I’m trying to lose weight. I was so happy when I saw my local pastry shop sold negative-mass doughnuts! But what happened next really freaked me out. Hear the tale here: This is the second of my Edinburgh Explorations videos. See if you can spot the dumb mistake at 14:25. In this video, I examine … … Continue reading →| Azimuth
‘Prussian blue’ is a crystal so blue you can’t accurately show it on most computer screens, since they can only display a limited region of color space. Its structure is really cool. It’s a cubical lattice made of iron atoms, each surrounded by 6 cyanides—carbon and nitrogen. But let Sean Silver explain it: The modern … … Continue reading →| Azimuth
I have a new job here in Scotland: I’m the Maxwell Fellow for Public Engagement. As part of that job I’m starting up a series of videos where I explain math and physics and also interview people here at the University. Right now I’m calling it Edinburgh Explorations, and here is the first one: I … … Continue reading →| Azimuth
I’ve been studying the physics of negative mass. Basically it doesn’t exist. But physicists are have run into a serious problem. They think they can use astronomical measurements to put…| Azimuth
Here is the video of my talk at Applied Category Theory 2023. While it has ‘epidemiology’ in the title, it’s mostly about general ways to use category theory to build flexible, ad…| Azimuth
How, exactly, can category theory help modeling in public health? I wrote a paper about this with two people who helped run Canada’s COVID modeling, together with a software engineer and a mathematician at the Topos Institute: • John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling […]| Azimuth
Is category theory being co-opted? These authors think so: • Esteban Montero and Brandon Baylor, Category theory is being co-opted, Holon Substack, 7 September 2025. Category theory offers a p…| Azimuth
In 2004, a team of scientists discovered hydrocarbons called anthracene and pyrene in an amazing structure called the Red Rectangle! Here two stars 2300 light years from us are spinning around each other while pumping out a huge torus of icy dust grains and hydrocarbon molecules. It’s not really shaped like a rectangle or X—it […]| Azimuth
Today I gave $10,000 to Doctors Without Borders, since they’re doing a lot of good work in Gaza. I made this gift in memory of Mariam Abu Dagga, a freelance photographer who was killed in the…| Azimuth
This image shows a filament of cosmic dust over ten light years long! It’s part of a giant cloud of cold gas and dust that’s starting to collapse under its own gravity to form stars. Newborn stars are hidden inside. The cosmic dust grains here are so cold that observations at millimeter wavelengths were needed […]| Azimuth
When a star first forms, it is powered not by nuclear fusion but simply by gravity. It shrinks, which causes a release of gravitational energy. This tends to heat it, which slows its shrinking. But…| Azimuth
You can cut a hole in a cube that’s big enough to slide an identical cube through that hole! Think about that for a minute—it’s kind of weird. Amazingly, nobody could prove any co…| Azimuth
When bad news gets me down, I often get insomnia. I wake up in the middle of the night, start thinking about how we’re all doomed, and can’t easily stop. To break out of these doom loop…| Azimuth
Today I’d like to dig a little deeper into some ideas from Part 2. I’ve been talking about causal loop diagrams. Very roughly speaking, a causal loop diagram is a graph with labeled edg…| Azimuth
Astronomers have found a truly huge black hole! It’s in the massive galaxy in the center here, called the Cosmic Horseshoe. The blue ring is light from a galaxy behind the Cosmic Horseshoe, s…| Azimuth
This week, 50 category theorists and software engineers working on “safeguarded AI” are meeting in Bristol. They’re being funded by £59 million from ARIA, the UK’s Advanced …| Azimuth
In Part 10 we saw that, loosely speaking, the theory of a hydrogen atom is equivalent to the theory of a massless left-handed spin-½ particle in the Einstein universe—a static universe where space is a 3-sphere. Today we’ll ‘second quantize’ both of these equivalent theories and get new theories that again are equivalent. ‘Second quantization’ […]| Azimuth
The poet Blake wrote that you can see a world in a grain of sand. Today we’ll see a universe in an atom! We’ll see that states of the hydrogen atom correspond to states of a massless spin-½ particle in the Einstein universe—a closed, static universe where space is a 3-sphere. The rotational symmetries of […]| Azimuth
Today I want to make a little digression into the quaternions. We won’t need this for anything later—it’s just for fun. But it’s quite beautiful. We saw in Part 8 that if we take the spin of the electron into account, we can think of bound states of the hydrogen atom as spinor-valued functions on […]| Azimuth
Now comes the really new stuff. I want to explain how the hydrogen atom is in a certain sense equivalent to a massless spin-½ particle in the ‘Einstein universe’. This is the universe Einstein believed in before Hubble said the universe was expanding! It has a 3-sphere for space, and this sphere stays the same […]| Azimuth
Did you know that Lawvere did classified work on arms control in the 1960s, back when he was writing his thesis? Did you know that the French government offered him a job in military intelligence? The following paper should be interesting to applied category theorists—for a couple of different reasons: • Bill Lawvere, The category […]| Azimuth
I’ve explained a cool way to treat bound states of the hydrogen atom as wavefunctions on a sphere in 4-dimensional space. But so far I’ve been neglecting the electron’s spin. Now let’s throw that in too! This will wind up leading us in some surprising directions. So far I’ve just been reviewing known ideas, but […]| Azimuth
I’ve been working with Adittya Chaudhuri on some ideas related to this series of blog articles, and now our paper is done! • John Baez and Adittya Chaudhuri, Graphs with polarities. Abst…| Azimuth
Back before satellites, to transmit radio waves over really long distances folks bounced them off the ionosphere—a layer of charged particles in the upper atmosphere. Unfortunately this layer…| Azimuth
In Part 10 we saw that, loosely speaking, the theory of a hydrogen atom is equivalent to the theory of a massless left-handed spin-½ particle in the Einstein universe—a static universe where space is a 3-sphere. Today we’ll ‘second quantize’ both of these equivalent theories and get new theories that again are equivalent.| Azimuth
In 1962, V. A. Antonov did some remarkable simulations showing that in Newtonian mechanics, gravitating systems can violate the usual rules of thermodynamics. Instead of reaching equilibrium they c…| Azimuth
guest post by Nathaniel D. Osgood Together with 4 students from our Computational Epidemiology and Public Health Informatics Lab (CEPHIL), I spent my Friday at one of our community group model buil…| Azimuth
In Part 2, I explained some stuff you can do with graphs whose edges are labeled by elements of a rig. Remember, a rig is like a ring, but it might not have negatives. A great example is the boolea…| Azimuth
I’m talking about ‘causal loop diagrams’, which are graph with edges labeled by ‘polarities’. Often the polarities are simply $latex +$ and $latex -$ signs, like here: But p…| Azimuth
Last time I explained ‘causal loop diagrams’, which are graphs with edges labeled by plus or minus signs, or more general ‘polarities’. These are a way to express qualitativ…| Azimuth
This is a progress report on some joint work with Xiaoyan Li, Nathaniel Osgood and Evan Patterson. Together with collaborators we have been developing software for ‘system dynamics’ mod…| Azimuth
guest post by David Spivak • Part 1: The problem. Creating a knowledge network In 2007, I asked myself: as mathematically as possible, what can formally ground meaningful information, includin…| Azimuth
Agent-based models are crucial in modern epidemiology. But currently, many of these models are large monolithic computer programs—opaque to everyone but their creators. That’s no way to…| Azimuth
I learned a lot from the comments on Part 3 and also this related thread on the Category Theory Community Server: • Coalgebras, operational semantics and the Giry monad. I’d like to than…| Azimuth
By next summer I need to understand agent-based models well enough to help people design better software for such models using category theory. And recently this project has been folded into an eve…| Azimuth
I’m working with Nate Osgood and other folks to develop better modeling tools for epidemiologists. Right now we’re trying to develop a category-based framework for agent-based models. I…| Azimuth
I’ve been explaining our software for agent-based models based on stochastic C-set rewriting systems. But my only example so far has been the Game of Life, which is not what most people would…| Azimuth
Our 6-week Edinburgh meeting for creating category-based software for agent-based models is done, yet my collaborators are still busy improving and expanding this software. I want to say more about…| Azimuth
Today I’d like to wrap up my discussion of how to implement the Game of Life in our agent-based model software called AlgebraicABMs. Kris Brown’s software for the Game of Life is here: …| Azimuth
Last time I began explaining how to run the Game of Life on our software for stochastic C-set rewriting systems. Remember that a stochastic stochastic C-set rewriting system consists of three parts…| Azimuth
We’ve been hard at work here in Edinburgh. Kris Brown has created Julia code to implement the ‘stochastic C-set rewriting systems’ I described last time. I want to start explainin…| Azimuth
Since May 1st, Kris Brown, Nathaniel Osgood, Xiaoyan Li, William Waites and I have been meeting daily in James Clerk Maxwell’s childhood home in Edinburgh. We’re hard at work on our pro…| Azimuth
Last time I presented a class of agent-based models where agents hop around a graph in a stochastic way. Each vertex of the graph is some ‘state’ agents can be in, and each edge is call…| Azimuth
Last time I presented a simple, limited class of agent-based models where each agent independently hops around a graph. I wrote: Today the probability for an agent to hop from one vertex of the gra…| Azimuth
Today I’d like to start explaining an approach to stochastic time evolution for ‘state charts’, a common approach to agent based models. This is ultimately supposed to interact we…| Azimuth
I’ve spent the last few weeks drawing pictures of tuning systems, since I realized this is a good way to show off their mathematical beauty. Now I’ll start deploying them! I’ve al…| Azimuth
In Part 3 of this series, we figured out all the scales where we choose 7 notes from the 12 notes in the usual chromatic scale, with the property that the biggest step between notes is a whole tone…| Azimuth
In Part 3, we figured out all the 7-note scales drawn from the usual 12-tone scale and obeying an extra constraint: there’s at most a whole tone between consecutive notes. We got a total of 2…| Azimuth
I want to tell you more about modes! I’ve been playing music and thinking about modes a lot, but now I have more to say. In Part 1, I described the 7 modes formed by starting the major scale …| Azimuth
If you start at any note of the piano and keep going up fifths until you’ve got a total of 7 notes, you get the Lydian mode. I explained this last time. Above I show how it works starting wit…| Azimuth
Some news! Nathaniel Osgood, Evan Patterson, Kris Brown, Xiaoyan Li, Sean Wu, William Waites and I are going to work together at the International Centre for Mathematical Sciences for six weeks sta…| Azimuth
The notes in the major scale are spaced in a funny way. Look at the white keys on the piano: some have black keys between them, other don’t. But you can understand the major scale using the c…| Azimuth
guest post by David Spivak The problem The idea that’s haunted me, and motivated me, for the past seven years or so came to me while reading a book called The Moment of Complexity: our Emerging Net…| Azimuth
When you first learn about the major scale it’s fairly straightforward, because they tell you about just one major scale. But the minor scale is more tricky, because they tell you about three…| Azimuth
I’ve been away from my piano since September. I really miss playing it. So, I’ve been sublimating my desire to improvise on this instrument by finally learning a bunch of basic harmony …| Azimuth
