It’s time to update you on the Azimuth Project. This project started out in 2010 when I moved to Singapore, had more time to think thanks to a great job at the Centre for Quantum Technologies, and decided to do something about climate change—or more broadly, the Anthropocene.

But do what? You can read my very first thoughts here. I rounded up some interested people, many of them programmers from outside academia, and we started a wiki to compile relevant scientific information. We thought a lot and wrote a lot about the huge problems confronting our civilization. We did some interesting stuff like making simple climate models—purely for educational purposes, not for trying to predict anything! We also recapitulated a network-based attempt to predict El Niños.

But it soon became clear to me that my own strengths lay not in climate science, and certainly not in leading a group of people outside academia trying to accomplish something practical. I got more and more interested in using category theory to study networks—and more generally in getting category theorists interested in practical things. I figured that category theory could really transform how we think about complex systems made of interacting parts.

I understand a bit about what motivates academics, and how to get them working on things. So, once I put my mind to it, I managed to speed up the trend toward applied category theory, which by now has its own annual conference. I’m on the steering committee of that conference, but luckily there are so many energetic people involved that I don’t have to do much. By now I can barely keep up with the progress in applied category theory, which is visible on the Category Theory Community Server, a forum set up by my student Christian Williams.

Indeed, part of how academia works is that if you get really good students, they go off and do things much better than you could do yourself!

For example, my former student Brendan Fong is an order of magnitude better at organizing things than I am. Together with Joshua Tan and Nina Otter he started the journal *Compositionality*, which has a strong emphasis on applied category theory, though it’s also open to other ways of thinking about compositionality (the study of how complex things can be assembled out of simpler parts). But even more importantly, Brendan now leads the Topos Institute, which brings together applied category theorists and people developing new technologies for the betterment of humanity. I’ll get back to that later.

Another amazingly successful student of mine is Nina Otter, now at Queen Mary University. At least I’ll gladly count her as a student, because she did a master’s thesis with me, on operads and the tree of life. But then she switched to topological data analysis, and she’s now using that to study weather regimes.

A big part of the Azimuth project’s focus on networks has always been studying Petri nets: a general formalism for studying chemical reactions, population biology and many other things.

A bunch of blog articles on Petri nets, written at the Centre for Quantum Technologies with Jacob Biamonte, eventually turned into our book *Quantum Techniques for Stochastic Mechanics*. But a new direction came when Brendan Fong developed decorated cospans, a general technique for studying open systems. My student Blake Pollard and I used these to study ‘open Petri nets’, which we called open reaction networks.

Later, my student Jade Master made the theory of open Petri nets really beautiful using structured cospans, a simplified version of Brendan’s decorated cospans developed by my student Kenny Courser.

Meanwhile something big was brewing. Two fresh PhDs named James Fairbanks and Evan Patterson came up with AlgebraicJulia, a software system that aims to “create novel approaches to scientific computing based on applied category theory”. And among many other things, they grabbed ahold of structured cospans and turned them into something you could write programs with!

In October 2020, together with Micah Halter, they used AlgebraicJulia to redo part of the UK’s main COVID model using open Petri nets. At the time I wrote:

This is a wonderful development! Micah Halter and Evan Patterson have taken my work on structured cospans with Kenny Courser and open Petri nets with Jade Master, together with Joachim Kock’s whole-grain Petri nets, and turned them into a practical software tool!

Then they used

thatto build a tool for ‘compositional’ modeling of the spread of infectious disease. By ‘compositional’, I mean that they make it easy to build more complex models by sticking together smaller, simpler models.Even better, they’ve illustrated the use of this tool by rebuilding part of the model that the UK has been using to make policy decisions about COVID19.

All this software was written in the programming language Julia.

I had expected structured cospans to be useful in programming and modeling, but I didn’t expect it to happen so fast!

Here’s a video about these ideas, from 2020:

Later Evan got a job at the Topos Institute and this work blossomed into the following paper:

• Sophie Libkind, Andrew Baas, Micah Halter, Evan Patterson and James Fairbanks, An algebraic framework for structured epidemic modeling, *Philosophical Transactions of the Royal Society A* **380** (2022), 20210309.

I should have blogged about this, but things are happening so fast I never got around to it! This illustrates why I’ve lost interest in the Azimuth Project as *originally* formulated, with this blog as the main communication hub and the wiki as the information depot: academics with their own modes of communication have been pushing things forward in their own ways too fast for me to blog about it all!

Another example: last summer in Buffalo I helped mentor a bunch of students at a program on applied category theory run by the American Mathematical Society. This led to two very nice papers on open Petri nets and related open networks:

• Rebekah Aduddell, James Fairbanks, Amit Kumar, Pablo S. Ocal, Evan Patterson and Brandon T. Shapiro, A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks.

• Benjamin Merlin Bumpus, Sophie Libkind, Jordy Lopez Garcia, Layla Sorkatti and Samuel Tenka, Additive invariants of open Petri nets.

I want to blog about these, and I will soon!

But at the same time, the use of category theory in epidemiological modeling keeps growing. The early work attracted the attention of a bunch of actual epidemiologists, notably my old grad school pal Nate Osgood, who now works at the University of Saskatchewan, both in computer science and also the department of community health and epidemiology. He helps the government of Canada run its main COVID models! This was a wonderful coincidence, made even sweeter by the fact that Nate was hankering to apply category theory to these tasks.

Nate explained that for modeling disease, Petri nets are less popular than another style of diagram, called ‘stock-flow diagrams’. But one can deal with *open* stock-flow diagrams using the same category-theoretic tricks that work for Petri nets: decorated or structured cospans. We worked this out together with Evan Patterson, Nate’s grad student Xiaoyan Li, and Sophie Libkind at the Topos Institute. And these folks—not me—converted these ideas into AlgebraicJulia code for making big models of epidemic disease out of smaller parts!

We wrote about it here:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams, to appear in the proceedings of *Applied Category Theory 2022*.

Alas, I’ve been too busy to properly blog about this paper, but I’ve given a bunch of talks about it, and you can see some on YouTube. The easiest is probably this one:

Since then we’ve made a huge amount of progress, due largely to Nate and Xiaoyan’s enthusiasm for converting abstract ideas into practical tools for epidemiologists. The current state of the art is pretty well reflected in this paper:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling with stock and flow diagrams.

In particular, Nate’s student Eric Redekopp built a graphical user interface for the software, so epidemiologists knowing nothing of category theory or the language Julia can collaboratively build disease models on their web browsers!

So, a lot of my energy that originally went into the Azimuth Project has, by a series of unpredictable events, become focused on the project of applied category theory, with the most practical application for me currently being disease models.

What happened to climate change? Well, a lot of these modeling methodologies could be applied to power grids or world economic models. In fact stock-flow diagrams were first developed for economics and business in James Forrester’s book *Industrial Dynamics*, and they were later used in the famous Limits to Growth model of the world economy and ecology, called World3. So there is a lot to do in this direction. But—I’ve realized—it would require finding an energetic expert who is willing to learn some category theory and teach me (or some other applied category theorist) what they know.

For now, a more instantly attractive option is working with someone I’ve known since I was a postdoc: Minhyong Kim. He’s now head of the International Center of Mathematical Sciences, and he’s dreamt up a project called Mathematics for Humanity. This will fund research workshops, conferences and courses in these areas:

A. Integrating the global research community

B. Mathematical challenges for humanity

C. Global history of mathematics

I’m hoping to coax people to run a workshop on mathematical epidemiology, but also get people together to tackle many other mathematical challenges for humanity. Minhyong has listed some examples:

The deadline to apply for funding is now June 1st, so if you know anyone who might be interested, please tell them about this—and tell me about them!

So, a lot is going on. But I’ve had very little time to do anything with the Azimuth Wiki or the Azimuth Forum (an online discussion forum for the Azimuth Project). Indeed I’ve largely ignored them for years now. David Tanzer has nobly been providing tech support for these sites. But after many conversations with him about this, I’ve decided that it’s time to close down those sites. So that’s what we plan to do on May 1st.

It’s a bit sad, but as I hope I’ve explained, the spirit of the Azimuth Project lives on. And at this moment I want to thank everyone who has been in involved with it in any way. There are many of you not mentioned above. If I tried to list all of you I’d leave some out, so please accept these collective thanks—and good luck with all your projects!

Thanks, John, and best of luck!

I’m retired from Akamai now. I live in greater Boston. I have spent a bit more a year recuperating from head/brain injury from falling down some stairs. I am looking forward getting better and returning to bryological studies and microphotography.

My world is pretty much my family. I won’t go into great detail now but am very proud of both my sons. My younger son is a professor of Mathematics at University College London.

https://www.ucl.ac.uk/~ucahalk/

Jeff’s wife has a doctorate in virology.

My older son is a successful ex-trader from Wall Street and Harvard alum.

My wife is a director of an organization devoted to managing waste and recycling for 18 towns in Massachusetts.

Best of luck, and thanks for all you have done and do and your help.

Hi! It’s great to hear from you—you were a key part of Azimuth for many years.

I’m terribly sorry to hear about your fall. My wife is very cautious around steps because she knows so many older people who have been injured by falls, up to and including serious brain damage. At first I was skeptical but now I’m convinced. I think the danger of falls tends to be underestimated, but they’re the second leading cause of injury and death after car accidents.

I didn’t know you were interested in bryology. When did that start, and what’s the main reason for your fascination?

It’s good to hear your family news. I’ll have to check out the math your son Jeff is doing.

Thanks, John.

I got interested in Bryology twice, once in college after taking two semesters of Botany, and then again 5-10 years back. The latter was a big re-learn. While I was a Physics major in college I continue to believe mosses offer lots of opportunity for new scientific physical explanation. I’ll continue later.

There are a couple of key features of mosses. First, whether in field or lab they are ideal experimental subjects, in lifespan and size. They have been studied for a long while but modern physical and mathematical models remain unapplied to them. For example AFAIK the behavior of spores aloft in breezes and the release of spores in breezes remain open to detailed modeling. Because mosses are seminal, having arisen some 0.5-0.8 Tya, many of the problems of life modern plants face were first faced by mosses, at least on land. So one wonders the degree to which modern plants solved these problems in the same or similar ways, down to the detailed physics. Lots of opportunity for study there. (More to come.)

So, I guess, what intrigues me is knowing enough about Bryophytes to be able to model them in these physical and mathematical ways.

So, the effort continues. What’s neat about it is that their scientific study appears to have no practical applications. In other words their study is an end unto itself. And that’s very nice.

Thanks John for the updates on the exciting progress in Azimuth Project Research! The Azimuth Forum surely had its day, but activity tapered off. I believe that the ideas and values which motivated its creation are vital, and I hope that they will live on and find expression in other settings.

For myself it’s time for a fresh start, and I will be regrouping here at the Azimuth Blog. I encourage other members of the forum to do so as well. All the Best, David Tanzer

Thanks for all your help over the years, David!

Another thing I wanted to say is this: I now plan to let myself blog about more things here on the Azimuth blog, because I think it’s silly to make people run around to lots of locations (like the n-Category Cafe, and Mathstodon) to read all my stuff. So: be ready for a bunch of articles on astrophysics, pure mathematics, and other things!

But I will also keep talking about things of interest to people who want to save the planet… starting with the new work I mentioned on Petri nets, stock-flow diagrams, and epidemiological models.

I’ve had a great time learning from you all, as USers are wont to say.

I hope you will archive all the forum discussions and the library articles, maybe on github, so we can continue to use them as an information source for non-academic, non-Azimuth people as I’ve been doing for some years.