Azimuth Project News

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 that to 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!

16 Responses to Azimuth Project News

  1. ecoquant says:

    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.

    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.

    • John Baez says:

      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.

      • ecoquant says:

        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.

        • ecoquant says:

          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.)

        • ecoquant says:

          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.

  2. David Tanzer says:

    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

    • John Baez says:

      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.

    • nad says:

      Hello David and John
      Unfortunately the certificate for the forum has expired before May 1st. Unfortunately I haven’t been fully done screenshooting the more relevant bits of my forum entries for archiving reasons. I had planned to finish this by May 1st, but now it seems it seems it is not possible to access the forum anymore. Is there a possibility to renew the certificate or make a pdf, which can be acquired at least by “Azimuth members”?

    • John Baez says:

      Hi, Nad! I suggest just ignoring the fact that the certificate has expired. For example Firefox gives me an “advanced” option to go ahead and view the forum here even though the certificate is expired.

      I don’t have a backup of the whole Azimuth Forum, but it seems that at least parts of it are saved on the Wayback Machine, e.g. here.

      • nad says:

        OK. I followed your suggestion – with the hope that David Tanzer would have told you if there would have been some major problems with the forum.
        Anyways, your announcement of closing down the Azimuth project and forum was fairly short of notice. Did you send out notices to “members” of the Project? If yes – I didn’t get a notice.

        If no – then as of today it seems the forum is already gone but the project pages are currently still there – I don’t know, but I think it would be good to at least contact the authors there. Some People had written a lot and they may think that their writings are well preserved at Azimuth.

        “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!”

        The outcome which I take from the Azimuth project is actually that “consulting” from experienced but “neutral” researchers may also improve academic works. With “neutral” I mean researchers which are considerably detached from the respective research areas, especially in terms of funding, i.e. in particular neither funded from within climate science nor from anti-climate science lobbyists. With experienced I mean sufficiently educated to be able to grasp the contents in a justifyable amount of time.

        That is in the “initial phase” of the project (the one roughly from 2010-2018 before the onset of your category classes), i.e. the climate science phase – the majority of Azimuthers were rather not students, but as you already mentioned programmers, post docs, engineers etc. Most of them appeared pretty neutral, but highly motivated to improve things.

        I think it would be good to summarize some results from that phase, things like the Azimuth back-up project for which you even founded a NGO (if I remember correctly) or the investigation of temperature data, which coincidentally had follow-up events of temperature improvements :

        As you may remember in 2015 I had also discussions with Gunnar Myhre from IPCC about missing shortwave spectral data in the climate models – there were now also coincidental follow-up events, as indicated in the new IPCC report in

        “The SARF for methane (CH4) has been substantially increased due to updates to spectroscopic data and inclusion of shortwave absorption (Etminan et al., 2016)….”

        I highlighted these follow-up events because one can see them either as that Azimuth comments were so substantial that they coincidentally anticipated upcoming research or that climate scientists revised their findings because of the Azimuth questions.

        So in short I think it would be worthwhile to summarize some achievements and to describe the climate science Azimuth project at least as a Wikipedia article. I am willing to contribute to this. Disclaimer: this is partially due to the fact that I am more and more pressured to justify my unpaid work at the Azimuth project and I have no records of any “achievements” at Azimuth that I could eventually use for generating an income and now it seems that I cannot even refer to the Azimuth project as such. What do you think?

        • John Baez says:

          Anyways, your announcement of closing down the Azimuth project and forum was fairly short of notice. Did you send out notices to “members” of the Project? If yes – I didn’t get a notice.

          No, I didn’t have an easy way to email everyone so I announced it here and on the Forum around March 9th, roughly two months before it closed down.

          The Azimuth Wiki still exists here, so use it now if you want, before it goes away. It will continue to exist on the Internet Archive, here:


          The Azimuth Forum seems to be gone now, but it will continue to exist on the Internet Archive, here:


          I think it would be worthwhile to summarize some achievements and to describe the climate science Azimuth project at least as a Wikipedia article. I am willing to contribute to this.

          It sounds like a good idea, but I don’t have the energy to help. As explained in this article, I’ve become quite involved in creating software for modeling systems. This seems to be working well, but precisely for that reason it eats up a lot of time.

        • nad says:

          OK. I put together a draft on Wikipedia:
          Please check if there are some mistakes, especially in attribution. In particular I wasn’t fully sure about who hosted what, when and where (Wiki, Forum). The draft page says:

          Review waiting, please be patient.

          This may take 4 months or more, since drafts are reviewed in no specific order. There are 4,729 pending submissions waiting for review.

          If the submission is accepted, then this page will be moved into the article space.
          If the submission is declined, then the reason will be posted here.
          In the meantime, you can continue to improve this submission by editing normally.

        • nad says:

          Some new developments from the Wikipedia submission: The draft had been rejected, but at least the rejecting person AngusWOOF has sofar been communicating about reasons for rejection:

        • nad says:

          I wrote a new shorter piece upon request of one of the wikipedia editors:

          This got rejected as well. I had also a discussion on the wikipedia chat about that.

          The problem is that no outside resource (outside of all people who participated) did mention the project by name in any independent “worthy” public english word resource. The Kickstarter funding doesn’t count, because Kickstarter is not deemed “notabilityworthy” within wikipedia that is you are not allowed to use it as a reference for “notability” of an outside crowd. “notability” has to be some kind of written statement, not just a payment. the fact that there were over 140 independent stem professionals who did participate voluntarily in the Azimuth project doesn’t count, because as soon they wrote something for the project they are not anymore regarded as independent from the project they write for. Likewise the Climate mirror webpage which lists the Azimuth project doesn’t count because no one (in particular journalists) did write (“noted”) about it. So I guess the Azimuth project will go down the gutter of history. This is bad for people who have to justify what they did at Azimuth.

  3. Jim Stuttard says:

    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.

  4. Noah Bernes says:

    I would recommend you look into the field of “economically unequal exchange”. The task of this field is to account for global flows of commodities, labor embodied in those commodities and to track environmental dynamics associated with these flows. In general, one accounts for key ecosystem dynamics, and witnesses ecological disturbances from the processes of production and circulation. People have been trying to come up with a way of “accounting” all these disparate forms of capital including geobiochemical dynamics AND social relations. This obviously requires at least a naive use of category theory to unify these disparate fields into one theory.

    However, all of the treatments I have seen use bulky spreadsheet methods and simply juxtapose heterogenous data in a naturalist manner which makes the theory simply glorified accounting as of now.

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