Formal Scientific Modeling

21 December, 2025

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In January I’m going to a workshop on category theory for modeling, with a focus on epidemiology.

• Formal scientific modeling: a case study in global health, 2026 January 12-16, American Institute of Mathematics, Pasadena, California. Organized by Nina Fefferman, Tim Hosgood, and Mary Lou Zeeman.

It’s sponsored by American Institute of Mathematics, the NSF, the Topos Institute, and the US NSF Center for Analysis and Prediction of Pandemic Expansion. Here are some of the goals:

1. Get a written problem list from a bunch of modelling experts, i.e. statements of the form “I’ll be interested in categorical approaches to modelling when they can do X”, or “how would category theory think about this specific dynamical behaviour, or is this actually not a category theory question at all?”, or … and so on.

2. Make academic friends. There will be people who are not at all category theorists (many of them haven’t even heard of the subject) but who have elected to spend 5 days at a working conference to actually work with some category theorists.

3. There will probably be a lot of conversations that are essentially 5–15 minute speed tutorials in “what is agro-ecology”, or “how do diabetes models work”, or “what does it mean to implement climate databases in a non-trivial way”.

I think looking at examples of existing successful collaborations between category theorists and modelers will help this meeting work better. I’m hoping to give a little talk about the one I’ve been involved in.

I really had very little idea how category theory could actually help modelers until Nate Osgood, Xiaoyan Li, Kris Brown, Evan Patterson and I spent about 5 years thinking about it. We used category theory to develop radically new software for modeling in epidemiology. It was crucial that Nate and Xiaoyan do modeling for a living, while Kris and Evan design category-based software for a living. And it was crucial that we worked together for a long, long time.

But I’m hoping that what we learned can help future collaborations. I’ve written up a few insights here:

• Applied category theory for modeling.

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Summer Research at Topos

27 November, 2025

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You can now apply for the 2026 Summer Research Associate program at the Topos Institute! This is a great opportunity.

Details and instructions on how to apply are in the official announcement.

A few important points:

• The application deadline is January 16, 2026.
• The position is paid and in-person in Berkeley, California.

These positions will last for 8 – 10 weeks, starting in June 2026 and ending in August. Each position will be mentored by Topos research staff or a select number of invited mentors. All positions are 40 hours/week, and the salary starts at $30-$50/hour.

There’s a research track and an engineering track. For the research track, possible topics include:

• Computational category theory using CatColab (Rust/Typescript skills recommended)
• Double category theory
• Categorical statistics
• Polynomial functors
• Interacting dynamical systems
• Hybrid dynamical systems, attractor theory and fast-slow dynamics
• Proof assistants, formal verification, or structure editors
• Philosophical and ethical aspects of applied category theory

For the engineering track, possible topics include:

• Delivery and support of mathematical technologies for various scientific disciplines and applications, and/or analysis, documentation, or guidance on their uses.
• Designing, implementing, testing, and maintaining software at the Topos Institute, in close collaboration with the research staff and in line with institute’s scientific strategy and mission.
• Contributing to developing the CatColab platform, including front end development in TypeScript and/or back end development in Rust. You might also contribute to the mathematical core, written in Rust, as your mathematical experience permits.

All positions require collaboration within a multi-disciplinary research environment. Each summer research associate will complete a specific Topos project, and will write a blog post by the last week of their employment. These projects may include an internal talk, software contribution, or paper. Go here to see the accomplishments of previous research associates.

Topos is committed to building a team with diverse perspectives and life experiences, so those with personal or professional backgrounds underrepresented at Topos are highly encouraged to apply. They are dedicated to shaping the future of technology to ensure a more equitable and just world, and believe that a technology that supports a healthy society can only be built by an organization that supports its team members.

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Beyond the Geometry of Music

22 November, 2025

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Yesterday I had a great conversation with Dmitri Tymoczko about groupoids in music theory. But at this Higgs Centre Colloquium, he preferred to downplay groupoids and talk in a way physicists would enjoy more. Click on the picture to watch his talk!

What’s great is that he’s not faking it: he’s really found deep ways in which symmetry shows up pervasively in music.

At first he tried to describe them geometrically using ‘orbifolds’, which are spaces in which some singular points have nontrivial symmetry groups, like the tip of a cone. But then he realized that the geometry was less important than the symmetry, which you can describe using ‘groupoids’: categories where every morphism is invertible. That’s why his talk is called “Beyond the geometry of music”.

I’m helping him with his work on groupoids, and I hope he explains his work to mathematicians someday without pulling his punches. I didn’t get to interview him yesterday, but I’ll try to do that soon.

For now you can read his books A Geometry of Music and Harmony: an Owner’s Manual along with many papers. What I’ve read so far is really exciting.

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Safeguarded AI (Part 2)

19 November, 2025

60 people, including a lot of category theorists, are meeting in Edinburgh for the £59 million UK project called Safeguarded AI. I talked about it before here.

The plan is to build software that will let you precisely specify systems of many kinds, which an AI will design, and verify that what the AI designed meets your specifications. So: it’s not about building an AI, but instead, building a way to specify jobs for it and verify that it did those jobs correctly!

The director of this project, David Dalrymple, has changed the plan recently. There were many teams of category theorists designing formalisms to get this job done. David Jaz Myers at Topos Research UK was supposed to integrate all these formalisms. That would be a huge job.

But recently all but a few teams have been cut off from the main project—they can now do whatever they want. The project will focus on 3 parts:

1) The “categorical core”: a software infrastructure that lets you program using category theory concepts. I think Amar Hadzihasanovic, my former student Owen Lynch, and two others will be building this.

2) “DOTS”: the double operadic theory of systems, a general framework for building systems out of smaller parts. This is David Jaz Myers’ baby—see the videos.

3) Example applications. One of these, building colored Petri nets, will be done by my former student Jade Master. I don’t know all the others.

By September 2026, David Jaz Myers, Sophie Libkind, Matteo Capucci, Jason Brown and others are supposed to write a 300-page “thesis” on how this whole setup works. Some of the ideas are already available here:

• David Jaz Myers and Sophie Libkind, Towards a double operadic theory of systems.

It feels funny that so much of the math I helped invent is going into this project, and there’s a massive week-long meeting about it just a ten minute walk away, but I’m not involved. But this was by choice, and I’m happier just watching.

I apologize for any errors in the above, and for leaving out many other names of people who must be important in this project. I’ve spoken to various people involved, but not enough. I’m going to talk to David Jaz Myers tomorrow, but he wants to talk about what I’m really interested in these days: octonions and particle physics!

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The Standard Model – Part 3

10 November, 2025

Physics is really bizarre and wonderful. Here I start explaining why the Standard Model has U(1) × SU(2) × SU(3) as its symmetry group. But I don’t assume you know anything about groups or quantum mechanics! So I have to start at the beginning: how the electromagnetic, weak, and strong force are connected to the numbers 1, 2, and 3. It’s all about quunits, qubits and qutrits.

You’ve heard of bits, which describe a binary alternative, like 0 and 1. You’ve probably heard about qubits, which are the quantum version of bits. The weak force is connected to qubits where the 2 choices are called “isospin up” and “isospin down”. The most familiar example is the choice between a proton and a neutron. A better example is the choice between an up quark and a down quark.

The strong force is connected to qutrits—the quantum version of a choice between 3 alternatives. In physics these are whimsically called “red”, “green” and “blue”. Quarks come in 3 colors like this.

The electromagnetic force is connected to “quunits” – the quantum version of a choice between just one alternative. It may seem like that’s no choice at all! But quantum mechanics is weird: there’s just one choice, but you can still rotate that choice.

Yes, I know this stuff sounds crazy. But this is how the world actually works. I start explaining it here, and I’ll keep on until it’s all laid out quite precisely.

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The Inverse Cube Force Law

5 November, 2025

Newton’s Principia is famous for its investigations of the inverse square force law for gravity. But in this book Newton also did something that remained little-known until fairly recently. He figured out what kind of central force exerted upon a particle can rescale its angular velocity by a constant factor without affecting its radial motion. This turns out to be a force obeying an inverse cube law.

Given a particle in Euclidean space, a central force is a force that points toward or away from the origin and depends only on the particle’s distance from the origin. If the particle’s position at time t is \mathbf{r}(t) \in \mathbb{R}^n and its mass is some number m > 0, we have

m \, \ddot{\mathbf{r}}(t) = F(r(t)) \,\hat{\mathbf{r}}(t)

where \hat{\mathbf{r}}(t) is a unit vector pointing outward from the origin at the point \mathbf{r}(t). A particle obeying this equation always moves in a plane through the origin, so we can use polar coordinates and write the particle’s position as \bigl(r(t), \theta(t)\bigr). With some calculation one can show the particle’s distance from the origin, r(t), obeys

\displaystyle{ m \ddot r(t) = F(r(t)) + \frac{L^2}{mr(t)^3} \qquad \qquad \qquad \qquad (1) }

Here L = mr(t)^2 \dot \theta(t), the particle’s angular momentum, is constant in time. The second term in equation (1) says that the particle’s distance from the origin changes as if there were an additional force pushing it outward. This is a “fictitious force”, an artifact of working in polar coordinates. It is called the centrifugal force. And it obeys an inverse cube force law!

This explains Newton’s observation. Let us see why. Suppose that we have two particles moving in two different central forces F_1 and F_2, each obeying a version of equation (1), with the same mass m and the same radial motion r(t), but different angular momenta L_1 and L_2. Then we must have

\displaystyle{ F_1(r(t)) + \frac{L_1^2}{mr(t)^3} = F_2(r(t)) + \frac{L_2^2}{mr(t)^3} }

If the particle’s angular velocities are proportional then L_2 = kL_1 for some constant k, so

\displaystyle{ F_2(r_1(t)) - F_1(r(t)) = \frac{(k^2 - 1)L_1^2}{mr(t)^3} }

This says that F_2 equals F_1 plus an additional inverse cube force.

A particle’s motion in an inverse cube force has curious features. First compare Newtonian gravity, which is an attractive inverse square force, say F(r) = -c/r^2 with c > 0. In this case we have

\displaystyle{ m \ddot r(t) = -\frac{c}{r(t)^2} + \frac{L^2}{mr(t)^3 } }

Because 1/r^3 grows faster than 1/r^2 as r \downarrow 0, as long as the angular momentum L is nonzero the repulsion of the centrifugal force will beat the attraction of gravity for sufficiently small r, and the particle will not fall in to the origin. The same is true for any attractive force F(r) = -c/r^p with p < 3. But an attractive inverse cube force can overcome the centrifugal force and make a particle fall in to the origin.

In fact there are three qualitatively different possibilities for the motion of a particle in an attractive inverse cube force F(r) = -c/r^3, depending on the value of c. With work we can solve for 1/r as a function of \theta (which is easier than solving for r). There are three cases depending on the value of

\displaystyle{ \omega^2 = 1 - \frac{cm}{L^2} }

vaguely analogous to the elliptical, parabolic and hyperbolic orbits of a particle in an inverse square force law:

\displaystyle{ \frac{1}{r(\theta)} } = \left\{ \begin{array}{lcl} A \cos(\omega \theta) + B \sin(\omega \theta) & \text{if} & \omega^2 > 0 \\ \\ A + B \theta & \text{if} & \omega = 0 \\ \\ A e^{|\omega| \theta} + B e^{-|\omega| \theta} & \text{if} & \omega^2 < 0 \end{array} \right.

The third case occurs when the attractive inverse cube force is strong enough to overcome the centrifugal force: c > L^2/m. Then the particle can spiral in to its doom, hitting the origin in a finite amount of time after infinitely many orbits, like this:

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All three curves above are called Cotes spirals, after Roger Cotes’ work on the inverse cube force law, published posthumously in 1722. Cotes seems to have been the first to compute the derivative of the sine function. After Cotes’ death at the age of 33, Newton supposedly said “If he had lived we would have known something.”

The subtlety of the inverse cube force law is greatly heightened when we study it using quantum rather than classical mechanics. Here if c is too large the theory is ill-defined, because there is no reasonable choice of self-adjoint Hamiltonian. If c is smaller the theory is well-behaved. But at a certain borderline point it exhibits a remarkable property: spontaneous breaking of scaling symmetry. I hope to discuss this in my next column.

For more on the inverse cube force law, see:

• N. Grossman, The Sheer Joy of Celestial Mechanics, Birkhäuser, Basel, 1996, p. 34.

For more on Newton’s work involving the inverse cube force law, see:

• Wikipedia, Newton’s theorem of revolving orbits.

• S. Chandrasekhar, Newton’s Principia for the Common Reader, Oxford U. Press, Oxford, 1995, pp. 183–200.

Cotes’ book is

• Roger Cotes, Harmonia Mensuarum, Cambridge, 1722.

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The Standard Model (Part 2)

3 November, 2025

Check out my video on the big ideas that go into the Standard Model of particle physics!

In the late 1800s physics had 3 main pillars: classical mechanics, statistical mechanics and electromagnetism. But they contradict each other! That was actually good – because resolving the contradictions helped lead us to special relativity and quantum mechanics.

I explain how this worked, or more precisely how it could have worked: the actual history is far more messy. For example, Planck and Einstein weren’t really thinking about the ultraviolet catastrophe when they came up with the idea that the energy of light comes in discrete packets:

• Helge Kragh, Max Planck: the reluctant revolutionary, Physics World, 1 December 2000.

Then, I sketch out how deeper thoughts on electromagnetism led us to the concept of ‘gauge theory’, which is the basis for the Standard Model.

This is a very quick intro, just to map out the territory. I’ll go into more detail later.

By the way, if you prefer to avoid YouTube, you can watch my videos at the University of Edinburgh:

• Edinburgh Explorations.

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Applied Category Theory 2026

29 October, 2025

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The next annual conference on applied category theory is in Estonia!

• Applied Category Theory 2026, Tallinn, Estonia, 6–10 July, 2026. Preceded by the Adjoint School Research Week, 29 June — 3 July.

The conference particularly encourages participation from underrepresented groups. The organizers are committed to non-discrimination, equity, and inclusion. The code of conduct for the conference is available here.

Deadlines

• Registration: TBA
• Abstracts Due: 23 March 2026
• Full Papers Due: 30 March 2026
• Author Notification: 11 May 2026
• Adjoint School: 29 June — 3 July 2026
• Conference: 6 — 10 July 2026
• Final versions of papers for proceedings due: TBA

Submissions

ACT2026 accepts submissions in English, in the following three tracks:

  1. Research

  2. Software demonstrations

  3. Teaching and communication

The detailed Call for Papers is available here.

Extended abstracts and conference papers should be prepared with LaTeX. For conference papers please use the EPTCS style files available here. The submission link is here.

Reviewing is single-blind, and we are not making public the reviews, reviewer names, the discussions nor the list of under-review submissions. This is the same as previous instances of ACT.

Program Committee Chairs

• Geoffrey Cruttwell, Mount Allison University, Sackville
• Priyaa Varshinee Srinivasan, Tallinn University of Technology, Estonia

Program Committee

• Alexis Toumi, Planting Space
• Bryce Clarke, Tallinn University of Technology
• Barbara König, University of Duisburg-Essen
• Bojana Femic, Serbian Academy of Sciences and Arts
• Chris Heunen, The University of Edinburgh
• Daniel Cicala, Southern Connecticut State University
• Dusko Pavlovic, University of Hawaii
• Evan Patterson, Topos Institute
• Fosco Loregian, Tallinn University of Technology
• Gabriele Lobbia, Università di Bologna
• Georgios Bakirtzis, Institut Polytechnique de Paris
• Jade Master, University of Strathclyde
• James Fairbanks, University of Florida
• Jonathan Gallagher, Hummingbird Biosciences
• Joe Moeller, Caltech
• Jules Hedges, University of Strathclyde
• Julie Bergner, University of Virginia
• Kohei Kishida, University of Illinois, Urbana-Champaign
• Maria Manuel Clementino, CMUC, Universidade de Coimbra
• Mario Román, University of Oxford
• Marti Karvonen, University College London
• Martina Rovelli, UMass Amherst
• Masahito Hasegawa, Kyoto University
• Matteo Capucci, University of Strathclyde
• Michael Shulman, University of San Diego
• Nick Gurski, Case Western Reserve University
• Niels Voorneveld, Cybernetica
• Paolo Perrone, University of Oxford
• Peter Selinger, Dalhousie University
• Paul Wilson, University of Southampton
• Robin Cockett, University of Calgary
• Robin Piedeleu, University College London
• Rory Lucyshyn-Wright, Brandon University
• Rose Kudzman-Blais, University of Ottawa
• Ryan Wisnesky, Conexus AI
• Sam Staton, University of Oxford
• Shin-Ya Katsumata, Kyoto Sangyo University
• Simon Willerton, University of Sheffield
• Spencer Breiner, National Institute of Standards and Technology
• Tai Danae Bradley, SandboxAQ
• Titouan Carette, École Polytechnique
• Tom Leinster, The University of Edinburgh
• Walter Tholen, York University

Teaching & Communication

• Selma Dündar-Coecke, University College London, Institute of Education
• Ted Theodosopoulos, Nueva School

Organizing Committee

• Pawel Sobocinski, Tallinn University of Technology
• Priyaa Varshinee Srinivasan, Tallinn University of Technology
• Sofiya Taskova, Tallinn University of Technology
• Kristi Ainen, Tallinn University of Technology

Steering Committee

• John Baez, University of California, Riverside
• Bob Coecke, University of Oxford
• Dorette Pronk, Dalhousie University
• David Spivak, Topos Institute
• Michael Johnson, Macquarie University
• Simona Paoli, University of Aberdeen

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Philip Gibbs – Black Holes and White Holes

27 October, 2025

A white hole is a purely hypothetical time-reversed black hole. What does general relativity say about them? Would they repel you? Could you fall into a white hole—or only fall out? Could the universe be a white hole?

Philip Gibbs answers all these questions and more in my first interview for Edinburgh Explorations—a series put out by the School of Mathematics and the School of Physics and Astronomy at the University of Edinburgh.

I met Philip online on sci.physics in the early 1990s, and together with a bunch of other folks we created the Physics FAQ, which answers a lot of the most fascinating questions about physics. At first I didn’t believe everything he said about white holes, but eventually I realized he was right!

What do I mean by “right”? Be careful: white holes are just solutions of the equations of general relativity, not things we’ve actually seen. But you can work out what general relativity predicts about them: that’s the game here, and that determines what’s “right”. It doesn’t mean white holes actually exist and do these things.

For the physics FAQ, much of it created by Philip Gibbs, go here.

The cover picture, showing the maximally extended Schwarzschild solution containing both a black hole and white hole, was made by Isaak Neutelings.

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The Standard Model (Part 1)

22 October, 2025

It’s our best theory of elementary particles and forces. It’s absolutely amazing: it took centuries of genius to discover that the world is like this, and it’s absolutely shocking. But nobody believes it’s the last word, so we simply call it The Standard Model.

But what does this theory say? I’ll try to explain part of it in this series of videos. I begin by introducing the cast of characters—the particles—and a bit about their interactions:

If you have questions, please ask—either here or on YouTube! Intelligent questions keep me motivated. Without them, I get bored.

By the way, these videos will contain mistakes. For example, this time I forgot to mention one key particle before saying “So I’ve introduced all the actors in the drama.” When I get better at editing videos, I will correct slips like this. But I will always try to point out errors in a “pinned” comment right below the video. So look down there.

Also: I don’t plan to explain the details of quantum field theory. So even if you watch all my videos, you’ll get just a taste of the Standard Model. But I will get into some of the math, so it will be much more than just chat. It will roughly follow this paper:

• John Baez and John Huerta, The algebra of grand unified theories, Bulletin of the American Mathematical Society 47 (2010), 483–552.

But I may explain more prerequisites, like a bit of quantum theory and group representation theory. That would let more people follow along.

This is part of my Edinburgh Exploration series, which will also include interviews.

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