:

$${\mu }_G(q)\in \mathbb{Z}[[q]].$$

I can’t figure out what the relationship between the two is — if any — or what information is contained in the magnitude function. It’s exasperating! I’ll be very pleased if someone can shed light on the situation.

In this post, “graphs” are always going to be finite and undirected. But some of the time I’ll allow loops and multiple edges, and some of the time I won’t. I’ll say which I’m doing.

(Formally, a graph with loops and multiple edges consists of a finite set $V$, a finite set $E$, and a map $E\to (V\times V)/S_2$. Here $(V\times V)/S_2$ is the quotient of $V\times V$ by the obvious action of the group $S_2$ of order $2$; it can be seen as the set of subsets of $V$ of cardinality 1 or 2. A graph without loops or multiple edges consists of a finite set $V$ and a subset $E\subseteq V\times V$ that, seen as a relation on $V$, is symmetric and irreflexive.)

The Tutte polynomial

The Tutte polynomial knows a lot about a graph. It knows how many edges the graph has, how many maximal forests it contains, and how many ways there are to colour each of its vertices from a palette of 38 colours without ever giving the same colour to adjacent vertices. It’s related to the Potts model in statistical physics and the Jones polynomial of knot theory.

I’ll give you the definition in a moment, but first we need some terminology.

For this part, graphs will be allowed to have loops and multiple edges. Given an edge $e$ of a graph $G$, there are two fundamental operations we can perform:

• Deletion. The graph $G-e$ is simply $G$ with the edge $e$ removed (but the vertices left untouched).

• Contraction. The graph $G/e$ is formed from $G$ by first removing $e$, then identifying the two vertices that it used to join.

An edge $e$ is called a bridge if $G-e$ has more connected-components than $G$, and a loop if it joins a vertex to itself.

The Tutte polynomial $T_G(x,y)\in \mathbb{N}[x,y]$ of a graph $G$ is characterized by the following two conditions:

• $T_G=T_{G-e}+T_{G/e}$ if $e$ is neither a bridge nor a loop;

• $T_G(x,y)=x^b{y}^{\ell }$ if $G$ consists of $b$ bridges, $\ell$ loops, and no other edges.

It takes some work to prove that there’s any polynomial with these two properties, since in the first condition there are usually many different edges $e$ we could choose. But this form of the definition has the advantage that it provides a nice algorithm for calculating Tutte polynomials. For instance, here’s a picture from Wikipedia of the algorithm at work:

The conclusion is that if $G$ is the five-vertex graph at the top of the tree then

$$T_G(x,y)=x^3+2x^2+y^2+2xy+x+y.$$

 

The magnitude function

I’ll say twice what the magnitude function of a graph is: once for people who know what the magnitude of a metric space is, and once for people who don’t. In this part, graphs are not allowed to have loops or multiple edges — or more exactly, they could do, but adjoining or deleting loops or duplicate edges doesn’t make any difference to the magnitude function.

First, here’s the explanation for if you know about magnitude of metric spaces. Every finite metric space $A$ has a magnitude function $t\mapsto \mid tA\mid$, where $tA$ is $A$ scaled up by a factor of $t$ and the bars indicate magnitude. (This “function” may have a finite number of singularities.) We can turn a graph $G$ into a metric space by taking the points to be the vertices and the distance between two vertices to be the number of edges in a shortest path joining them. The magnitude function of a graph is the magnitude function of the resulting metric space. It will be convenient to make the substitution $q=e^{-t}$ and write ${\mu }_G(q)=\mid tG\mid$.

Now here’s a direct explanation.

List the vertices in some order: $v_1,\dots ,v_n$. (The choice of order won’t affect anything.) Let $d_{ij}$ be the number of edges in a shortest path from $v_i$ to $v_j$ (assuming there is one). Let $Z_G(q)$ be the square matrix over $\mathbb{Q}[q]$ whose $(i,j)$-entry is $q^{d_{ij}}$, or $0$ if there is no path from $v_i$ to $v_j$.

We’re going to want to invert $Z_G(q)$, so let’s consider its determinant. It’s a polynomial in $q$. Also, $Z_G(0)=I$, so $\det Z_G(0)=1$; hence $\det Z_G(q)$ is not the zero polynomial. It follows that the matrix $Z_G(q)$ is invertible over $\mathbb{Q}(q)$, the field of rational functions with rational coefficients. We define

$${\mu }_G(q)\in \mathbb{Q}(q)$$

to be the sum of all $n^2$ entries of the inverse matrix $Z_G(q{)}^{-1}$. This is the magnitude function of the graph $G$.

It’s a fact that the magnitude function of a graph can always be expanded as a power series with integer coefficients. This is a special property of ${\mu }_G$: your average rational function over $\mathbb{Q}$ is merely a Laurent series with rational coefficients.

(The key to the proof is that the constant term of the polynomial $\det Z_G(q)$ is $1$, which is invertible in $\mathbb{Z}$. It follows that $1/\det Z_G(q)$ can be expanded as a power series over $\mathbb{Z}$. Hence ${\mu }_G(q)$ can be too.)

What does the magnitude function tell us about a graph?

I’ll come clean: I have no direct evidence that the magnitude function of a graph tells us anything interesting about it. However, in several other settings, magnitude and related invariants have proven to be very fruitful, producing important quantities associated to topological spaces, metric spaces, sets, groupoids, orbifolds, associative algebras, etc. So it’s a good bet that it’s going to be fruitful for graphs too.

Moreover, in the particularly interesting setting of metric spaces, magnitude has been particularly reluctant to give up its secrets. So, the lack of obvious interpretation for the magnitude function of a graph doesn’t make me think it’s less likely to be interesting: it makes me wonder what it’s hiding!

But what does the magnitude function tell us? It’s hard to see. For instance, take the 4-cycle $C_4$:

We have

$${\mu }_{C_4}(q)=\frac{4}{(1+q{)}^2}=4-8q+12q^2-16q^3+\cdots .$$

How can we interpret this? It’s tempting to try to understand a power series over $\mathbb{Z}$ as the generating function of something, but what? Even ignoring the negative signs, what’s being counted here?

For another example, take $G$ to be a forest with vertex-set $V$ and edge-set $E$. Then

$$\begin{array}{rl}{\mu }_G(q)& =\mid {\pi }_0(G)\mid +\mid E\mid \frac{1-q}{1+q}\\ & =\mid V\mid -2\mid E\mid q+2\mid E\mid q^2-2\mid E\mid q^3+\cdots ,\end{array}$$

where ${\pi }_0(G)$ is the set of connected-components of $G$. What is this function telling us about the graph?

Tutte vs. magnitude

In some ways, the information contained in the magnitude function seems to be complementary to that contained in the Tutte polynomial. For example:

• the magnitude function knows how many vertices a graph has, but the Tutte polynomial doesn’t (at least if we’re allowing disconnected graphs)

• the Tutte polynomial knows how many edges a graph has, but the magnitude function doesn’t (at least if we’re allowing multiple edges).

In other ways, the magnitude function and the Tutte polynomial seem more similar than complementary. Let’s consider some families of graphs known to have the same Tutte polynomial:

• all trees with a given number of edges have the same Tutte polynomial; they also have the same magnitude function.

• The three graphs known as the bull, the (3,2)-tadpole and the cricket (shown below) all have the same Tutte polynomial; they also all have the same magnitude function.

In fact, it’s consistent with the evidence below that for connected graphs, the Tutte polynomial determines the magnitude function.

However, in examples such as those above, I see no way of transforming the Tutte polynomial into the magnitude function. One way of proving that such a transformation exists would be to use the so-called “universal property” of the Tutte polynomial. (I don’t know whether it’s a universal property in the categorical sense.) Roughly, this says that Tutte determines magnitude if ${\mu }_G$ is determined by ${\mu }_{G-e}$ and ${\mu }_{G/e}$ whenever $e$ is an edge of $G$ that is neither a bridge nor a loop. But I don’t know whether that’s the case; I suspect not.

All in all, I don’t think the magnitude function is likely to be a specialization of the Tutte polynomial, even for connected graphs. In other words, the magnitude function seems to contain extra information about the graph that Tutte doesn’t. But what?

Properties and examples

I’ll now try to convey an idea of how the magnitude function behaves, by way of some basic properties and examples. For comparison, I’ll show the behaviour of the Tutte polynomial alongside.

1. Disjoint unions Any two graphs $G$ and $H$ have a disjoint union $G+H$.

   Tutte: $T_{G+H}=T_G{T}_H$.

   Magnitude: ${\mu }_{G+H}={\mu }_G+{\mu }_H$.

2. Joins Suppose we have a graph $G$ with a distinguished vertex $x$, and another graph $H$ with a distinguished vertex $y$. We can form a new graph $G*H$, the “join”, by first taking $G+H$ then identifying $x$ with $y$.

   Tutte: $T_{G*H}=T_G{T}_H$.

   Magnitude: ${\mu }_{G*H}={\mu }_G+{\mu }_H-1$.

3. Knows the number of vertices?

   Tutte: no, if we’re allowing graphs to have multiple connected-components: see “discrete graphs” below. Yes, if we stick to connected graphs: then the degree of $T_G(x,y)$ as a polynomial in $x$ (with $y$ regarded as constant) is $\mid V\mid -1$.

   Magnitude: yes: it’s ${\mu }_G(0)$.

4. Knows the number of edges?

   Tutte: yes: $T_G(2,2)=2^{\mid E\mid }$. (Here and later, I’ll use $E$ for the set of edges of a graph $G$, and $V$ for the set of vertices.)

   Magnitude: no, if we’re allowing loops or multiple edges, since adding a loop or a duplicate edge doesn’t change the magnitude. Yes, if we stick to graphs without loops or multiple edges: the coefficient of $q$ in the power series expansion of ${\mu }_G(q)$ is $-2$ times the number of edges. In other words, $\mid E\mid =-\mu {\prime }_G(0)/2$. (See this comment, and the reply to it, below.)

5. Discrete graphs Let $G$ be a discrete graph, that is, one with no edges at all.

   Tutte: $T_G(x,y)=1$.

   Magnitude: ${\mu }_G(q)=\mid V\mid$.

6. Forests Let $G$ be a forest, that is, a disjoint union of trees.

   Tutte: $T_G(x,y)=x^{\mid E\mid }$.

   Magnitude: ${\mu }_G(q)=\mid {\pi }_0(G)\mid +\mid E\mid \cdot \frac{1-q}{1+q}=\mid V\mid -2\mid E\mid q+2\mid E\mid q^2-2\mid E\mid q^3+\cdots$.

7. Cycles Let $C_n$ be the $n$-cycle:

   

   Tutte: $T_{C_n}(x,y)=x^{n-1}+x^{n-2}+\cdots +x+y=\frac{x^n-x}{x-1}+y$

   Magnitude: ${\mu }_{C_n}(q)=\frac{n(q-1)}{q^{\lfloor (n+1)/2\rfloor }+q^{\lceil (n+1)/2\rceil }-q-1}$. (Those are the floor and ceiling functions in the denominator.) This naturally splits into two cases. If $n$ is even then

   $${\mu }_{C_n}(q)=\frac{n(q-1)}{(q^{n/2}-1)(q+1)}.$$

   If $n$ is odd then

   $${\mu }_{C_n}(q)=\frac{n(q-1)}{2q^{(n+1)/2}-q-1}.$$

8. Complete graphs Let $K_n$ be the complete graph on $n$ vertices (that is, every vertex is joined to every other by a single edge).

   Tutte: it seems that there is no neat formula for $T_{K_n}$, and that computing it is nontrivial. A single example:

   $$T_{K_4}(x,y)=(x^3+y^3)+(x^2+y^2)+2(x+y{)}^2+2(x+y).$$

   Magnitude: ${\mu }_{K_n}(q)=\frac{n}{1+(n-1)q}$.

9. Complete bipartite graphs I’ll just do one. Let $K_{2,3}$ be this graph:

   

   Tutte: $T_{K_{2,3}}(x,y)=x^4+2x^3+3x^2+3xy+y^2+x+y$.

   Magnitude: ${\mu }_{K_{2,3}}(q)=\frac{5-7q}{(1+q)(1-2q^2)}=5-12q+22q^2-36q^3+56q^4-\cdots$.

10. Bull, tadpole, cricket These three graphs all have the same Tutte polynomial and the same magnitude function:

           

    Let $G$ be any one of them.

    Tutte: $T_G(x,y)=x^2(x^2+x+y)$.

    Magnitude: ${\mu }_G(q)=\frac{5+5q-4q^2}{(1+q)(1+2q)}=5-10q+16q^2-28q^3+52q^4-\cdots$.

    There’s no mystery as to why they all have the same Tutte polynomial. It’s simply because all three graphs are the join of $C_3$ and two copies of the graph consisting of a single edge, and the Tutte polynomial of the join of a bunch of graphs is determined by their individual Tutte polynomials. The same goes for the magnitude functions.

11. Petersen graph Don Knuth observed that the Petersen graph “serves as a counterexample to many optimistic predictions about what might be true for graphs in general”. So even though I don’t have a prediction to be counterexampled, I thought I’d better check its magnitude function for disturbing behaviour. I’m not disturbed.

    

    Tutte: according to Wikipedia, the Petersen graph $P$ has Tutte polynomial

    $$\begin{array}{rl}{T}_P(x,y)=& 36x+120x^2+180x^3+170x^4+114x^5+56x^6+21x^7+6x^8+x^9\\ & +36y+84y^2+75y^3+35y^4+9y^5+y^6+168xy+240x^{2}y+170x^{3}y+70x^{4}y\\ & +12x^{5}y+171x{y}^2+105x^2{y}^2+30x^3{y}^2+65x{y}^3+15x^2{y}^3+10x{y}^4.\end{array}$$

    Magnitude: ${\mu }_P(q)=\frac{10}{1+3q+6q^2}=10-30q+30q^2+90q^3-450q^4+\cdots$.

    Actually, this does defeat a conjecture! Having read the previous examples, you might have wondered whether the coefficients of the magnitude function (when expanded as a power series) always alternate in sign. The Petersen graph shows that they don’t.

12. Linear codes I’m going to be very sketchy here. (If you want to know more, just say so in the comments.) Let $F$ be a finite field. A linear code over $F$ is simply a linear subspace of $F^n$, for some $n$.

    Tutte: Every linear code $C$ has a Tutte polynomial $T_C$. (More exactly, the definition of Tutte polynomial extends smoothly from graphs to matroids, and every linear code gives rise to a matroid.) But also, with every linear code is associated an important polynomial called its “weight enumerator”. It’s a fact that the weight enumerator of a code $C$ is determined by its Tutte polynomial together with the cardinalities of $C$ and $F^n$.

    Magnitude: On the other hand, every linear code can be regarded as a metric space (with the Hamming distance inherited from $F^n$), and so has a magnitude function. It’s a fact that the weight enumerator of a code is essentially the same thing as its magnitude function; each determines the other once you know the cardinality of $C$.

    So in the case of linear codes, the Tutte polynomial (plus the cardinalities of $C$ and $F^n$) determines the magnitude function.

Questions

The magnitude function of a graph remains mysterious to me. I have very little intuition for what it means. I understand the magnitude of a metric space moderately well for subspaces of ${ℝ}^n$, but metric spaces coming from graphs are about as un-Euclidean as can be.

Here are some questions I’d like answering.

1. What information can be read off from the magnitude function of a graph?

2. Is there a useful way to regard the power series expansion of the magnitude function as a generating function? In other words, do the coefficients count something?

3. It’s consistent with the evidence above that two graphs with the same Tutte polynomial and the same number of vertices have the same magnitude function. Is it true?

4. It’s consistent with the evidence above that two connected graphs with the same Tutte polynomial have the same magnitude function. Is it true?