The n-Category Café

Back to Kenny Courser’s thesis:

• Kenny Courser, Open Systems: A Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020.

Last time I explained the problems with decorated cospans as a framework for dealing with open systems. I vaguely hinted that Kenny’s thesis presents two solutions to these problems: so-called ‘structured cospans’, and a new improved approach to decorated cospans. Now let me explain these!

The University of San Diego invites applications for a postdoctoral research fellowship in homotopy type theory beginning Fall 2021, or earlier if desired. This is intended as a two-year position with potential extension to a third year, funded by the second AFOSR MURI grant for HoTT, entitled “Synthetic and Constructive Mathematics of Higher Structures in Homotopy Type Theory”.

Now that John gave an overview of the Petri nets paper that he and I have just written with Jade and Fabrizio, I want to dive a bit more into what we accomplish. The genesis of this paper was a paper written by Fabrizio and several other folks entitled Computational Petri Nets: Adjunctions Considered Harmful, which of course sounds to a category theorist like a challenge. Our paper, and particularly the notion of $\Sigma$-net and the adjunction in the middle column relating $\Sigma$-nets to symmetric strict monoidal categories, is an answer to that challenge.

I’ve been thinking about Petri nets a lot. Around 2010, I got excited about using them to describe chemical reactions, population dynamics and more, using ideas taken from quantum physics. Then I started working with my student Blake Pollard on ‘open’ Petri nets, which you can glue together to form larger Petri nets. Blake and I focused on their applications to chemistry, but later my student Jade Master and I applied them to computer science and brought in some new math. I was delighted when Evan Patterson and Micah Halter used all this math, along with ideas of Joachim Kock, to develop software for rapidly assembling models of COVID-19.

Now I’m happy to announce that Jade and I have teamed up with Fabrizio Genovese and Mike Shulman to straighten out a lot of mysteries concerning Petri nets and their variants:

• John Baez, Fabrizio Genovese, Jade Master and Mike Shulman, Categories of nets.

This paper is full of interesting ideas, but I’ll just tell you the basic framework.

Take a copy of this!

• This Week’s Finds in Mathematical Physics (1-50), 242 pages.

These are the first 50 issues of This Week’s Finds of Mathematical Physics. This series has sometimes been called the world’s first blog, though it was originally posted on a “usenet newsgroup” called sci.physics.research — a form of communication that predated the world-wide web. I began writing this series as a way to talk about papers I was reading and writing, and in the first 50 issues I stuck closely to this format. These issues focus rather tightly on quantum gravity, topological quantum field theory, knot theory, and applications of n-categories to these subjects. There are, however, digressions into elliptic curves, Lie algebras, linear logic and various other topics.

The Department of Mathematics at Johns Hopkins University solicits applications for one two-year postdoctoral fellowship beginning Summer 2021 (with some flexibility in the start and end dates). The position is funded by the Air Force Office of Scientific Research (AFOSR) through the Multidisciplinary University Research Initiative (MURI) program. This position is open to anyone who is able to obtain a visa to come and work in the US, but it is necessary to be physically in the US to receive funding from this grant. (Johns Hopkins will sponsor and pay for a visa application, if required.)

The $n$-Category Café has recently hosted a lively discussion on the ethics of military funded mathematics and US military funding in particular. This is the first time I’ve collaborated on a military funded grant, so I have limited experience in this area. But every year, I’m heartbroken to disappoint the dozens of highly-qualified postdoctoral applicants I come in contact with. My department also offers university-funded postdoctoral positions (though one could argue that military funding provides some support for all employees at Johns Hopkins) but at some point I calculated that it would be “my turn” to make an offer to my first choice candidate exactly once a decade, and I wanted to try to find a way to hire others in the meanwhile.

Do you want to get involved in applied category theory? Are you willing to do a lot of work and learn a lot? Then this is for you:

• Applied Category Theory 2021 – Adjoint School. Applications due Friday 29 January 2021. Organized by David Jaz Myers, Sophie Libkind, and Brendan Fong.

There are four projects to choose from, with great mentors. You can see descriptions of them below!

By the way, it’s not yet clear if there will be an in-person component to this school — but if there is, it’ll happen at the University of Cambridge. ACT2021 is being organized by Jamie Vicary, who teaches in the computer science department there.

Azat Miftakhov is a finishing graduate student in Mathematics at Moscow State University, and is a political activist. In February 2019 he was arrested and charged with terrorist activity and the production of explosives. These charges were quickly dropped, but he is nevertheless still in pre-trial detention, now under the charge of having participated in a group act of vandalism resulting in a broken window on a building belonging to the United Russia party.

Many disturbing signs of violation of his due legal process have been reported by the press and by human rights activists. These include torture, harassment of his relatives by local police, and a smear campaign involving homophobic slurs in the media. He has also been denied access to his scientific work. It is difficult to see how the charge of minor vandalism could warrant a year of pre-trial detention and this mistreatment. “Memorial”, the oldest Russian human rights organization, lists Azat Miftakhov as a political prisoner.

Please join many prominent mathematicians and sign a petition protesting Azat Miftakhov’s treatment here! The text above is not my own, but copied from the American Mathematical Society, who is also protesting this outrage. For more information go here.

We can think of the exceptional Jordan algebra as a funny sort of spacetime. This spacetime is 27-dimensional, with light rays through the origin moving on a lightcone given by a cubic equation instead of the usual

$$t^2 - x^2 - y^2 - z^2 = 0$$

in 4-dimensional Minkowski spacetime. But removing this lightcone still chops spacetime into 3 connected components: the past, the future, and the regions you can’t reach from the origin without exceeding the speed of light. The future is still a convex cone, and so is the past. So causality still makes sense like it does in special relativity.

At some point I got interested in seeing what physics would be like in this funny spacetime. Greg Egan and John Huerta joined me in figuring out the very basics of what quantum field theory would be like in this world. Namely, we figured out a bit about what kinds of particles are possible.

One difference is that we must replace the usual Lorentz group with the 78-dimensional group $\mathrm{E}_6$. But an even bigger difference is this. In 4d Minkowski space, every point in your field of view acts essentially like every other, if you turn your head. But in our 27-dimensional spacetime, the analogous fact fails! There is a ‘sky within the sky’: some particles moving at the speed of light can only be seen in certain directions. Thus, the classification of particles that move at the speed of light is much more baroque.

This is a big digression from my main quest here: explaining how people have tried to relate the octonions to the Standard Model. But it would be a shame not to make our results public, and now is a good time.

The Dynkin diagram of $\mathrm{E}_6$ has 2-fold symmetry:

So, this Lie group has a nontrivial outer automorphism of order 2. This corresponds to duality in octonionic projective plane geometry! There’s an octonionic projective plane $\mathbb{O}\mathrm{P}^2$ on which $\mathrm{E}_6$ acts. But there’s also a dual octonionic projective plane $(\mathbb{O}\mathrm{P}^2)^*$. Points in the dual plane are lines in the original one, and vice versa. And these two projective planes are not isomorphic as spaces on which the group $\mathrm{E}_6$ acts. Instead, there’s a bijection

$$\alpha : \mathbb{O}\mathrm{P}^2 \to (\mathbb{O}\mathrm{P}^2)^*$$

such that acting by $g \in \mathrm{E}_6$ and then applying $\alpha$ is the same as applying $\alpha$ and then acting by $g' \in \mathrm{E}_6$, where $g'$ is what you get when you apply the outer automorphism to $g$.

Similarly, the group $\mathrm{E}_6$ acts on the exceptional Jordan algebra and its dual, but these are not isomorphic as representations of $\mathrm{E}_6$. Instead they’re only isomorphic up to an outer automorphism.

Today I want to tell you about invariant structures on the exceptional Jordan algebra and its dual. But a lot of this stuff applies more generally.

Jade Master at Riverside has written a short, important and lucid article about military funding of math, The Lie of “It’s Just Math”, accompanied by a call to action:

Fellow mathematicians, it’s time to stop letting the military benefit from our work.

Military involvement in math is particularly an issue in applied category theory, and particularly an issue in the USA. But the principles that Jade pithily expresses are universal:

• The [US Department of Defense’s] real goal is not just the math you produce, they want to gain access to your mathematical community.

• Your math is not too abstract to be useful.

• The DoD wants to normalize themselves in your non-mathematical communities.

• The DoD will lie to you.

Mathematicians are generally highly reluctant to talk about the human impact of what we do and the choices we make. For that reason, we’re not very practised at it. But Jade’s article deserves wide discussion, and I hope it gets it.

I’m planning to stop teaching at U. C. Riverside in June 2021. I’ll only be 60, but what’s the use of quitting work when you’re too old to have fun?

I want to spend more time doing research and writing expository papers and books, and I’ve saved up enough money to do this. I’ll still do serious work, like trying to save the planet with applied category theory. But I’ll also delve into all sorts of puzzles that I haven’t had enough time for yet.

Here’s one. You may have heard about the funny way the number 24 shows up in the homotopy groups of spheres:

$$\pi_{n+3} (S^n) \cong \mathbb{Z}_{24}$$

whenever $n$ is big enough, namely $n \ge 5$. If you try to figure out where this comes from, you’re led back to a map

$$S^7 \to S^4$$

called the quaternionic Hopf fibration. This by itself doesn’t make clear where the 24 is coming from — but you can’t help but notice that when you pack equal-sized balls as densely as is known to be possible in the quaternions, each one touches 24 others.

Coincidence? Maybe! But it’s also true that

$$\pi_{n+7} (S^n) \cong \mathbb{Z}_{240}$$

when $n$ is big enough. And if you try to figure out where this comes from, you’re led back to a map

$$S^{15} \to S^8$$

called the octonionic Hopf fibration. And you can’t help but notice that when you pack equal-sized balls as densely as possible in the octonions, each one touches 240 others!

The category of groups $\mathbb{Z}^n$ and isomorphisms between these is symmetric monoidal under $\oplus$. You can build a space out of simplexes where the 0-simplexes are objects of this category, the 1-simplexes are morphisms, the 2-simplexes are commutative triangles, the 3-simplexes are commutative tetrahedra, and so on forever. This space has an operation, coming from $\oplus$, that obeys the commutative monoid axioms up to homotopy. If you ‘group complete’ this space by throwing in formal inverses, you get a space that’s an abelian group up to homotopy. It’s called the algebraic $K$-theory spectrum of the integers.

The algebraic $K$-theory spectrum of the integers has homotopy groups $\pi_0 = \mathbb{Z}$, $\pi_1 = \mathbb{Z}/2$, $\pi_3 = \mathbb{Z}/48$, and so on. These groups are called the algebraic $K$-theory groups of the integers, $K_n(\mathbb{Z})$.

Hey! There’s a new online course coming up!

• Applied Compositional Thinking for Engineers. January 7, 8, 11-15, 20-22, and 25-29, 2021. Taught by Andrea Censi, Jonathan Lorand and Gioele Zardini.

It’s not an accident that the acronym for “Compositional Thinking” is “CT”.

A ‘mathematical phantom’ is a mathematical object that doesn’t exist in a literal sense, but nonetheless acts as if it did, casting a spell on surrounding areas of mathematics. The most famous example is the field with one element. Another is Deligne’s S_t, the symmetric group on $t$ elements, where $t$ is not a natural number. Yet another is G₃, a phantom Lie group related to G₂, the automorphism group of the octonions.

What’s your favorite mathematical phantom? My examples are all algebraic. Does only algebra have enough rigidity to create the patterns that summon up phantom objects? What about topology or combinatorics or analysis? Okay, G₃ is really a creature from homotopy theory, but of a very algebraic sort.

Last night I met another phantom.