There’s some exciting very recent work by Alpöge and Buckmaster, building upon prior work by Córdoba and Martínez-Zoroa, in the general topic around the infamous global regularity problem for the incompressible three-dimensional Navier-Stokes equations. It is now widely expected that it should be possible to construct smooth initial data and smooth forcing term that would make these equations develop singularities in finite time; and it should even be possible to do without the forcing term. While these authors do not quite achieve these goals yet, they have made enough of a breakthrough that it looks very feasible to complete these goals in the near future. In particular, they have demonstrated such finite time blowup for three simpler model equations: the incompressible porous medium (IPM) equation, the two-dimensional Boussinesq equation, and the three-dimensional incompressible Euler equations. (The first of these equations was already handled by Córdoba and Martínez-Zoroa, but Alpöge and Buckmaster found a variant of their method that also extended to the other two equations, and has a high likelihood of also extending to Navier-Stokes as well.) As is now remarkably feasible in the modern era of autoformalization agents, their work has also been formalized in Lean.

As one may expect nowadays, the arguments here are heavily AI-assisted, but the authors have been working over the last few weeks to simplify and rewrite the proofs from what they literally call “the worst writeup we had ever seen in the history of mathematics” into something far more readable and of professional quality. This is still a work in progress: unfortunately, they were forced to release their preliminary preprints before they were completely digested and polished, due to external events that are documented on the above link. Nevertheless, the introduction to the Boussinesq paper at least is in pretty good shape, and can serve as an initial starting point. I was also fortunate to have Tristan Buckmaster explain the main ideas of the paper in a half-hour phone conversation, although I still need to work some more (probably with some combination of a blackboard and modern AI tools) to digest things more. For now I will try to write a quick summary of some of the main ideas, the highlighting of which I view as the main value of such work; the actual solving of these problems is only a proxy goal for the primary goal of developing mathematical understanding and insight. Without such understanding, even a problem as infamous as the Navier-Stokes regularity problem of far less intrinsic significance to mathematics than is sometimes promoted in popular media.

The basic strategy, due to Cordoba and Martínez-Zoroa, is to iteratively build up the solution to such equations in stages, repeatedly adding small high frequency corrections to a previous (forced) solution in a manner that makes the solution more singular towards the blowup time while keeping the forcing term well behaved. Rather than work with any specific equation, let’s work with a completely abstract equation

where is the solution, is the nonlinear differential operator representing the equation of motion, and is the forcing term. Of course this is far too general a setting to perform a full analysis, but it should suffice for this brief post.

Suppose that one has already managed to construct a low frequency solution

to this equation, and would like to perturb it to create a new solution

that adds a high frequency correction to the solution that starts emerging near the blowup time, at the cost of some (presumably also) high frequency correction to the forcing term. If one can make the amplitude of the solution correction relatively large while keeping the amplitude of the correction very low, and the frequencies of the corrections increase rapidly with each iteration, then one can hope to iterate this procedure and pass to a limit to obtain a solution

where now exhibits blowup in finite time, while remains smooth.

To make this strategy work, should approximately solve the difference equation

to keep small. If we can somehow neglect nonlinear effects, this basically amounts to solving a linearized equation

The game is then to design the background solution in such a way that the evolution equation exhibits some sort of exploitable instability, in which a solution to such an equation can start off exponentially small at early times, but become large near the blowup time. At this point one may expect nonlinear effects to kick in and make the solution extremely difficult to analyze; but if one can time the emergence of large amplitudes just right, one can hope to arrive at a sweet spot where, by the blowup time, the amplitude has become large enough to disrupt smoothness, but not so large to destabilize the analysis.

In the case of the Boussinesq equation at least, there is an explicit ansatz, described in the introduction to the relevant paper, in which, at times close to blowup and locations close to the origin, behaves linearly in space, and behaves like a high frequency plane wave. Remarkably, this ansatz can be solved exactly (without any nonlinear correction terms), leading to an explicit system of ODE modulation equations that have the required instability property. This is the basic mechanism for blowup; however there are an enormous number of technical complications, for instance relating to spatial cutoffs, that are needed to make the full argument rigorous. The Alpöge–Buckmaster construction has some technical improvements over the older Córdoba–Martínez-Zoroa construction that allow them to treat more general fluid equations; I have not yet digested the precise differences, but the ODEs seem to be more unstable and the high frequency corrections appear to have better spatial localization properties.

Hopefully there will be some better expositions and talks by the authors on this nice result in the future. If I have time and am able to digest the results better, I may also be able to give more details in a followup blog post.

EDIT: there is now also an independent preprint of Ganeshram, Duruisseaux, and Anandkumar that has made a significant advance on the other major approach to finite time blowup, which is to first use numerical or machine learning tools to locate an approximately self-similar blowup profile ansatz, and then demonstrate that it is stable enough to be perturbed to an actual solution. For the Euler equations (with no forcing term or boundary), they have used a physics-informed neural network (PINN) to locate a numerically stable candidate solution; though actually establishing its stability to within the tolerance of the residual error in the solution remains a major challenging task to carry this result all the way through to a full rigorous demonstration of finite time blowup.

Interesting developments tonight, as Levent Alpöge and Tristan Buckmaster announce that after a fair amount of work they have constructed examples of finite-time blowup for a broad class of PDEs including 3-d incompressible Euler, inspired by of Diego Córdoba and Luis Martínez-Zoroa, and using plenty of LLM iteration in order to get the details right. This is, of course, a problem in the neighborhood of Navier-Stokes (in the negative direction of finding a counterexample to the conjecture, which I have over the years heard many PDE folks saying was the right way to bet), and Terry Tao says in a Mastodon thread that in principle this method doesn’t seem so far from showing blowup for Navier-Stokes too, though a large amount of compute and detail-checking would be involved.

At least part of this has already been Lean-formalized, though perhaps eccentrically I find I care a little less about that. What matters is not whether there’s an example but whether the example has something to teach us. An interesting but incorrect example would surely be of more value than an uninteresting but correct one. Well, I suppose the latter would have more financial value. Though even on that Millennium Prize page, one sees: “Why ask for a proof? Because a proof gives not only certitude, but also understanding.” Very true! We mustn’t settle for mere certitude. Certainly the work of Alpöge, Buckmaster, Córdoba, and Martínez-Zoroa seems to offer understanding as well.

I’ve been thinking about the exceptional Lie algebra E₆, as a spinoff of my project on E₇, so I want to get a good mental picture of the E₆ root polytope. This is 6-dimensional polytope with remarkable symmetry.

Let’s climb up to it, starting with some of its 4-dimensional faces, which are called 4-demicubes because you get them by taking a 4-dimensional cube, or tesseract, and removing every other corner. The 3-demicube is just a tetrahedron, since you can fit two tetrahedra in a 3-dimensional cube like this:

The 4-demicube builds on this fact in a surprising way.

I’m going to use the technology of Dynkin diagrams, or technically Coxeter diagrams: they’re closely related, and the difference is invisible here. I won’t explain them, just use them. I explained them here:

• Symmetry and the fourth dimension: part 3, part 4, part 5, part 6.

Let’s dive in!

The 4-demicube lives in 4 dimensions. It has 8 vertices.

You get it from a 4-dimensional cube, which has 2⁴ = 16 vertices, by keeping every other vertex, throwing away half. That leaves 8.

What are its top-dimensional faces, aka ‘facets’? Surprise: there’s only one kind! All of them are regular tetrahedra.

In higher dimensions the demicube has two kinds of facet. You get a simplex-shaped facet from every other vertex, formed when you remove it. And you get a demicube-shaped facet from each of the cube’s facets. But in 4 dimensions the two kinds happen to be the same shape!

Eight of them are tetrahedra. These appear at the 8 corners you sliced off: one per removed corner.

Eight more come from the 8 faces of the 4-dimensional cube. These are 3-demicubes. But as we’ve seen, the 3-demicube is also a tetrahedron!

So the 4-demicube is especially symmetric: it has 16 tetrahedral facets. You can find coordinates where its vertices are

It’s actually one of the 4-dimensional regular polytopes, sometimes called the 4-orthoplex. It’s also called the 16-cell because it has 16 facets. It’s the 4-dimensional cousin of the octahedron, which has 8 triangular facets.

You can read some of these facts off the D₄ Dynkin diagram, if you know what you’re doing. As you can see above, this diagram has a central node with three arms, each just 1 edge long: a perfectly symmetric three-pronged star. To get the 4-demicube, you ring the tip of any one arm.

To get the facets of the 4-demicube, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives. There are two choices: you can delete the tip of either other arm. But either way, what’s left is a straight chain of 3 nodes—the so-called A₃ diagram—with a ring at one node at the end. This gives the tetrahedron.

Both choices give the same shape of facet, a tetrahedron, because all three arms of the D₄ Dynkin diagram are interchangeable. That ceases to be true in higher dimensions!

Next, the 5-demicube. This lives in 5 dimensions and has 16 vertices.

You get it from a 5-dimensional cube—which has 2⁵ = 32 vertices—by keeping every other vertex, throwing away half. That leaves 16.

What are its top-dimensional faces, or ‘facets’? There are two kinds!

Sixteen of them are 4-dimensional analogues of the regular tetrahedron, called 4-simplexes. These appear at the corners you sliced off: one per removed corner.

The other ten come from the ten faces of the 5-dimensional cube. After you take every other vertex, they become 4-demicubes. These are precisely the 4-demicubes we saw in the last section!

You can also read these two kinds of facets from the D₅ Dynkin diagram. As you can see above, this diagram has three arms of lengths 2, 1, 1 (edges from the central branch node). To get the 5-demicube, you ring the tip of either length-1 arm. That ringed diagram encodes the whole polytope.

To get the facets, delete an unringed node so the piece still holding the ring stays connected, and see what diagram survives.

There are two choices.

If you delete the tip of the other length-1 arm, what’s left is a straight chain of 4 nodes—the diagram whose polytope is the 4-simplex. That gives the 4-simplex faces.

Or you can delete the tip of the length-2 arm. Then what’s left is a shorter branching diagram, the one I showed you in my last post! That gives the 4-demicube faces.

So the 5-demicube has both 4-simplex and 4-demicube faces.

Next let’s go up to the 6th dimension, which was my goal all along.

The E₆ root polytope lives in 6 dimensions. It has 72 vertices.

What are its facets? You can read them straight off the E₆ Dynkin diagram, using the same procedure we’ve been using so far.

As you can see, the E₆ Dynkin diagram has three arms of lengths 2, 2, 1 (edges from the central branch node). To get the root polytope, you ring the node that’s the tip of a length-1 arm. That fact is not obvious, but let’s go ahead and do that.

Then, to get the facets, delete any unringed node such that the piece still holding the ring stays connected, and see what diagram survives.

There are two choices: the two other nodes at tips of the Dynkin diagram.

However, deleting either of these nodes leave a D₅ diagram with a ring on one node, and this gives the 5-demicube we saw last time: a 5-cube with alternate vertices removed.

So the facets of the E₆ root polytope are all the same shape: 5-demicubes!

With more work, we can count the facets of the polytopes we’ve been studying:

• The E₆ root polytope has 54 facets, all 5-demicubes. They come in two kinds, because we had two choices of which node to delete, so there are really 27 ‘positive’ 5-demicube facets and 27 ‘negative’ 5-demicube facets.

• The 5-demicube has 16 4-simplex facets, one for each vertex that we removed from the 5-cube to create this demicube, and 10 4-demicube facets, one for each facet of that 5-cube.

• The 4-demicube has 8 3-simplex facets, one for each vertex that we removed from the 4-cube to create this demicube, and 8 3-demicube facets, one for each facet of that 4-cube. But both the 3-simplex and the 3-demicube are the familiar tetrahedron. So in fact the 4-demicube has 16 tetrahedral facets. Indeed, the 4-demicube is the 4-dimensional analogue of an octahedron: the so-called 4-orthoplex, or 16-cell.

Using some fancier math I explained here, we can count all the faces of the E₆ root polytope:

I’m well underway on revising Don’t Be Too Sure, which I finished a first draft of right before surgery. A lot of people make appearances in this book, most of all William James and John von Neumann, who became the two main characters despite not being in my original plans for the book at all. Some other people: Felix Hausdorff, Alfred Kroeber and his daughter Ursula Kroeber Le Guin, John Keats, Sheila Heti, Katharine Briggs and her daughter Isabel Briggs Myers, Jakob Bernoulli, Elbert Hubbard, Anna Kiesenhofer, Grace Hopper, Thomas Jefferson, Caroline Hoxby, David Hilbert, Michel Adanson… well, there are a lot of people in it, who do a lot of things.

As we descend from the base of Earth’s crust through the mantle, the rock does not remain unchanged. Pressure and temperature rise inexorably, and the minerals that thrive at the surface are forced, step by step, into new and denser crystallographic arrangements. This is the story of those transformations.

In this tale, I’ll act like I know a bit about minerals. I actually don’t: there are a bewildering variety, and I can never remember them. So don’t worry: when you come across a jargon-filled patch of prose, just power through it. You might learn a little… or you can just ignore it. The overall point here is that the Earth is made of beautiful crystalline structures that change character in complex ways as we descend.

The Mohorovičić discontinuity

Our story begins at the boundary where Earth’s crust, rich in feldspar and quartz, gives way to the denser mantle beneath. We see this boundary through its effect on seismic waves, and it’s called the Mohorovičić discontinuity or “Moho”. The Moho does not lie at one fixed depth: it’s 5–10 kilometers below the seafloor, but 30–50 kilometers below most continents, and as much as 70–80 below young mountain belts like the Himalayas.

The mantle just below the Moho mainly consists of a rock called peridotite, which is made mostly of olivine and pyroxene, with smaller amounts of garnet (or, at shallower depths, spinel). Peridotite has a delicious coarse green appearance:

More precisely, this is what peridotite looks like up here. But when geochemists talk about the bulk composition of the upper mantle, they often use an idealized model called pyrolite—not a rock you can pick up, but a hypothetical recipe Ted Ringwood proposed in the 1960s for the primitive upper mantle.

Why? Since the Earth has had a convecting mantle, solid mantle rock wells up in places. As it does, the pressure drops, and a bit of it melts: the minerals with lower melting points. This melt flows upward. It’s called basalt. It builds the Earth’s crust. But it leaves a residue behind, made of minerals with higher melting points.

In Ringwood’s theory, which for expository purposes I’ll assume is true, pyrolite is what mantle rock is like before any partial melting depletes it of basaltic ingredients. The name is a portmanteau of pyroxene and olivine, the two dominant minerals. Pyrolite is about 60% olivine; the remaining 40% is mostly pyroxenes plus garnet.

• A pyroxene is a mineral built from single, unbranched chains of corner-sharing SiO₄ tetrahedra, with metal cations—chiefly Mg, Fe, and Ca—linking the chains together. The general formula is XY(Si,Al)₂O₆, where X and Y are those cations.

• Olivine is a green silicate, (Mg,Fe)₂SiO₄:

Its crystal structure in the upper mantle is an orthorhombic arrangement of isolated SiO₄ tetrahedra knit together by magnesium and iron in octahedral sites. It’s called the α-phase because we’ll see some more compressed phases as we descend.

• A garnet is built from separate SiO₄ tetrahedra held together by cations, but assembled into a dense, hard, characteristically cubic-symmetry crystal. There are different kinds of garnet, but the general formula is X₃Y₂(SiO₄)₃: three divalent X cations, two trivalent Y cations, and three isolated silica tetrahedra. The mantle’s garnet is largely pyrope, Mg₃Al₂(SiO₄)₃.

As we descend, the pyroxenes and garnet gradually dissolve into each other, producing a new high-pressure mineral called majorite. Here’s a rare sample from a meteorite fall in Canada:

So even before the dramatic change 410 kilometers down, the rock is no longer the simple olivine-pyroxene-garnet assemblage we had further up.

The 410-kilometer discontinuity

Roughly 410 kilometers down, the pressure reaches about 13,000 atmospheres and the temperature hovers around 1,400°C. Olivine can no longer hold its familiar shape. It transforms to its β form: wadsleyite, a mineral with the same chemical formula but a fundamentally different atomic arrangement. Instead of isolated SiO₄ tetrahedra, wadsleyite contains paired Si₂O₇ groups, and the oxygens pack more densely. The density jump is sharp enough to be detected globally by seismologists as a reflector of earthquake waves.

Wadsleyite has a remarkable property: it can hold several weight percent of water locked within its crystal structure. The transition zone may thus contain more water than all the oceans combined! However, very little wadsleyite has been seen on the Earth’s surface. Here’s a bit from that same meteor fall in Canada:

The 520-kilometer discontinuity

Descend further, to around 520 kilometers, and the temperature goes up only a little, to roughly 1500–1600°C, since convection here is strong. The pressure goes up to about 175,000 atmospheres. At this point wadsleyite transforms into the γ form of olivine: ringwoodite. This is denser, still chemically Mg₂SiO₄, but now with cations packed into tetrahedral and octahedral holes in a close-packed oxygen framework—the most efficient packing geometry that nature offers for this composition:

Ringwoodite is named for the great Australian geochemist Ted Ringwood, who studied these transitions. Here’s an artificially manufactured sample:

For a long time the mineral’s existence in the mantle was purely hypothetical. But in 2014, a tiny grain was discovered as an inclusion inside a diamond brought up from the deep mantle by an eruption, providing the first direct proof of its existence in Earth’s interior.

The 660-kilometer discontinuity

At a depth of 660 kilometers and a pressure of roughly 230,000 atmospheres, the most dramatic phase transition of all occurs. Ringwoodite does not merely rearrange into a still more dense form! Instead, it decomposes into two entirely new minerals: bridgmanite (MgSiO₃) and ferropericlase (MgO). The majorite garnet also decomposes, yielding davemaoite (CaSiO₃), which is stable through the rest of the lower mantle:

The 660-kilometer discontinuity is sharp, globally consistent, and marks the conventional boundary between the upper and lower mantle. One reason it’s important is that enormous slabs of colder, denser rock sink through the upper mantle until they hit this discontinuity, where the phase change between ringwoodite and bridgmanite creates a kind of barrier.

These slabs are 30–100 kilometers thick and hundreds to a thousand kilometers across! Some punch straight through into the lower mantle and keep sinking. But many flatten out when they hit the barrier, sometimes lying there and piling up for tens of millions of years. You can see this in seismic images beneath Japan and the Marianas. Numerical models suggest that they pile up until they overwhelm the barrier and flush down in a comparatively sudden avalanche—lasting mere millions of years.

The lower mantle

This is the realm of bridgmanite, probably the most abundant mineral in the Earth. Bridgmanite is a beautifully symmetric cage of corner-sharing SiO₆ octahedra, with Mg tucked into the large cavities between them. It accommodates enormous pressure because there is very little void space left to compress.

It is a striking fact that while bridgmanite is the most abundant mineral on the planet, it went unnamed until 2014, simply because no natural hand-sized specimen had ever been recovered. Everything we know about it comes either from high-pressure laboratory synthesis, from microscopic grains in shocked meteorites, or from the indirect testimony of earthquake waves that have traveled through 2,000 kilometers of it.

For over 2,000 kilometers of descent, from 660 to roughly 2,700 kilometers down, bridgmanite and its companion ferropericlase reign without significant further phase change. Seismic velocities increase steadily, but there are no dramatic discontinuities.

The D″ discontinuity

As we approach the core-mantle boundary—at depths around 2,700 kilometers, pressures of approximately 120,000–125,000 atmospheres, and temperatures of 2,200–3,7000°C—even bridgmanite yields. It transforms into the post-perovskite phase. Post-perovskite is a layered, sheet-like structure of SiO₆ octahedra, quite different from bridgmanite’s three-dimensional cage, making it potentially much weaker and more prone to flow.

This transition is believed to be responsible for the seismic D″ discontinuity observed at 2,900 kilometers depth. The D″ layer is a highly dynamic region, likely the site of storage of subducted materials and the source of deep mantle plumes.

A summary of the descent

The table below summarizes the major transitions:

The interesting thing about this story is that it was told first by seismology—the sharp jumps in wave speeds at 410 and 660 kilometers were detected long before geologists could reproduce those pressures in the lab—and only later checked by diamond-anvil cell experiments squeezing tiny mineral samples to millions of atmospheres. The rocks never rise to the surface to tell their story directly, so much of the tale above is just theory.

Which minerals are there the most of?

We can estimate how much of the Earth is made of wadsleyite, ringwoodite, and bridgmanite using known shell volumes, estimated densities, and mineral proportions from the pyrolite model.

Step 1: Earth’s mass budget by layer

The Earth’s total mass is M_⊕ ≈ 5.972 × 10²⁴ kg. The mass budget by layer is approximately:

•    Crust: ~0.4% of Earth’s mass
•    Upper mantle + transition zone (35–660 km): ~18% of Earth’s mass
•    Lower mantle (660–2,891 km): ~49% of Earth’s mass
•    Core (outer + inner): ~32.5% of Earth’s mass

Step 2: The transition zone (410–660 km)

Using PREM densities averaging ~3,760 kg/m³ across the transition zone, and the volume of each spherical shell:

Wadsleyite zone (410–520 km):
Shell volume ≈ 4.8 × 10¹⁹ m³
Shell mass ≈ 1.76 × 10²³ kg
Fraction of Earth’s mass ≈ 2.9%

Ringwoodite zone (520–660 km):
Shell volume ≈ 5.9 × 10¹⁹ m³
Shell mass ≈ 2.24 × 10²³ kg
Fraction of Earth’s mass ≈ 3.8%

In the pyrolite model of mantle composition, forms of olivine (wadsleyite and ringwoodite) make up roughly 60% of the transition zone by mass, with the remaining ~40% being majoritic garnet. Applying this correction:

Wadsleyite: 0.60 × 2.9% ≈ 1.8% of Earth’s mass
Ringwoodite: 0.60 × 3.8% ≈ 2.3% of Earth’s mass

These estimates carry roughly 20–30% uncertainty, mainly from the assumed 60% olivine proportion in the transition zone, which varies with local temperature and bulk composition.

Step 3: Bridgmanite (660–2,700 km)

The lower mantle holds about 49% of Earth’s mass—it is an enormous shell! Bridgmanite constitutes approximately 80% of the lower mantle mineral assemblage (by mass) in the pyrolite model:

0.80 × 49% ≈ 39% of Earth’s mass

This is consistent with the well-cited literature figure that bridgmanite comprises approximately 38% of the planet’s mass—making it the single most abundant mineral in the Earth by a vast margin.

Thus, these three minerals—all members of the same Mg₂SiO₄/MgSiO₃ chemical lineage—together constitute roughly 42% of Earth’s entire mass. All other named minerals on Earth, including quartz, feldspar, calcite, diamond, and the roughly 3,800 others known to mineralogists, divide up the remaining scraps.

Rethinking the electric field

Have you ever wondered what an electric field actually is? 

The electric field is the foundation of most technologies that we rely on every day. From power grids and electronic devices to radio communication and the internet, the electric field is extremely relevant to our daily lives. However, despite its importance, I have always felt that the common explanations of the electric field leave something unanswered. 

Most textbooks define the electric field as a property of space or a physical entity surrounding electric charges, or with the equation of force per unit charge. These definitions help us understand what the electric field does and its effect on electrically charged particles, but they do not fully answer what an electric field actually is and how it influences charges. Thus, I started thinking about the question: what allows charges to influence each other without touching?

This question led me down a path that began with a simple observation in everyday life, and it eventually pointed toward much deeper ideas in modern physics.

Objects Influenced by Their Surroundings

Before talking about electric fields, let’s consider a more basic question: Does it seem reasonable that objects can be influenced by their surroundings? 

Most people would answer yes. We have all seen examples of objects responding to something else nearby, such as the Earth orbiting the Sun, a compass needle reacting to a magnet, and our phones responding to signals from a WiFi router. But what is the mechanism behind these interactions? 

A simple physical phenomenon that we can look at is a balloon rubbed on a piece of clothing that can pick up strands of our hair. Many of us have seen this demonstration in kindergarten or first grade of elementary school. This might seem completely ordinary, but if we pause and think about it, something strange is happening – the balloon is influencing the hair without touching it. 

How is that possible? One answer is simply that the balloon “pulls” on the hair, but this raises more questions. How does the balloon reach the hair? What is happening in the space between them? These questions suggest that something is missing from the picture of objects pulling on each other directly through contact.

To put this in the context of physics, we might all have learned that “like charges repel, and opposite charges attract”. We might have solved equations on how fast charges would move away from or toward each other. We were always told to just accept it because these motions result from the electric field. But why do these observations happen? What is happening between the charges?

Historically, physics encountered the same problem. If one object can influence another at a distance, it is natural to ask what is happening in the space between them. One guiding principle that physicists often use is the concept of locality. Locality is the idea that an object can only be directly influenced by its immediate surroundings. Thus, an influence should not simply leap across space from one object to another, and changes should propagate through intermediate regions step by step. 

At first glance, locality seems reasonable because it matches many of our everyday experiences. If I push a book across a table, my hand influences the book through direct contact. The influence does not appear to jump instantaneously across the table. 

However, locality creates a tension when we return to the scenario of the balloon pulling on strands of hair. If locality is true, something must be happening in the space between the balloon and the hair. But from a standard electromagnetic perspective, the space between the balloon and the hair is empty. Therefore, we have encountered a contradiction: if what is between the balloon and the hair is empty space, then what is responsible for transmitting the influence? Neither the usual electromagnetism nor locality tells us the answer to these questions.

The Classical Electric Field

In the usual electromagnetic picture, the answer to the puzzle is the electric field. Rather than allowing charges to influence one another directly across space, the theory assigns an electric field to the space surrounding charges. The field acts as the intermediary through which influence is transmitted. 

A useful way to think about the electric field is that it assigns information to every point in space. If we imagine a charged particle that is placed at a particular location, the electric field tells us how that particle would move. This charged particle is what physicists call a test charge. By observing how the test charge behaves, we can infer information about the electric field at that location. 

This could seem like a satisfying answer as the electric field tells us how influence is transmitted, but it does not tell us what kind of thing is doing the transmission. Is the electric field a physical substance? Is it a mathematical tool? Or is it something else? 

It might be easy to fall back on the idea that the electric field ultimately works through tiny particles physically touching one another. After all, contact interactions are among the most familiar interactions that we experience. 

But physics challenges this intuition as well. It is surprisingly difficult to define what it means for two objects to “touch”. We usually think of the balloon attracting hair as an example of action at a distance, whereas pressing a hand on a table feels like direct physical contact. However, at the microscopic level, the two situations are fundamentally similar. If we could zoom in on our fingertip and the table with a microscope, we would find that the atoms in our skin never make contact with the atoms in the table. This is because of the repulsion between the electron clouds surrounding the atoms, which prevents the two atomic nuclei from overlapping. Say if we scale the atom in the table up to be the size of a marble, then the nearest atom in our fingertip would still be separated from it by a few centimeters. In the end, nothing is truly “touching” in the intuitive, physical sense. 

Thus, the idea of contact does not solve our problem. We are forced to ask the same question again: what is it that allows these interactions to occur? To answer that question, I turned to a different perspective of thinking about electric fields.

The Electric Field as A Dynamical Structure

From our intuition, it is natural to imagine the electric field as some invisible substance filling space. This is often the picture suggested by the common field line diagrams in physics textbooks, which make the field appear to flow outward or inward from charges, almost like a moving fluid. 

A useful analogy is the ocean. A boat floating on water can move because waves pass beneath it. The boat responds to changes in its surrounding waves rather than to some direct push from a distant object. Similarly, charged particles respond to changes in the electric field around them. We can then view the electric field as a dynamical structure that governs how the state of the world can evolve.

However, the ocean analogy can only take us so far. Ocean waves are made of water molecules. Sound waves are made of vibrating air molecules. But what is the electric field made of? When light travels through empty space, it seems that there is no material medium at all. 

This brings us back to the mystery: if locality suggests that something must exist in the space between interacting objects, and if the electric field is not made of the ordinary matter that we understand, then what exactly is occupying the space? 

To answer this question, we have to rethink what we mean by “empty” space itself.

Empty Space is Not Empty

Conventionally, we have always imagined empty space as exactly what the name suggests—empty. Just like if all the particles were removed and nothing was remaining. But modern physics suggests a very different picture. 

In quantum field theory, what we call “empty space” is not truly empty. Empty space is filled with underlying quantum fields that permeate all of space and time, even in the absence of particles. Even when the surface of the ocean looks perfectly still, the water is still there. The ocean is not defined only by visible waves, but by the underlying medium that can support waves in the first place. The waves are patterns of motion of the ocean itself, just like the electric field. These fields are part of the fundamental structure of the universe from which physical phenomena emerge. Quantum field theory suggests that particles are not independent objects moving through an otherwise empty space. Rather, they are localized patterns or excitations of underlying fields that already exist throughout the universe. 

From this perspective, the electric field is not something that is added to empty space. It is part of the fundamental dynamical structure of space itself.

Conclusion

At the beginning, I asked a simple question: how can objects influence each other without touching? The straightforward answer is the electric field. Charges create electric fields, and those fields determine how other charges move. But what is an electric field? Is it an invisible material filling space between objects, or is it a dynamical structure that governs how physical systems in the world evolve? 

From the perspective of quantum field theory, quantum fields permeate all of space and time. Particles are not separate objects moving through an empty space, but are excitations of these underlying fields; electric fields are not secondary matter surrounding charged particles, but are particular configurations of the underlying electromagnetic quantum fields. What we observe as the motion of a charged particle is the result of its interaction with the electromagnetic field, whose local state determines how the particle evolves. 

In the end, our original question may not have a single definitive answer. But asking it revealed a shift in perspective, and physics has repeatedly shown that every explanation opens the door to an even more fundamental question. Stopping at this step, a new mystery emerges: what are these underlying quantum fields themselves? What are they made of, and where do they arise from? 

While technically the funds don't go away at the end of September, this kind of underspending raises the possibility of a pocket rescission. OMB and the executive branch have been pushing for massive cuts to the agency; Congress has disagreed. It sure looks like all the "see, don't worry, Congress didn't allow big cuts to the NSF" palliative statements don't hold up very well to scrutiny, if the majority party is content to just give up Article I power to the executive branch. 

In this period of complete flood-the-zone craziness, the mainstream news media seemingly doesn't have the bandwidth or interest to report on this; they seem to have judged that it's too obscure, it doesn't play in Peoria, the public doesn't really care. This kind of disruption will have ripple effects that last for many years and affect US scientific and economic competitiveness, and it's happening without much notice.

This week's news about an agreement between NIH and DOD to funnel NIH funds for infectious disease to DOD (or, in the official statement, to work together on projects of mutual interest), is at least getting some public attention.  Agencies agreeing to pass around at minimum hundreds of millions of dollars outside congressional oversight or what the appropriations acts say is another example of an Article I crisis, when the majority party basically hands over what are supposed to be congressional powers to executive branch.

(An additional sciencey blog post coming soon!)

Academics and journalists have a lot in common, at least in principle.

Whether you’re a reporter or a professor, your job is to go out into the world and figure out the truth. You’re supposed to be careful, to check and correct for how you might be wrong. And at the end of the day you’re supposed to communicate what you found.

The differences mostly come in how you’re paid.

You could imagine some sort of pure truthseeker, paid purely by how well they tell the truth. People would ask them to find out the truth about something, and pay them for the service. And the truthseekers with the best track record would get the most clients. But neither profession really works like this.

Journalism comes closest. Once upon a time, people bought newspapers in order to be the first to know when something important happened. While there’s still a little bit of that going on (I guess this is what Bloomberg Terminals are for?), it’s a lot less central because of the internet. Now, there are hundreds of ways to find out about things, from a multitude of news sites to social media. More and more, people expect to be able to get information for free.

In that environment, the news has to compete not on the facts themselves, but on how it presents them. People pay for news that’s curated well to match their interests, or news that feels more respectable. And more than either of those, they pay for news that’s entertaining. So while truthseeking skills pay, writing skills often end up mattering more. In a sense, it’s why it’s possible for me to do journalism at all. I was trained in the academic truthseeking tradition, not the journalistic one. I got into journalism by impressing editors with my writing, not my ability to suss out the truth.

That academic truthseeking tradition is quite different, in part because the rewards for it are much more indirect. Academics pay comes from two main sources: research grants, and student tuition. Students are mostly there to learn old facts, not new ones, so that source of money supports research only in so far that students believe that a successful researcher with time for research will also be a better teacher.

Research grants, in principle, pay for truthseeking. But they’re typically paid by governments, which often don’t have a clear idea of what they’d like to learn, since the more practical questions are already being researched by private companies. So the decision gets delegated out to other academics, who have a vague shared sense of what’s worth knowing and what’s not. Accuracy should have an impact: that is, it should be easier to get grants if you’re better at finding the truth. But in practice, unless someone does so badly they trigger a scandal, academics don’t usually get things all that wrong. So grants are mostly based on other factors.

Paying someone purely to deliver the truth, not to entertain or match a culture, seems tricky. You could imagine sci-fi scenarios. What if we could track the logic people used to make decisions, and demand payment if those decisions were based on facts we uncovered, like a journalist getting a percentage of every short made in response to bad news they dug up about a company? What if governments paid in proportion to how valuable academic ideas turned out to be, centuries after they were discovered, and modern-day academics sold shares in that future payout to fund themselves? What if prediction markets something something?

For the moment, academics and journalists are both in a weird middle space. They’re truthseekers, still, by culture and inclination and desire. But they’re paid for something else.

I woke up yesterday with the following thoughts, which are probably either obvious or dumb.

A central thesis that many readers, including me, took from Douglas Hofstadter’s Gödel Escher Bach when young was that the secret of intelligence (and therefore, of AI) was going to have a lot to do with self-referentiality and “strange loops.”

Even Roger Penrose’s The Emperor’s New Mind, which in some ways was the anti-GEB, ironically agreed with GEB about the fundamental importance of self-reference to the success or failure of the whole AI project. It claimed (incorrectly, in my view and in most experts’) that AI could never work because there was something about Gödel’s Theorem and self-reference that no computer program could ever capture, but that could be captured by exotic physics accessible to the human brain.

Now, in 2026, we’ve succeeded at building AIs that outperform most humans at most intellectual tasks that are well-defined enough to judge. And at no point in the tech stack of those AIs — neither in the transformer neural nets, nor in the GPU clusters they run on, nor in the training process, nor anywhere else — did anyone need to build in anything about self-reference. (Excepting, eg, the system instructions that tell the model about its role and identity, which aren’t needed for intelligent behavior. Also, I’m not going to count the autoregressive nature of LLMs as “self-referential”; that’s just dynamical feedback.)

Of course, GPT 5.6 Pro and Fable can talk about themselves, about Gödel’s Theorem, about self-reference, about what we’re talking about right now, all of it, better than most humans. But at no point did anyone need to build self-referential abilities in. They popped out as a byproduct of the same pretraining that let the models talk about Pokémon and long-chain polymers and cognitive behavioral therapy and plate tectonics and everything else.

No wonder Hofstadter says he’s been stunned by the success of LLMs, and has seemed depressed about current AI capabilities in essays like this one. He’s way too smart to deny what’s happened or invent reasons why it doesn’t really count (the approach many have taken). But he realizes that we now have true conversational intelligence from a path that the GEB worldview would’ve regarded as far too cheap and simple, and that certainly has no “strange loops” built in anywhere.

Of course, a Hofstadterian could argue that a strange loop emerges in LLMs — indeed, nothing in GEB ever said that strange loops would need to be explicitly engineered at the outset. But would anyone who hadn’t been brought up on GEB arrive at this as a useful way of thinking about LLMs?

What can we say about this with hindsight? While the ideas of diagonalization and self-reference of course played a central role in the birth of modern mathematical logic and computer science, the most famous uses were negative: there is not a bijectjon between the natural numbers and the reals. There is not a complete sound proof system for arithmetic. There is not an algorithm to solve the halting problem.

If your goal was only to build the axioms of ZFC and the rules of first-order inference, or build an electronic computer, you wouldn’t explicitly need self-reference for that. You would just … start building, taking care that your instruction set didn’t fall short of universality.

Yes, ZFC can formalize and prove theorems about itself. Yes, electronic computers can run programs that take their own code as input. But no one ever needed to build those abilities in, any more than self-reference needed to be built in to the alphabet or the rules of grammar. It popped out as a free byproduct of universality.

In the same way, LLMs’ ability to talk about themselves popped out as a byproduct of their ability to talk about anything in the discourse universe they were trained on. The big, old ideas about intelligence that ended up basically vindicated were the ideas about how intelligence is about prediction, and prediction is about compression, and compression is about finding better and better upper bounds on Kolmogorov complexity. Not the self-reference stuff. (Although, if you wanted to know why Kolmogorov complexity can’t be computed perfectly, that negative statement would again require a self-referential argument.)

What’s left? Consciousness and subjective experience of course remain extremely mysterious. For all we know, Hofstadter could be right that those have something to do with self-reference. (For all we know, even Penrose could be right that they have something to do with exotic physics accessible to biological brains but not digital computers!)

But the idea that you’d need explicit self-referentiality before you could get convincing and world-changing conversational intelligence? Let it be buried in a Westminster Abbey or Arlington National Cemetery for the most important wrong ideas in human history — geocentrism, Aristotle’s teleological physics, aether, phlogiston, Freud’s psychology, Marx’s prediction of a workers’ uprising followed by a classless utopia, etc. But buried it needs to be.

It’s long been a mystery why there are 3 generations of quarks and leptons: three sets of particles, apparently identical except for how they interact with the Higgs boson. It would be nice if there were some good physical explanation. Nobody knows one. Barring that, it would be nice if some beautiful mathematical structure made this pattern seem natural. That’s what my new paper is about.

I’ll keep this nontechnical. I’ll say a bit about what the paper does, what it does not do, what led up to it, and how I wrote it.

This is my third paper about exceptional algebraic structures and the Standard Model. When you classify famous gadgets in algebra, beautiful gadgets with fancy names like ‘simple Lie algebras’ and ‘Euclidean Jordan algebras’ and ‘positive hermitian Jordan pairs’, you tend to get infinite series of them — together with a few exceptions that can be built using the octonions. This is a bit spooky, so I’ve been interested in this for a long time.

A few physicists have hoped that these exceptions are good for something. For example, maybe the quirky features of our best theory of particle physics, the Standard Model, aren’t accidental. Perhaps they fall out naturally from some exceptional algebraic structure.

It’s a long shot, but we’ve been stuck on figuring out new fundamental laws of particle physics for so long — roughly since the early 1980s — that it’s worth a try.

In 2018, Michel Dubois-Violette and Ivan Todorov noticed that the gauge group of the Standard Model falls out as symmetries of the so-called ‘exceptional Jordan algebra’ together with some ordinary Jordan algebras sitting inside it. I tried to clarify that here, with a huge amount of help from an excellent young mathematician:

• John Baez and Paul Schwahn, The Standard Model gauge group from the exceptional Jordan algebra. (Blog article here.)

It’s very nice, because the Jordan algebras in question arise naturally when you try to axiomatize the foundations of quantum physics. It would be so cool if something about quantum physics made the Standard Model seem mathematically natural!

But really this result only concerns the gauge bosons in the Standard Model: the photon, gluons, and the W and Z bosons. It says nothing about the fermions — that is, the quarks and leptons. And it seems quite hard to get those into the picture.

In 2020, Latham Boyle tried to solve this problem by tensoring the exceptional Jordan algebra with the complex numbers. This made one generation of fermions appear quite naturally! But the connection to the foundations of quantum physics seemed lost: tensoring the exceptional Jordan algebra with the complex numbers seems at first like it might be just a formal trick.

This spring, Latham and his student Endre Bokor and I showed the connection to quantum physics is not lost:

• John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model. (Blog article here.)

The idea is to work, not with Jordan algebras, but with more general things called Jordan pairs, which have been studied by mathematicians since at least 1975. We showed that you can still do quantum physics with Jordan pairs. And we showed that there’s an ‘exceptional’ Jordan pair that naturally contains the Standard Model gauge group and one generation of fermions!

This Jordan pair is built from the bioctonions: the octonions tensored with the complex numbers. And it’s closely related to an exceptional Lie algebra called $\mathfrak{e}_6$.

This is nice because the work of Dubois-Violette and Todorov used a smaller exceptional Lie algebra called $\mathfrak{f}_4$. Going up to $\mathfrak{e}_6$ gives the room to include one generation of fermions.

There’s an even larger exceptional Lie algebra you can use to build a Jordan pair: it’s called $\mathfrak{e}_7$. Bokor, Boyle and I tried using this to get three generations of fermions. There are things that make this tempting: not just the fact that $\mathfrak{e}_7$ is bigger, but the fact that the Jordan pair you get from it has a kind of three-fold symmetry. But we couldn’t get it to work.

Around this time I got very interested in some work that someone had sent me in October 2025. My inbox is packed with new theories of physics. Since the rise of large language models the inflow has increased: I get about two emails a day from someone telling me they’ve made a revolutionary discovery in physics. Practically none of these theories appeal to me. But this paper, and this thesis, were different:

• Benjamin Nasmith, An exceptional combinatorial sequence and Standard Model particles, 2020.

• Benjamin Nasmith, Tight Projective 5-Designs and Exceptional Structures, Ph.D. thesis, Royal Military College of Canada, 2023.

He claimed to fit three generations of fermions into the exceptional Lie algebra $\mathfrak{e}_7$.

When I started seriously trying to understand this paper, I wound up translating it into a language I’m more comfortable with, and expanding on the ideas a bit. So I wrote this:

• John Baez, Three generations in $\mathfrak{e}_7$.

Here’s the basic idea.

The idea

There is a standard way to fit the Lie algebra of the Standard Model gauge group, which I call $\mathfrak{g}_{\text{SM}}$, into the Lie algebra $\mathfrak{e}_7$. You can construct a Lie algebra $L$ that fits between them:

$$\mathfrak{g}_{\text{SM}} \subset L \subset \mathfrak{e}_7$$

As a vector space we have

$$\mathfrak{e}_7 \; \cong \; L \oplus V$$

for some vector space $V$ of dimension $3 \times 32$.

Moreover, the Lie algebra $\mathfrak{g}_{\text{SM}}$ acts on $V$, via the $\mathfrak{e}_7$ Lie bracket, precisely as it does on three generations of Standard Model fermions and their antiparticles, including right-handed neutrino and its antiparticle — but ignoring spin!

There is, in fact, a very interesting three-fold symmetry built into $\mathfrak{e}_7$, which is revealed when we put the Standard Model Lie algebra $\mathfrak{g}_{\text{SM}}$ into it. It permutes the three generations.

Like Nasmith, I am not proposing a theory of physics. I’m only observing a fascinating mathematical pattern that might (or might not) be of some use in physics.

There are lots of things this pattern does not include: basically, everything I didn’t already mention. It does not include the spin of the fermions and gauge bosons. It does not include the Higgs boson, though in some sense it comes close (see the paper). It does not include a Lagrangian, so it doesn’t say anything at all about particle masses or interactions.

I could say a lot more about what my paper does do… most importantly, where this Lie algebra $L$ comes from! The details are very interesting. There’s also the curious role of the right-handed neutrinos. But I’ve already spent weeks explaining all these things in my paper, so I won’t do it here. Instead let me say a bit about how I wrote the paper.

Writing the paper

I’ve been wanting to keep up with how AI is transforming math. About a year ago a friend gave me a subscription to Claude Pro. I wanted to test it out, despite my many misgivings, including how large language models are contributing to global warming and income inequality. Given the amazing things that people have recently done in math using large language models, I didn’t think that never trying them out would put me in the best position to make good decisions about the future.

So, I wrote this paper with help from Claude Opus 4.8.

I started by giving it Nasmith’s paper and asking a long series of questions about that paper over several days. The results were very interesting and helpful. Eventually I asked it to summarize and expand on our conversation. It quickly spat out a 10-page paper.

This paper was written in a breezy, pleasant style — but also quite hard to understand in detail, since it mixed Nasmith’s terminology with the Lie algebra terminology I prefer, and the proofs skipped over some steps.

It took me about three weeks of hard work to fully understand and re-express all the ideas a way that I like. For a while I felt dumb and frustrated, because when I asked Claude to fill in the gaps in proofs, it used math I was not very competent in, like the theory of regular subalgebras, and the theory of minuscule representations. But I learned this math, and everything turned out to be basically correct — in part, I’m sure, because Nasmith’s original work was correct.

For several weeks I checked, reorganized, expanded and completely rewrote this material. By the end everything was written in a style I like, emphasizing the ideas I consider important, proving things fairly carefully, and adding a lot of expository material — for example, explaining the theory of regular subalgebras.

Almost no traces of Claude’s original writeup remain, even though I was deeply influenced by them. My proofs make few references to deep theorems, though they assume solid familiarity with simple Lie algebras and their root systems. The proofs also require no brutally hard computations — though Claude was eager to do such computations to check things.

Any mistakes in this paper are my own.

I’m not sure what conclusions I draw from writing this paper. I’m writing another math paper now, with a human coauthor, and I have no desire to get help from a large language model. For work on my own it could be very helpful. Fields medalist Jacob Tsimerman says it roughly doubles his productivity. Would using it be so bad for the environment, or so bad for society, that I should avoid it? Maybe. I deliberately stuck with Claude Opus 4.8 instead of something more powerful, to see what I could do with what you get from a $20/month subscription. But maybe that’s still bad.

I avoid flying to conferences, which in some ways cripples my ability to keep up with new trends and influence people — but I don’t mind that. It gives me more time to think.

I will think carefully about my next move.

This past week was a genuinely horrific reminder of the importance of lab safety, especially in the realm of hazardous chemicals.  

First, a graduate student at Hokkaido University was killed due to some kind of large-scale exposure to hydrofluoric acid.  For those who don't know, HF is used at some rate in semiconductor-related work, because it's a way to etch SiO\(_2\) from silicon surfaces and leave a hydrogen-terminated surface.  (Usually this is done using buffered oxide etch, which is less concentrated than the pure acid but still must be handled with great care and appropriate personal protective equipment.)  Accidental exposure to small amounts of HF is not always immediately obvious, because it is not that aggressive in damaging human skin (unlike, say, nitric or sulfuric acid).  Rather, it attacks the calcium in bones (as well as screwing up many other biological processes).  The topical treatment is calcium gluconate gel, which can help by being a much more readily accessible source of calcium ions to bind with the fluoride ions.  There are no real details out yet about how someone had a massive amount of HF splash on their head/face, but that sure sounds like a terrible case of poor storage and handling practices.  

Then it came out that this past Wednesday a doctoral student at MIT had been exposed to dimethyl mercury.  Here is a reddit discussion thread in r/mit, and here is another one on r/chemistry.  Apologies for the reddit links, but there doesn't seem to be any news reporting about this yet.  From the MIT announcement in those threads, it was in building 18, and decontamination of the space is ongoing.  Any scientist of my generation knows about dimethylmercury because of the horrifying death of Dartmouth chemistry professor Karen Wetterhahn in 1997.  She was exposed to tiny drops of this stuff, which diffused through her latex gloves (which she did not realize at the time).  Prior to her death, people still occasionally used dimethylmercury as a standard in NMR measurements.  Organic mercury compounds are widely recognized as incredibly dangerous because tiny amounts can lead to mercury crossing the blood-brain barrier, leading to terrible neurological systems and death.  Once mercury is into organic tissues, it is very difficult to chelate the metal ions.  Again, there is a shortage of official information about this incident, but MIT's announcement made it clear that any synthesis or use of this compound is not permitted and was unauthorized.  

Update:  the latest from MIT’s emergency response page raises the possibility that there may not have been any exposure or dimethylmercury present.  Fingers crossed that this turns out to be a false alarm.

Update 2:  The always excellent Derek Lowe with a further discussion of what seems now to have (thankfully, hopefully) been a false alarm.

To students reading this:  PLEASE be careful in the lab.  Know the hazards of what you're doing, and use appropriate procedures and protective equipment.  If you ever have questions about safety, for goodness' sake please ask.  Your PI and your environmental health and safety team would far rather have you ask questions and be cautious then to do something dangerous.  No PI should ever make students feel like thinking about safety is unnecessary or overly cautious, and no PI should ever be hesitant about supplying or letting students purchase PPE.  

In the beginning there was a question. Actually, no, in the beginning there was language.  The question came later, presumably as one of its side effects. Since then we have asked about nearly everything, though the answers have done little to alter our circumstances. Human existence has always seemed strange to me. We arrive without consent, in a place not of our choosing, then spend decades asking why, until we close our eyes and enter the abyss of nothingness.

Some people accept this situation very well. I did not.

 Until my mid-teens, I tenaciously found one solution in fashion. If I could not understand who I was, I could at least decide what version I could become. Clothes gave form to something otherwise difficult to locate. Desire, after all, begins with a lack. 

It was only later that physics presented a solution to the same lack, only with more elaborate mathematics.

Quantum mechanics tells us that the world beneath the familiar world does not behave as we experience it.  A thing can resist being one thing, certainty begins to dissolve, and reality becomes strangely unfamiliar. The uncanny begins very close to home.

It is in this uncanniness that fashion and physics, liminally, meet for me: both begin with a human being standing before something they cannot understand, the universe in one case, the self in the other, and trying to make a form out of it. 

But even this distinction, on closer inspection, begins to collapse; for our desire to understand the universe has always concealed a deeper desire to understand the self that stands within it. 

All roads eventually lead back to the interests you had as a child, or so I say. 

I would squint into the dark, my mind already at work. I would imagine the next thing I would wear, project its color and fabric onto the ceiling. 

Then,  go to the bazaar, Rangrizano Dana, where, as the Kandaharis like to say, everything is sold except one’s mother and father.

The designs that had existed only in my head, after night upon night of theorizing and calculating, were finally beginning to take shape. The alleys of Rangrizano Dana were full of fabric of different textures and colors. 

Once the fabric was bought, came the tailoring. Tailors hold a special place in Afghanistan. Some are stars in their own right, the kind you have to book an appointment with. And once you get there, another matter comes up: making the tailor promise not to show your design to another woman. Everyone wants to wear something unique. So, naturally,  there is a lot of secrecy.

The designs themselves would either be described or sketched. Somewhere between what was imagined and what the tailor could make, something would emerge. And the day the tailor shipped your clothes was really a day of revelations.

Time passed, which is another way of saying the object of my desire changed. The same attention once spent on cut, texture, and appearance turned, little by little, toward the fabric of space and time.

Studying physics, I organized my process in the same way. There was again an idea that existed first in the mind, and then the problem of giving it form. 

Only now the materials were different. Instead of cloth, there was mathematics; instead of the tailor’s table, the blackboard and chalk. And, standing before it, more often than not, a badly dressed physicist covered in chalk dust. All the libidinal energy of the universe, you might say, had been sublimated into equations.

Fashion had, in some ways, been the more serious pursuit. There was something at stake: you wanted to look better than everyone else. Physics, by comparison, could be surprisingly playful.

Alice sends a particle to Bob. The sound of those names would tickle my Pashto ear then, and still cracks me up now. Cats are placed in boxes, both dead and alive. Observers hover near black holes or sometimes even, whooooooosh, fall into them.

So, in this way, in my mind’s eye, the physicist has become a rather more amusing figure: a tailor of space and time. The universe, of course, is a notoriously difficult client (perhaps even more difficult than a Kandahari woman making her tailor promise not to show her design to anyone else). 

He begins, as any tailor must, with an imagined shape. Mathematics is his chalk, his scissors, his needle. With it he marks the fabric, cuts it, folds it, joins one piece of reality to another. Some constructions fall upon the universe with elegance. Others bunch at the shoulder, pull at the waist, or refuse altogether to button, and must be altered, or thrown away completely.

A tailor works against the resistance of the human body; a physicist against the resistance of the universe.

Imagine, then, a physicist seated at the edge of a black hole, spectacles low upon his nose, sewing little cloaks from the fabric of spacetime.

“How many dimensions will you need?” asks Alice.

“That depends upon the suit,” he replies.

    For AdS/CFT, suppose we are making a three-dimensional garment.  The tailor of the universe never begins with three dimensions. Before him lies only the flat, two-dimensional fabric, with all its degrees of freedom spread upon the board.

 It is then, in a less sartorial language, we might call the disentangling and entangling of degrees of freedom; the tailor coaxes a third dimension out of the flat cloth. This is the secret embroidery of the holographic idea: what appears to the wearer as a three-dimensional world is encoded in a fundamentally two-dimensional way.

Bob tries this newly fashioned quantum garment on before entering the black hole. 

 “Too tight,” he says. “See if there is more degree of freedom.” 

The physicist frowns, takes up his chalk, and makes another small mark, with some disentangling and entangling here and there.

What could have been wrong? Well, perhaps the group theory has been chosen badly, the symmetry is broken, and a representation must be changed, a seam opened, a dimension added, or one cunningly concealed.

Alice watches him work, the almighty tailor of the universe,  making change without a change.

“But how do you know when the suit is right?”

And with that, the physicist looks at her.

“What do you want? Do you want to put me out of work? Unknowing is the lack that drives us to make more suits. If we knew the suit was right, that would be the end of everything.”

Dark matter bugs people.

I’ve talked before about why, and why it, and other beyond-the-standard-model proposals like those inspired by MOND, are nonetheless credible with physicists. But beyond the logic in that post, there’s a deeper reason people find dark matter strange. It’s that they don’t know what kind of an explanation dark matter is.

Dark matter sounds very lazy. If you can’t explain the movements of stars based on the matter you can see, then proposing invisible matter sounds like the easy way out. But it’s actually a lot less easy than it sounds, because matter is something quite specific. Matter gravitates and bends light. Matter moves. Matter can be described with a pressure, one like gas and dust and not like other things like light or the Higgs field. If you propose a new type of matter, you have to check and see that all of those consequences hold, with detailed implications for almost every observation every astronomer takes.

For the most part, those consequences have been checked, and they do hold. Sometimes they fail, and it’s those failures, and not the idea that dark matter is “lazy”, that drive dark matter’s critics in the physics profession. Physicists who oppose dark matter have other explanations with their own consequences, for example new types of quantum fields that often get described to the public as “modified gravity”. When they argue against dark matter, they do it by comparing those consequences in detail, working through the implications and seeing which phenomena hold.

Dark matter, as it turns out, is a very constraining explanation, one with strict consequences. There are other corners of physics where the explanations may seem less lazy, but actually have fewer consequences, and thereby less scientific heft.

For example, consider the debate about evidence for dark energy I wrote about last month. A key question there was how to interpret light from supernovae. Some groups argued that supernovae change in brightness with distance, others that they change based on how old their galaxies are. Sabine Hossenfelder glossed the debate by saying it comes down to how you model supernovae. And while that’s true, it can give the wrong impression.

You might think that these people are comparing detailed computer models of supernovae, and making different assumptions when they set their models up. But in reality, it’s much less detailed. The people on both sides of this debate are looking at correlations, trying to draw statistical lines through supernova datasets. The difference between one model and another isn’t a complicated physical setup you can put into a simulation, it’s just which lines on a graph you account for and which you ignore.

Because of that, while these models may sound much more sophisticated than dark matter, they actually have much less scientific weight. The different supernova models don’t have grand, widespread consequences, they’re not mucking with the laws of physics or proposing new classes of object that every astronomer needs to account for. They’re pretty much just proposing tweaks to how to interpret one very specific type of data. That makes their questions much harder to resolve, and their answers much less universally convincing.

If you’re not a scientist, if you read science news, it can be hard to tell the difference. Some ideas in science may sound simple, but have a whole raft of consequences that distinguish them from other ideas. Others may sound sophisticated, but are much more like “fudge factors”, only distinguished by statistical arguments, not by a rich trail of qualitative evidence.

For the most part, as an outsider, you’ll never know which is which. But as always, it’s best to be aware of your limits.

Freshman year of college, I took a writing seminar from German-literature professor Ellis Shookman. Professor Shookman loved Mozart’s music, he told us early in the term. He listened to Mozart on the radio while driving from campus to Boston. Static might mar the transmission, but he could often turn up the volume and continue enjoying the program. Sometimes, the static worsened during the drive. It could worsen and worsen, until Professor Shookman’s frustration outweighed his delight at listening. He’d switch off the radio.

As Professor Shookman loved listening to Mozart’s music, he loved reading about students’ ideas. Yet static can mar a piece of writing: infelicities in grammar, structure, composition, word choice, and more. If enough infelicities obscure the writing, the frustration of reading outweighs the benefits. Professor Shookman will quit reading.

Professor Shookman marked up our essays with a blue pencil that achieved the status of legend among his students. If you’ve written a paper I’ve coauthored, you’ve probably received PDF drafts replete with green highlighting.¹ A sticky note explains the reason for each highlighting: “Singular–plural mismatch.” “Active voice >> passive voice.” “Let’s clue the reader in as to this formula’s meaning before lobbing the math at them.” 

Over the past year, I’ve catalogued the suggestions I write most often on paper drafts. The comments embody principles gleaned from Strunk and White’s The Elements of Style; the Physical Review style guide; other writing guides I esteem; literature whose writing I esteem;² collaborations with professional editors; and writing instructors, including Professor Shookman. Each section below begins with more-important principles, shading into more-nuanced ones.

Please use and disseminate these principles. Train your favorite large-language model (LLM) on them, and have the LLM critique your manuscripts. Instruct it to use green highlighting if you wish. Even if the LLM suggests fixes initially, tell it to stop offering suggestions later, so that you can devise the solutions: train not only the LLM, but also yourself. I hope to enjoy your papers as much as Professor Shookman enjoyed his sonatas.

1. Organization
   1. Motivate your work; then, present it; and then, explain its physical significance.
   2. Begin each paragraph with a topic sentence.
   3. Begin each section, apart from the introduction and conclusion, with (i) a statement of the takeaway and (ii) an outline of the section. When outlining a section, hyperlink to each subsection. Similar guidelines concern subsections and subsubsections.
   4. Before presenting a piece of math, sketch its meaning and origin. This strategy enables the reader to understand the math as soon as they encounter it. If you throw math at the reader without introducing it, the reader will have to squint at the symbols for a while to figure out what the expression means and where it comes from.
      • Example: To calculate the average work, we substitute the Hamiltonian formula (10) into the definition (12): [equation].
   5. Most citations belong at the ends of (i) sentences and (ii) phrases concluded with commas. Put a citation elsewhere only if you have a compelling reason for doing so.
   6. Bridge each component of your writing to the next component; the next shouldn’t sound like a non sequitur.
      • Suppose that the next sentence refers to (i) a topic mentioned in the previous sentence and (ii) a new topic. Mention (i) before (ii).
        • Example: Smith et al. applied control theory to the extent possible. The attempt led to intractable equations, unlike our approach.
        • Example of a broken bridge: Smith et al. applied control theory to the extent possible. Our approach does not involve intractable equations, unlike theirs.
   7. Whenever you tell a story, tell it from start to finish, step by step. Derivations, proofs, and descriptions of experiments qualify as stories.
      • This guideline extends to descriptions of experimental setups and of mathematical objects. For example, imagine referring to an element of a subgroup of the group generated by some operators. Did you have to read the preceding sentence multiple times to process it? The sentence begins at the end of a story, then rewinds to the story’s beginning. This structure impedes understanding. The subgroup forms the context for the subgroup element, which one can’t grasp until hearing about the subgroup. The subgroup participates in a similar relationship with the group, as does the group with its generators. Therefore, one should introduce the generators, then the group, then the subgroup, and then the subgroup element.
2. Word choice
   1. Use strong, specific words, rather than weak words.
      1. Verbs and nouns are stronger than adjectives and adverbs.
      2. Choose specific verbs (e.g., “prepare,” “evolve,” and “measure”), rather than vague, general verbs (e.g., variants of “to be” and “take,” as in “take a measurement”).
   2. Avoid statements such as “we investigate,” “we study,” and “we analyze.” Such statements don’t relate that you’ve accomplished anything. State what you’ve accomplished. Verbs such as “prove,” “test,” “confirm,” “discover,” and “find” achieve this goal.
   3. Adverbs such as “importantly” and “remarkably” pollute scientific writing with the authors’ opinions. Demonstrate that a claim is important or that a result is remarkable; then, leave readers to draw their own conclusions. Those conclusions will coincide with yours if you’ve demonstrated your point.
   4. Use the active voice, rather than the passive voice. Take responsibility for your work. Editors of high-impact scientific journals have endorsed this advice.
   5. Refer to yourself when necessary and only when necessary.
      • Example of unnecessary reference to self: We use the superscript “max” to signify the maximal Fisher information.
        Preferable alternative: The superscript “max” signifies the maximal Fisher information.
      • Example of unnecessary reference to self: Our results establish several opportunities for future research. First, we can implement the experimental proposals.
        Preferable alternative: Our results establish several opportunities for future research. First, one can implement the experimental proposals.
      • You may use the first-person plural when escorting the reader through a derivation.
      • Example: We substitute from Eq. (1) into Eq. (2).
   6. If you’re the only author, don’t use the plural (“we,” “our,” etc.). The usage is inaccurate and misleading. It portrays you as dodging responsibility for your work by dispersing that responsibility across the scientific community.
   7. Avoid dangling modifiers.
   8. Pair every verb with the appropriate noun.
      • Example of grammatically incorrect text: Equation (1) follows by calculating the sum.
        • One should pair the verb “calculate” with the noun “we,” because “we” undertook the calculating. However, this example’s author omitted the noun out of squeamishness about using the first person in a scientific document. Hence the sentence says that the equation calculates the sum. Equations can’t calculate sums.
      • Examples of correct alternatives
        • We derived Eq. (1) by calculating the sum.
        • Equation (1) follows from the evaluation of the sum.
        • Calculating the sum yields Eq. (1).
   9. Avoid empty subjects.
   10. Include no unnecessary words.
       1. “So-called” is unnecessary.
       2. “Note that” and “We note that” are unnecessary.
       3. “We have that,” used as a preface to a mathematical statement, is unnecessary. One can better serve the reader by prefacing the mathematical statement with (i) a derivation or (ii) a prose description of the statement.
       4. Never write “is equal to”; “equals” is more concise.
       5. Never write “is able to”; “can” is more concise.
       6. Never write “gives an upper bound to” or “places an upper bound on”; “upper-bounds” is more concise. Analogous statements concern lower bounds.
       7. Never write “a large number of”; “many” is more concise. Never write “a small number of”; “few” is more concise.
       8. Never write “We refer to [symbol] as [name]”; “we call [symbol] [name]” is more concise.
       9. Never write “as long as”; “if” is more concise.
       10. The symbol means “greater than”; and , “greater than or equal to.” Don’t translate into “strictly greater than”; the “strictly” is unnecessary. Analogous statements concern and .
   11. Avoid contractions, which are too informal for professional writing.
   12. The possessive is not a contraction and belongs in professional writing. It facilitates conciseness.
   13. Use the word “for” only when it belongs. Physicists often write “for” when they mean “if,” “per,” “at,” or something else.
       • Example of inappropriate use: The function vanishes for odd arguments.
         Corrected statement: If the argument is odd, the function vanishes.
       • Example of inappropriate use: We performed 10 trials for each parameter value.
         Corrected statement: We performed 10 trials per parameter value.
       • Example of inappropriate use: The function is smaller for small x values.
         Corrected statement: The function is smaller at small x values.
   14. Write “we evolve the state,” “we measure,” etc. only if you’re an experimentalist who undertakes those actions. Alternatives include “Consider measuring,” “Suppose the system evolves,” and the command tense (e.g., “One can measure this quantity as follows: prepare the qubit in . Evolve it under …”).
   15. The condition defines a regime, not a limit. The conditions and define limits and are inequivalent to .
   16. Write “first,” “second,” “last,” etc., not “firstly,” “secondly,” “lastly,” etc. (I defer in this matter to The Elements of Style.)
   17. Humans can assume, suppose, etc. Mathematical expressions, protocols, etc. can’t.
   18. One multiplies factors together and sums terms. Don’t call factors terms and vice versa.
   19. If you mean “X equals Y,” say so. Don’t write “X agrees with Y,” “X matches Y,” or “we identify X with Y.” The latter three phrases are vaguer, and two of them contain more words, than “X equals Y.”
   20. Regarding the words “general” and “generally”:
       1. A general object subsumes every example of that object. If any example behaves unlike a supposedly general object, don’t call the object general.
       2. Many claims contain the term “general,” “generally,” or “in general” but don’t need the term.
          • Example of a sentence that contains “generally”: The terms generally commute.
          • Equivalent, more concise sentence: The terms commute.
       3. Physicists tend to use the words “general” and “generic” differently. By “general,” physicists usually mean “subsuming every example.” By “generic,” we usually mean “typical,” or “common.”
   21. “Then” makes sense (i) in discussions of chronology and (ii) in if–then statements. Don’t use “then” outside these contexts.
       • Example of inappropriate use: “Define Then Y.”
       • Examples of appropriate alternatives
         • Define This definition implies Y.
         • If then Y.
         • Define such that Y.
   22. Don’t justify any equation with “we used that [such-and-such is true],” which violates the rules of grammar. Grammatically correct alternatives include “We applied [a property],” “The equation follows from [a property],” and “…since [such-and-such is true].”
   23. Regarding tense:
       1. When describing what you’ve accomplished, use only one tense.
       2. Experiments happened in the past, so describe them in the past tense.
       3. When describing a proof’s steps, use the present tense.
          • Example: We Taylor-approximate the function about . Substituting into Eq. (1) yields [equation].
   24. Nouns, verbs, and adjectives should agree about whether a quantity is singular or plural.
       • Example of singular–plural mismatch: The equations are a rule for evolving the cellular automaton.
       • Example alternative: The equations form a rule for evolving the cellular automaton.
   25. “Admit of” means “allow for,” or “permit.” The phrase needs the “of.”
       • Example: The formula admits of the following interpretation.
3. Punctuation
   1. Consider any list that contains at least three items. If no item contains a comma, separate the items with commas. If any item contains a comma, separate the items with semicolons.
   2. In American English, periods and commas belong inside quotation marks. (Example: She told me, “Have a good day.”) In British English, periods and commas belong outside quotation marks. (Example: She told me, “Have a good day”.)
   3. To write quotation marks in LaTeX, don’t use your keyboard’s quotation-mark key; use the appropriate keys.
   4. Regarding hyphens:
      1. The hyphen (-) feeds into punctuation of three types: the hyphen (-), the en dash (–), and the em dash (—).
      2. The hyphen appears in some compound words, as in “non-negative.”
      3. In American English, em dashes can separate ideas within a sentence. Don’t separate any em dash from neighboring text with a space.
         • Example of appropriate use: The sample—the only product of this experiment—barely survived.
         • Example of inappropriate use: The sample — the only product of this experiment — barely survived.
         • Example of appropriate use: He told me only one sample had survived—hardly what I wanted to hear.
      4. This article specifies how to use the en dash. One use is “to separate the names of two or more people used as a compound modifier.”
         • Example: Feynman–Kitaev clock
      5. Hyphenate compound adjectives.
      6. If an adverb ends in “-ly,” it probably shouldn’t precede a hyphen.
         • Example of inappropriate hyphenation: strongly-coupled systems
      7. Follow a prefix with a hyphen if and only if the Physical Review style guide indicates that you should.
   5. A complete clause must follow any semicolon (unless the semicolon separates items in a list).
4. Math
   1. Introduce only necessary notation, which readers will have enough trouble remembering. If a mathematical symbol appears only once, eliminate it. If a symbol appears only twice, try to eliminate it.
   2. Every sentence must obey the rules of English grammar, punctuation, and syntax, regardless of whether the sentence contains mathematical symbols. All math-containing sentences must end with punctuation marks. If a sentence contains a list of mathematical expressions, precede the final expressions with an “and.” If the list contains at least three mathematical expressions, separate them with commas.
   3. Introduce almost every mathematical symbol before you use it. If you introduce a symbol after using it, the reader will encounter the first use, stop, feel confused for a while, tentatively continue, find the definition, return to the earlier use to understand it, and then progress again. This back-and-forth breaks up the reading process. You may define a mathematical symbol after using it only if (i) the symbol is very common, known to nearly all physicists, and unmistakeable and (ii) defining the symbol earlier would disrupt the text’s flow.
   4. If you define a new function, denote it by only one letter. (I defer in this matter to the Physical Review style guide.)
      • Example:
      • Examples of disallowed notation: ,
   5. Suppose that a superscript or subscript stands for a word or phrase without representing any variable or constant. The superscript/subscript must not be italicized. (I defer in this matter to Physical Review style guide.)
      • Example: Let denote the measurement outcome.
   6. If a variable or constant appears in a superscript, parenthesize it. The parentheses communicate that the superscript isn’t an exponent.
      • Example: Let denote the Pauli- operator of qubit
      • If a superscript is not italicized (stands for a word or phrase), don’t parenthesize it.
   7. Regarding the definition of a symbol :
      1. If you write alone on one side of a defining equation, use \coloneqq or \eqqcolon: \coloneqq [expression], or [expression] \eqqcolon . The symbols \coloneqq and \eqqcolon relate more information than does , encoding directionality.
      2. Use if does not appear alone on its side of the equation: [function of ] [result of replacing with its definition in the equation’s left-hand side].
   8. Refer to the Cartesian axes using the formatting “[italicized letter]-axis.” Don’t include any hat, boldface, or \vec symbol.
      • Example: -axis
   9. Avoid denoting any index by , which means to physicists. Use instead, unless you’re writing for engineers (who denote by ).
   10. Don’t use the lowercase letter l (“ell”) as an index; readers might mistake it for a one. Use (\ell) instead.
   11. Give every set-off equation a number. Readers (and coauthors) may want to refer to the equation easily when discussing the paper. Save them (and us) from having to say, e.g., “that equation halfway down page three.”
   12. When writing a set-off mathematical expression, use the align environment, not the equation environment. Using the align environment, one can easily extend an expression across multiple lines.
   13. Regarding a set-off mathematical expression that extends across multiple lines:
       1. Format the expression as follows by default.
          1. Put an & symbol immediately leftward of the first = sign or analogous symbol (e.g., ).
          2. If any subsequent line begins with another = sign (or analogous symbol), put an & immediately leftward of the symbol. (I’ll stop writing “or analogous symbol.”)
          3. Suppose that a subsequent line begins with a , –, , or . Find the symbol immediately rightward of the initial = sign. Begin the new line directly below that symbol.
          • Example:
       2. Modify the default formatting if necessary (a) to reduce the number of lines used in a PRL submission or (b) if the initial = appears awkwardly far to the right.
          • Example of (b):
       3. Suppose a new line begins with a term or factor, such as the in the example under (A). Put the corresponding , –, , or at the beginning of the new line, not at the end of the previous line.
          • Examples of inappropriate placement:
   14. The symbol means “approximately equals”; and ~, “scales as.” Approximations convey more information than scaling relations do.
   15. Use big-O-type notation or ~ symbols, not both; they’re partially redundant.
   16. \ldots, rather than \cdots, should stand in for elements that fit a pattern.
       • Example:
   17. When using \ldots as in the previous rule, present at least two initial examples of the pattern. One can’t define the pattern.
       • Contains insufficient examples: For example, if is odd, then and fit the template.
   18. Parentheses (), square brackets [], and curly braces {} are delimiters. If you nest them, do so in the order dictated by the Physical Review style guide.
   19. If delimiters enclose a symbol, it shouldn’t protrude above or below them (unless the delimiters would have to be grotesquely enormous). Use the \left and \right commands if the delimiters appear on the same line.
   20. An operator isn’t a matrix; a matrix represents an operator in terms of a particular basis. Therefore, no equals sign should interrelate an and a matrix. An arrow can.
       • Example:
   21. Every real number is complex. Don’t say “complex” if you mean “nonreal.”
   22. Consider introducing a mathematical symbol in a prose sentence without using a comma or colon. Put the symbol immediately after the word that names the object represented by the symbol.
       • Example of inappropriate placement: the set of real numbers
       • Examples of appropriate placements
         • the set of real numbers
         • the set of real numbers and
         • Recall the set of real numbers, , in Lemma 1.
5. More mechanics of writing
   1. Use concise sentences, as advocated for in The Elements of Style. The reader can hold only so many ideas in their head at once.
   2. Structure sentences simply, as advocated for in The Elements of Style. The reader should be able to grasp each sentence easily.
      • Avoid nesting ideas within a sentence, to avoid convoluting the sentence’s structure.
      • Example of sentence with convoluted, nested structure: Any model of equilibrium and nonequilibrium behaviors of systems observed in tabletop experiments and high-energy colliders must obey the laws of relativistic quantum mechanics.
      • Visualization of the nesting: [Any model of ([(equilibrium and nonequilibrium) behaviors] of {systems observed in [(tabletop experiments) and (high-energy colliders)]})] must obey [the laws of (relativistic quantum mechanics)].
   3. The ideal paper title has the structure of a newspaper headline: it presents a claim, containing a subject and a predicate.
   4. Regarding abbreviations:
      1. Don’t abbreviate the first word in any sentence.
      2. Abbreviate “Figure,” “Section,” “Professor,” and “Appendix” if such a word appears partway through a sentence.
      3. Don’t abbreviate “Sections.”
   5. Regarding acronyms:
      1. Write every acronym in capital letters, as per the Physical Review style guide.
      2. Introduce each acronym the first time you use it.
      3. Thereafter, use only the acronym, not the spelled-out phrase, throughout the rest of the document’s main text. You may spell out the phrase in section, figure, and table titles if doing so improves the document’s clarity.
   6. Every paragraph should contain at least three sentences.
   7. Wherever you insert a blank line into your LateX code, a new paragraph begins in the corresponding PDF. Insert a blank line only if you wish to begin a new paragraph. This advice applies immediately before and after set-off equations.
   8. Never begin a subsection immediately after a section title. Between the two titles, overview the section. Analogous rules govern subsections and subsubsections.
   9. Put the word “only” in the appropriate place.
      • For example, suppose you’ve sampled data at a point in parameter space and sampled data at no other points. “We sampled data only at ” is correct; “We only sampled data at ” is probably not. The latter claim means that (i) you might have sampled data at and (ii) you did nothing to the data apart from sample it: you didn’t analyze the data, discuss the data, etc.
6. When in doubt, consult the Physical Review style guide or The Elements of Style.
   • If those references don’t contain the information you seek, search for it in online writing guides. Not all such guides have equal merit, however. Lean toward guides written by human editors or published by college writing centers.

¹ Collaborators have wondered why I use green; a student guessed it’s my favorite color. It isn’t; but I bleed green, having graduated from the Big Green, also known as Dartmouth College. Sometimes, I highlight certain pieces of text for one reason (e.g., to point out logical inconsistencies) and other text for another reason (e.g., to point out grammatical inconsistencies). Green distinguishes the first highlightings, while orange distinguishes the second: when not bleeding Dartmouth green, I bleed Caltech orange.

² Don’t learn how to write from physics papers. 

A couple of new videos of me talking to people, which feature some themes that are going to be in Don’t Be Too Sure, as well as some stuff I’ve written about before. Both of these are long, so only watch if you, I dunno, have a long series of physical therapy exercises you have to do or something!

On the Particles of Thought podcast for PBS:

And talking to the Wisconsin Mathematics Council about the virtue of uncertainty:

I think it's safe to say that most people who've considered the issue think that a doctoral degree in the sciences and engineering in the US often takes too long.  

How long?  According to the latest data (see here, Table 1-12), the median time to degree in the physical sciences, for example, is 5.7 years after starting the program, while in all of engineering it's 5.3 years.  

Too long for what?  Well, life, basically.  Any decision to go to grad school is inherently a trade-off with opportunity costs.  Graduate stipends remain low compared to expected wages in entry-level (bachelors degree-qualified) positions in the sciences and engineering in industry.  The long duration of doctoral programs is certainly a powerful disincentive for many who might be interested but are under financial pressures.  Family considerations are also a major factor.  From the perspective of basically any career path, thanks to the time value of money and ideas of seniority, it's better to get going earlier if you have the qualifications for the particular job.  Companies would rather hire younger (cheaper) people.

So, there are already strong reasons to think about shortening doctoral programs.  Now, with the proposed change in duration of status of student visas (rule here, with plenty of editorializing; legal challenges very likely forthcoming in September) to four years, there is additional pressure. 

Why do US programs take so long?  Don't they give PhDs in three years in the UK and Europe?  In the UK and Europe, a student enters a doctoral program after already pursuing and receiving a masters degree, with grad level coursework taking place there.  Thus they go directly into research.  In the US, in contrast, it is far more common for students to go directly into the doctoral program.  Likewise, in the US, it is far more common for funding for students to go through PI-written research proposals, while in the UK, the students come funded, so to speak.  

This is certainly a very interesting pilot program.  There are rumors that the DOE Genesis Mission is going to put something extremely similar in place as well.   The implementation details will be enormously important.  (For example:  Who is eligible?  Who handles the coordination between industry and the university - that is, who does the match-making and how?   At the department-company level and at the particular academic/industrial advisor level?   How will intellectual property be handled?  Publications?  Project design? If there are economic challenges, how committed are the companies?)  Given that this is a form of NSF fellowship, it seems highly likely that it will only be open to US citizens and permanent residents.  Obviously, not every discipline is well-suited to this, in terms of there being a ready supply of companies set to buy in.  Still, it is absolutely worth seeing how this works.

Update:  Thanks to one of my colleagues for pointing out the fine print, which is here.  In brief, as expected this is only open to US citizens and permanent residents.  No indirect costs allowed.  There is a $16K cost-of-education piece that looks like a substitute for grad tuition.  The intellectual property issues have to be ironed out between the university and the company before the start.  Perhaps not unexpectedly, this is most likely to work well for programs and PIs who already have close collaborations with particular companies.  Engineering disciplines are most likely to fit well here, it seems, while basic research farther away from applications will have more challenges.  (Question:  will finance companies or AI materials companies be interested in supporting theorist/computational scientists through this mechanism?)

A recent tradition of the ICM is to have a non-technical popular article written for each of its medallists. (This is separate from the older tradition of having a laudatio for the winner, which is similar but aimed at a more advanced audience.) I was approached to write such an article for Hong Wang on the occasion of her Fields Medal. I have uploaded my initial version of this article, “Rotating needles in space: the road to the Kakeya conjecture, and why it matters“, to the arXiv. Chris Sogge gave the corresponding laudatio; both should eventually appear in the Proceedings of the ICM.

So yeah, Anthropic has announced that it’s now watermarking the outputs of Claude, using a scheme based on Google’s SynthID, which is in turn based on the Gumbel Softmax scheme that I proposed at OpenAI back in 2022—as far as I know, the first LLM watermarking proposal, though far from the last one. I’m gratified that Anthropic credits me for this, even though I shirked my duty by never publishing a paper about it (by the time I sat down to write one, it seemed like the whole field had already assimilated my scheme and moved beyond it—AI just moves too fast for me!).

For those who don’t know, watermarking means slightly changing the way that an LLM operates to insert a subtle signal that lets you prove later, with high statistical confidence, that a text indeed came from your specific LLM. It uses the randomness that’s already present anyway in LLM outputs, replacing some of it by pseudorandomness that favors certain word combinations over others in a way that’s later detectable, given only the sequence of tokens itself (not the prompt or the probabilities) along with the key of the pseudorandom generator. Christ, Gunn, and Zamir then substantially improved my scheme to get true cryptographic indistinguishability, and there have been other improvements since.

I’d been meaning to blog about this for days. Thankfully, Zvi Mowshowitz, the world’s foremost blogger about AI, has now written a wonderful post, entitled AI Text Watermarking Is Free And Good, which saves me from the need to write my own long post. In particular, Zvi masterfully explains the central point that I needed to explain to everyone back in 2022-23: why, contrary to many people’s intuitions, there’s no inherent tradeoff between watermarking and the quality of LLM output. Basically, nearly every LLM output was already a sample from a cloud of exponentially many possibilities, all of them about equally good, so there’s plenty of room to steer within that cloud without affecting anything that an ordinary user would notice. As my kids would put it, the math mathes.

As Zvi explains, the central technical drawback of watermarking schemes like the one I proposed, and what Anthropic is now using, is that it’s possible to remove the watermarks with a little extra work (even stuff as simple as, e.g., translating between English and French, asking the LLM for words interspersed with emojis and then removing the emojis, or using an open model to paraphrase the output). Zvi gives detailed arguments for why he expects watermarking to remain a net positive in practice despite this vulnerability.

I could add that, in addition, there’s recent progress (see here for example) on what I’ve called “semantic watermarking,” or watermarking at the level of the underlying concept vectors rather than the tokens themselves. This actually seems to work, albeit with no theoretical guarantees, and will hopefully make removing watermarks a lot harder—although the Barak et al. impossibility result suggests that under plausible assumptions, no LLM watermarking method will be completely foolproof.

Anyway, I worked out my scheme in Fall 2022, then gave lots of talks about it (including, as it happens, at Anthropic), and also worked with Hendrik Kirchner at OpenAI, who actually implemented and tested my scheme. Unfortunately, OpenAI leadership decided against deploying watermarking, worried mostly about risks to the product (i.e., customers disliking the idea, and leaving for a competing LLM that doesn’t watermark). You can read this Wall Street Journal investigation from two years ago for more. I was hopeful that the State of California was going to solve the collective-action problem by mandating watermarking for AI models, but then they decided to do that for audiovisual content only, for some reason exempting text.

Nevertheless, Google DeepMind implemented something very similar to my proposal in its SynthID, deployed in all its Gemini text models. But they heavily restricted who gets to detect the watermark, which made their admirable decision of limited use to my academic colleagues, who’ve been begging me for a way to detect whether their students are using AI to cheat. (For now, I mainly send them to Pangram, a leading AI detector not based on watermarking, as a first line of defense.)

And now, apparently to comply with EU regulations, Anthropic says they’ve deployed a watermarking scheme like mine where anyone will be able to do detection (though they also say in their FAQ that they’re still working on the detection API). Even OpenAI suggests that it plans to follow suit. So, four years after I seriously thought about this, it looks to my surprise like this is actually happening. Thanks, EU!

Tell you what: read Zvi’s post, and then whatever questions you still have, you can come here and ask in the comments. Just please don’t use Claude to write the comments. With any luck, I’ll eventually be able catch you if you do.

Mathematical Discourse is a new online, peer-reviewed mathematics journal, publishing videos of mathematics research talks of the highest quality.  Our inaugural scientific and editorial boards are listed at the end of this message.

Our goal is to promote and nourish a culture which values communication as an essential part of the research process. Giving a mathematical talk captures the human side of mathematics in a unique way, and supporting these human aspects of our discipline will be ever more important in the coming years. 

We are now accepting submissions for our first issue.  If you have seen (or given!) a truly excellent, mathematically stimulating research talk which was video recorded, we hope you will encourage the speaker (possibly yourself!) to submit it.  

Our current call for submissions is limited to ~1 hour research seminar or colloquia style talks in pure mathematics. A detailed description of the scope of the journal and submission requirements are available at https://www.mathematicaldiscourse.org/submit.

We hope you will spread word of this new initiative to your colleagues and collaborators!

Sincerely yours, 

Katie Mann, Akshay Venkatesh, Rachel Webb

Managing Editors of Mathematical Discourse

Scientific Board: Hugo Duminil-Copin, Larry Guth, Bryna Kra, Yair Minsky, Bjorn Poonen, Terence Tao, Ravi Vakil, Geordie Williamson

Editors: Matt Baker, Richard Bamler, Andrej Bauer, Alexei Borodin, Yaiza Canzani, Daniel Cristofaro-Gardner, Jordan Ellenberg, Hélène Eynard-Bontemps, Jessica Fintzen, Sergey Fomin, Zaher Hani, Peter Hintz, Shrawan Kumar, Justin Moore, Samuel Taylor, Richard Thomas, Isabel Vogt, Yilin Wang, Alex Wright, Zhiwei Yun

mathematicaldiscourse.org

I had a recurring “elevator pitch” at Lancefest earlier this summer. After explaining that I’m a science journalist now, I’d end with “so if you run into a story, let me know!”

One person had a question that left me stumped: “What counts as a story?”

I didn’t have a good response then. I’ve got a better one now, though I’m afraid it doesn’t fit in an elevator pitch. This is all based on my experience, so take it with a grain of salt. But here are the criteria that seem to matter:

First, a story usually needs a news hook. News is, in particular, supposed to be “new”. That doesn’t mean I can’t write about history, or established science. But editors like those stories a lot better if there is some recent development, within the past year or so, to tie it to. The new development doesn’t have to be all that important, the story can mostly focus on something else. But it needs to be somewhere in there.

Second, news stories are usually qualitative, not quantitative. I need to be able to tell a story about what happened, what actions people took and why they mattered. Quantitative developments usually only make the news if they’re so big that they shade into the qualitative: something doubling unexpectedly, for example.

Third, ideally a science news story is something that is getting the experts excited. Journalists aren’t supposed to judge the scientific merit of ideas on their own, they’re supposed to rely on experts. The most solid stories, the ones that are easiest to pitch, are ones where there’s a community of experts that largely think something is cool. That makes it easier to get good quotes, and easier to justify its relevance. If you accomplished something and you’re having trouble convincing anyone it matters, don’t start with me, start with your colleagues!

Fourth: less importantly, it helps when stories have a human angle. If you can tell a tale about how you came up with an idea, if you came from an unusual background, if something was hotly debated but now is deemed essential: these things sweeten a story, they capture readers’ interest, and editors see their value.

Finally, stories involve something changing. It can be something that just changed now, for a news piece, but it can also be something that changed over time, for a feature in a magazine. The key is change. “Old method still works” is not going to excite people, and it won’t count as news.

After writing all that out, I’m still not sure I answered the original question. But hopefully I’ve at least given some tips that can get you started. If you’re a scientist, and you see something that hits most of the boxes on this list but hasn’t been covered in the news yet, consider reaching out to me. You may have run into a story!

What’s about the only thing more badass than a 17-year-old winning a gold medal at the International Olympiad in Informatics (IOI)?

That 17-year-old intentionally forfeiting his gold medal by wearing an Israeli flag while the medal was announced, defying the IOI’s boycott of Israel (for background on this boycott, see my post from 2024).

Kol HaKavod (mad respect) to Yotam Budnik, who incredibly, has also won a Gold Medal (which he was allowed to keep, apparently) at the International Math Olympiad. And congratulations to the entire Israeli team, which (incredibly) would apparently have had a higher overall score than the US team, had it been allowed to compete as an official team at all.

Ben Krause, Hamed Mousavi, Joni Teräiväinen, and I have just uploaded to the arXiv our paper Quantitative bounds for sets lacking polynomial progressions with shifted prime difference. The purpose of this paper is to obtain quantitative versions of this theorem of Wooley and Ziegler:

Theorem 1 Let be polynomials of one variable with integer coefficients with zero constant term, and let be a set of integers of positive density. Then there exist infinitely many primes such that contains a progression of the form

for some integer .

This generalizes the famous theorem of Szemerédi in two ways: firstly, by considering “polynomial progressions” instead of arithmetic progressions, and secondly by requiring the shift parameter to be one less than a prime . The first extension of Szemerédi’s theorem is a theorem of Bergelson and Leibman; and the second extension is also obtainable by combining the arguments of Frantzikinakis, Host and Kra with the results of Green, Ziegler, and myself.

The proof of the above theorem uses ergodic theory, which makes it difficult to extract quantitative bounds from it; and standard methods of “finitizing” ergodic theory results, for instance by replacing Host–Kra seminorms by their Gowers uniformity norm counterparts, run into a number of technical difficulties here due to the need to work with multiple scales due to the presence of polynomials, as well as the fact that many of the conjectural uniformity properties of the prime numbers at small scales remain unproven.

Nevertheless, we are able to get reasonable quantitative results (with density bounds that are roughly single or doubly logarithmic in scale) in the following special cases:

• linear polynomials;
• polynomials of distinct degree; and
• multiples of a fixed polynomial.

Previous quantitative bounds in the linear case were obtained by Leng and by Teräväinen and myself, but our new method improves upon these bounds by roughly one iterated logarithm. On the other hand, in the case of two term progressions of spacing , there is a much stronger quantitative result (with polynomial dependence of constants) due to Green; our method do not recover that result.

Our methods use a variety of old and new methods in the subject. For instance, we use the (now quite standard) “-trick” to restrict the primes to a single congruence class to improve their uniformity properties (which, thanks to the recent work of Leng and Matthiesen–Teräväinen–Wang, are now quite strong quantitatively); and for good configurations of polynomials, one can also use a recent transference theorem of Altman and Sawhney to compare the polynomial averages with simpler linear ones, without having to pass to short scales. In order to get relatively strong bounds unconditionally, a “Siegel approximation” for the primes is used taking into account the potential influence of a Siegel zero. It will not be surprising to the experts that quantitative inverse Gowers theorems and nilsequence equidistribution theorems also play a major role.

A key technical difficulty is the presence of the modulus in the coefficients of the polynomial progressions after changes of variable, which forced us to make several of the existing estimates uniform over such coefficients (assuming they are not unreasonably large).This caused several complications that made a correct argument to more time-consuming to locate than initially planned.

AI usage in this work was fairly light, being restricted to proofreading and literature search only.

Yes, I was there! The craziest game I’ve ever seen. Seattle utility infielder Leo Rivas throwing 40mph eephus after 40mph eephus, the closest I will ever get to seeing what it would look like if I through some bizarre chain of circumstance had to take the mound against major league hitters.

We missed a lot of this game because of a tornado warning that kept us off the road. Got there just in time to see old man Christian Yelich hit a three-run homer to put the Brewers up 6-0; I thought that was going to be the memorable thing about this game.

Mayhem when Gary Sanchez came in to close it out for the Crew, clinging to a 22-run lead. “Gary! Gary!” Ever louder as he kept getting guys out on his way to a scoreless frame. Changing speeds? Try 70mph fastball after a 35mph curve. How are you supposed to adjust to that?

Friend-of-the-blog (well, mainly just friend) Adi Akavia has asked me to publicize that she’s helping to organize an exciting CS conference called Mind-IL at Tel Aviv University on October 26, in memory of the Israeli-American Turing Award winner Michael O. Rabin, who passed away in April. Please note that October 26 is the day before the Israeli election, for any Israeli citizenship holders living abroad who might want an academic excuse to come to Israel and vote.

Update (August 19): Avi Wigderson also asked me to advertise a conference, to be held September 16-18 at Bletchley Park in the UK, to commemorate the 90th anniversary of Alan Turing’s “On Computable Numbers” paper.

In recent months there has been a proliferation of AI-generated proofs of various old and new results, some of which have been formalized in the proof assistant language Lean. However, checking that a given Lean repository actually proves the claimed statement is somewhat non-trivial, especially for an audience which is not expert in the use of Lean: one has to first check that the claimed formal Lean statements have proofs that typecheck, that the proofs do not contain any “cheats” such as adding additional axioms, and that the formal statements also match (in a semantic sense) the informal description of the claimed results.

To help bring some clarity to this situation, I am happy to announce that Palomar registry of Lean verified mathematics, which is an initiative incubated by the Lean FRO and by ICARM, is now open for submissions. I am serving in several roles on this registry, including on the scientific advisory board, together with Jeremy Avigad, Matthew Ballard, Jaume de Dios, Nestor Guillen, Bryna Kra, Kim Morrison, Ravi Vakil, and Akshay Venkatesh.

A detailed motivation for Palomar can be found here, and further information about Palomar can be found here. A zeroth approximation of what Palomar intends to be is the analogue of a preprint server for Lean proofs. More precisely, Palomar (which is named after the astronomical observatory) is a registry of external Github repositories (or more precisely, “snapshots” of such repositories, as represented by a specific Github commit) containing Lean code adhering to the current best practices for such formalizations, in particular containing

• A “challenge file” containing a short, human readable description in Lean of the results claimed.
• A “solution module” containing an (arbitrarily long) proof of the results claimed in the challenge file.
• A “formalization.yaml” file describing the results in informal language, and also containing a number of other relevant metadata and disclosures.

(There are also some additional technical requirements for the repository which I will omit here.) If a snapshot of a repository is submitted to Palomar, it will check both (a) that the solution module typechecks and proves exactly the results claimed in the challenge file, and that (b) the informal description of the result in the formalization.yaml file appears to match the result claimed in the challenge file, and that the repository meets various minimal standards required for a registry entry. The first check (a) is purely mechanical, using the Lean tool Comparator; the second check (b) is non-deterministic, being performed by a large language model. If a repository passes both checks, it can be registered on Palomar. It is worth stressing that the checks in (a) and (b) fall well short of what a proper human peer review of a submission for novelty, interest, and accuracy would give; in particular, Palomar is not a peer-reviewed journal.

The submission process is thorough, but achievable: as a test, I successfully managed to submit my own recent formalization of the proof of Sendov’s conjecture to Palomar, and also plan to submit some older formalizations to the registry soon.

In any event, the registry is now open for formalizations of both old and new results. Submissions (whether human-generated, AI-generated, or some mixture of both) are welcome; please read the (somewhat detailed) instructions here before starting a submission. (I will however note that modern AI agents are quite helpful in assisting with the mechanical details of the submission, though a human review is still strongly recommended.)

Discussion and feedback on Palomar will occur on this Zulip channel.

It’s long been a mystery why there are 3 generations of quarks and leptons: three sets of particles, apparently identical except for how they interact with the Higgs boson. It would be nice if there were some good physical explanation. Nobody knows one. Barring that, it would be nice if some beautiful mathematical structure made this pattern seem natural. That’s what my new paper is about.

It’s my third paper about exceptional algebraic structures and the Standard Model. When you classify famous gadgets in algebra, beautiful gadgets with fancy names like ‘simple Lie algebras’ and ‘Euclidean Jordan algebras’ and ‘positive hermitian Jordan pairs’, you tend to get infinite series of them—together with a few exceptions that can be built using the octonions. This is a bit spooky, so I’ve been interested in this for a long time.

A few physicists have hoped that these exceptions are good for something. For example, maybe the quirky features of our best theory of particle physics, the Standard Model, aren’t accidental. Perhaps they fall out naturally from some exceptional algebraic structure.

It’s a long shot, but we’ve been stuck on figuring out new fundamental laws of particle physics for so long—roughly since the early 1980s—that it’s worth a try.

In 2018, Michel Dubois-Violette and Ivan Todorov noticed that the gauge group of the Standard Model falls out as symmetries of the so-called ‘exceptional Jordan algebra’ together with some ordinary Jordan algebras sitting inside it. I tried to clarify that here, with a huge amount of help from an excellent young mathematician:

• John Baez and Paul Schwahn, The Standard Model gauge group from the exceptional Jordan algebra. (Blog article here.)

It’s very nice, because the Jordan algebras in question arise naturally when you try to axiomatize the foundations of quantum physics. It would be so cool if something about quantum physics made the Standard Model seem mathematically natural!

But really this result only concerns the gauge bosons in the Standard Model: the photon, gluons, and the W and Z bosons. It says nothing about the fermions—that is, the quarks and leptons. And it seems quite hard to get those into the picture.

In 2020, Latham Boyle tried to solve this problem by tensoring the exceptional Jordan algebra with the complex numbers. This made one generation of fermions appear quite naturally! But the connection to the foundations of quantum physics seemed lost: tensoring the exceptional Jordan algebra with the complex numbers seems at first like it might be just a formal trick.

This spring, Latham and his student Endre Bokor and I showed the connection to quantum physics is not lost:

• John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model. (Blog article here.)

The idea is to work, not with Jordan algebras, but with more general things called Jordan pairs, which have been studied by mathematicians since at least 1975. We showed that you can still do quantum physics with Jordan pairs. And we showed that there’s an ‘exceptional’ Jordan pair that naturally contains the Standard Model gauge group and one generation of fermions!

This Jordan pair is built from the bioctonions: the octonions tensored with the complex numbers. And it’s closely related to an exceptional Lie algebra called

This is nice because the work of Dubois-Violette and Todorov used a smaller exceptional Lie algebra called Going up to gives the room to include one generation of fermions.

There’s an even larger exceptional Lie algebra you can use to build a Jordan pair: it’s called Bokor, Boyle and I tried using this to get three generations of fermions. There are things that make this tempting: not just the fact that is bigger, but the fact that the Jordan pair you get from it has a kind of three-fold symmetry. But we couldn’t get it to work.

Around this time I got very interested in some work that someone had sent me in October 2025. My inbox is packed with new theories of physics. Since the rise of large language models the inflow has increased: I get about two emails a day from someone telling me they’ve made a revolutionary discovery in physics. Practically none of these theories appeal to me. But this paper, and this thesis, were different:

• Benjamin Nasmith, An exceptional combinatorial sequence and Standard Model particles, 2020.

• Benjamin Nasmith, Tight Projective 5-Designs and Exceptional Structures, Ph.D. thesis, Royal Military College of Canada, 2023.

He claimed to fit three generations of fermions into the exceptional Lie algebra

When I started seriously trying to understand this paper, I wound up translating it into a language I’m more comfortable with, and expanding on the ideas a bit. So I wrote this:

• John Baez, Three generations in

Here’s the basic idea.

The idea

There is a standard way to fit the Lie algebra of the Standard Model gauge group, which I call into the Lie algebra You can construct a Lie algebra that fits between them:

As a vector space we have

for some vector space of dimension

Moreover, the Lie algebra acts on , via the Lie bracket, precisely as it does on three generations of Standard Model fermions and their antiparticles, including right-handed neutrino and its antiparticle—but ignoring spin!

There is, in fact, a very interesting three-fold symmetry built into which is revealed when we put the Standard Model Lie algebra into it. It permutes the three generations.

Like Nasmith, I am not proposing a theory of physics. I’m only observing a fascinating mathematical pattern that might (or might not) be of some use in physics.

There are lots of things this pattern does not include: basically, everything I didn’t already mention. It does not include the spin of the fermions and gauge bosons. It does not include the Higgs boson, though in some sense it comes close (see the paper). It does not include a Lagrangian, so it doesn’t say anything at all about particle masses or interactions.

I could say a lot more about this… most importantly, where this Lie algebra comes from. The details are very interesting. There’s also the curious role of the right-handed neutrinos. But I’ve already spent weeks explaining all these things in my paper, so I won’t do it here. Instead let me say a bit about how I wrote the paper.

Writing the paper

I’ve been wanting to keep up with how AI is transforming math. About a year ago a friend gave me a subscription to Claude Pro. I wanted to test it out, despite my many misgivings, including how large language models are contributing to global warming and income inequality. Given the amazing things that people have recently done in math using large language models, I didn’t think that never trying them out would put me in the best position to make good decisions about the future.

So, I wrote this paper with help from Claude Opus 4.8.

I started by giving it Nasmith’s paper and asking a long series of questions about that paper over several days. The results were very interesting and helpful. Eventually I asked it to summarize and expand on our conversation. It quickly spat out a 10-page paper.

This paper was written in a breezy, pleasant style—but also quite hard to understand in detail, since it mixed Nasmith’s terminology with the Lie algebra terminology I prefer, and the proofs skipped over some steps.

It took me about three weeks of hard work to fully understand and re-express all the ideas a way that I like. For a while I felt dumb and frustrated, because when I asked Claude to fill in the gaps in proofs, it used math I was not very competent in, like the theory of regular subalgebras, and the theory of minuscule representations. But I learned this math, and everything turned out to be basically correct—in part, I’m sure, because Nasmith’s original work was correct.

For several weeks I checked, reorganized, expanded and completely rewrote this material. By the end everything was written in a style I like, emphasizing the ideas I consider important, proving things fairly carefully, and adding a lot of expository material—for example, explaining the theory of regular subalgebras.

Almost no traces of Claude’s original writeup remain, even though I was deeply influenced by them. My proofs make few references to deep theorems, though they assume solid familiarity with simple Lie algebras and their root systems. The proofs also require no brutally hard computations—though Claude was eager to do such computations to check things.

Any mistakes in this paper are my own.

I’m not sure what conclusions I draw from writing this paper. I’m writing another math paper now, with a human coauthor, and I have no desire to get help from a large language model. For work on my own it could be very helpful. Jacob Tsimerman says it roughly doubles his productivity. Would using it be so bad for the environment, or so bad for society, that I should avoid it? Maybe. I deliberately stuck with Claude Opus 4.8 instead of something more powerful, to see what I could do with what you get from a $20/month subscription. But maybe that’s still bad.

I avoid flying to conferences, which in some ways cripples my ability to keep up with new trends and influence people—but I don’t mind that. It gives me more time to think.

I will think carefully about my next move.

I don’t usually talk about particle physics here. I have a whole series of articles about octonions and the Standard Model on my other blog. But I’m kind of excited about this new paper, so I’ll talk about it here too:

• John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model.

Jordan algebras were introduced by Jordan, von Neumann and Wigner in 1934 in an attempt to formalize algebras of observables in quantum theory. They come in 4 infinite series—but there’s one more, the ‘exceptional Jordan algebra’, consisting of 3 × 3 self-adjoint matrices of octonions. For years physicists sought to find some use for it.

In 2018, Todorov and Dubois–Violette noticed that the symmetries of the exceptional Jordan include the Standard Model gauge group in a nice way. But it was unclear how to bring in the fermions—the quarks and leptons. That’s what our new paper does.

To do this, we need to go beyond Jordan algebras. Jordan pairs and Jordan triples are two closely linked formalisms that generalize Jordan algebras. Our paper explains them in detail—and how they’re connected to geometry and quantum mechanics. But here I will mostly skip that wonderful story, so I can quickly explain the connection to the Standard Model.

Here’s how the Standard Model gauge group, together with its representation on one generation of fermions, drops out of a Jordan triple.

The bi-Cayley triple

Let

be the bioctonions: octonions with complex coefficients. Write for the space of column vectors with two bioctonion entries.

has a certain triple product

which obey the axioms of a gadget called a ‘positive hermitian Jordan triple’. It’s called the bi-Cayley triple.

Now, every positive hermitian Jordan triple gives rise to a -graded real Lie algebra

Not a Lie superalgebra: a plain old-fashioned Lie algebra with a -grading!

How does this work? We take the hermitian Jordan triple itself to be The Lie algebra consists of all linear maps from to itself that are of this form:

for some These maps are called real inner derivations. They form a Lie algebra since the commutator of two such maps is another such map. With a bit more work we can define other operations making all of into a -graded Lie algebra.

So, we get a big Lie algebra and a Lie subalgebra sitting inside it. From this we get two Lie groups: a big one whose Lie algebra is and a subgroup whose Lie algebra is

The quotient is is a nice kind of manifold called a hermitian symmetric space. Conversely, any compact hermitian symmetric space give rise to a positive hermitian Jordan triple!

This geometric picture is revealing. The group acts transitively as symmetries of our hermitian symmetric space, while the stabilizer of any point is isomorphic to Our original Jordan triple, is then the tangent space of that point! So, acts on this Jordan triple. This action preserves the triple product, and we call the real inner automorphism group of our Jordan triple.

Here’s another great thing about the geometric picture: hermitian symmetric spaces were classified by Eli Cartan (who seems to have spent his life classifying things). As a result we also know the classification of positive hermitian Jordan triples. They come in four infinite series together with two exceptions. One is the bi-Cayley triple, and other is the Albert triple, which is the complexification of the exceptional Jordan algebra. The bi-Cayley triple is a subtriple of the Albert triple. It’s these two exceptions that are connected to the Standard Model. But we’ll start with the bi-Cayley triple.

The 3-graded Lie algebra coming from the bi-Cayley triple is the compact real form of :

The even part of this Lie algebra is in brackets. The corresponding hermitian symmetric space is called the bioctonionic plane The even part of our 3-graded Lie algebra, generates the stabilizer of a point in the bioctonionic plane. The odd part, our friend is the tangent space of that point.

Here’s the first big surprise. The even part transforms as the adjoint representation of while the odd part itself transforms as the 16-dimensional complex spinor representation of Ignoring the extra for a moment, this is exactly what we see in a grand unified theory: gauge bosons in the adjoint representation, and one generation of fermions in the 16-dimensional spinor representation.

So before we do anything, the bi-Cayley triple already smells like it contains the ingredients of an grand unified theory.

Tripotents

In a Jordan algebra the important elements are the idempotents, In a Jordan triple their role is played by tripotents: elements with

A tripotent always lets us split into three parts via something called its Peirce decomposition. The operator has eigenvalues 0, 1/2, and 1, so splits into the corresponding eigenspaces

which are called the Peirce 0-space, Peirce 1/2-space and Peirce 1-space of A tripotent is called minimal when its Peirce 1-space is one-dimensional. Two tripotents are called colinear when each lies in the other’s Peirce 1/2-space.

I can’t resist explaining some of the quantum physics here. In a hermitian Jordan triple, the triple product is linear in the first and last slot, but conjugate-linear in the middle slot. So, if you multiply a tripotent by a phase you get a new tripotent:

This should remind you of how when you multiply a unit vector in a Hilbert space by a phase, you get a new unit vector. In Jordan triple quantum mechanics, minimal tripotents take the place of these unit vectors. The hermitian symmetric space that I was talking about earlier is the same as the space of minimal tripotents mod phase! So, it generalizes the familiar space of ‘pure states’ in quantum mechanics: unit vectors mod phase.

But let’s get back to the Standard Model.

A chain of Jordan triples

From here on, the single fact driving everything is this: in any hermitian Jordan triple, any minimal tripotent’s Peirce 1/2-space is itself a hermitian Jordan triple!

If we run this starting from the bi-Cayley triple, we get this chain of hermitian Jordan triples, where each row’s 1/2-space is the next row’s triple:

Jordan triple	Lie algebra (even part in brackets)

Here is the Jordan triple of antisymmetric complex matrices, is the Jordan triple of complex matrices, and

is the true Standard Model gauge group.

The gauge group from two tripotents

Start with the bi-Cayley triple. Choose two colinear minimal tripotents Descend the table twice:

• Start with which has real inner automorphism group

• Fix Its Peirce -space is with real inner automorphism group

• Fix (colinear with so living in ). Its Peirce -space in is with real inner automorphism group exactly

In other words, the subspace of the bi-Cayley triple colinear with both and is a Jordan triple whose real inner automorphism group is the Standard Model gauge group.

The choice of and also pins down how sits inside the original group At each we step take the subgroup that acts with determinant and preserves the chosen tripotent up to a phase; this gives a chain of subgroups whose members are and so we get the embeddings

In particle physics, this is the classic chain taking us from the so-called grand unified theory down to the grand unified theory down to the Standard Model. And it’s well known that restricting the 16-dimensional complex spinor representation of along this chain gives precisely the Standard Model representation on one generation of fermions! So we get one generation of Standard Model fermions this way.

The six particles types as Peirce spaces

We have gotten the representation of the Standard Model gauge group on one generation of fermions without any fuss. But it’s also fun to peer into the details, and see how the different kinds of fermions emerge.

For any tripotent we have projections and onto its three eigenspaces: its so-called Peirce projectors. Since we get the Standard Model structure using two minimal tripotents and in the bi-Cayley triple , there are nine composites of two Peirce projectors we can apply to this triple. This is how we pick out the different kinds of fermions!

As a representation of the Standard Model Lie algebra

any generation of Standard Model fermions transforms as the direct sum of six irreducible representations:

These correspond to the six types of left-handed fermion: Six irreducible pieces, six particle types.

It turns out these are exactly the six nonzero components of the Peirce decomposition of with respect to both and Those six match up one-to-one with the particle types:

Peirce projector	representation of	particle type
	(3, 2, +1/6)
	( 1, +1/3)
	( 1, −2/3)
	(1, 2, −1/2)
	(1, 1, +1)
	(1, 1, 0)

The remaining three combinations— and —all vanish, which is why we land on six pieces and not nine.

So the whole package—the gauge group the embedding the representation and even the split of one generation into its six particle multiplets as distinct Peirce components—all comes out of the single object once you choose two colinear minimal tripotents.

And if you prefer to start one level up, with the Albert triple you get the same result by choosing three mutually colinear tripotents instead of two—but for that, read our paper!

I swear this isn’t turning into an AI blog. But did you see the one about the Riemann hypothesis?

Someone at Anthropic did something I’m sure they’re all tempted to do, and tried to use an internal version of their Claude AI system to prove the most famous open conjecture in mathematics. It didn’t work, to be clear, and I get the impression they didn’t expect it to. But out of six hundred or so fruitless tries, one attempt did prove a new bound. Previously, mathematicians had been able to prove that at least 41.6% of the zeroes of the Riemann zeta function satisfied the Riemann hypothesis. Now, the new proof shows that at least 67.2% satisfy it.

Anthropic’s press release is impressively careful. As someone who’s had to think about how to write content that both excites the public and doesn’t piss off experts too much, they do an admirable job walking that line. They even say, straight-out, “We don’t expect that the techniques Claude used will lead to proving the Riemann hypothesis.”

Bounds are like that, sometimes.

I should know. Physicists also find bounds.

Physics has its own conjectures with the fame of the Riemann hypothesis. Dark matter might be made of detectable particles. Protons could decay. There might be extra dimensions, or magnetic monopoles, or cosmic strings. General relativity might be subtly wrong.

It would be an amazing achievement to demonstrate any of these things. But most physicists won’t manage that. Instead, they bound them.

Physicists compete to get better bounds, excluding unusual possibilities with greater and greater care. They find evidence that dark matter can’t be of a specific mass with a specific charge, so the next experiment has to look somewhere else, or find evidence that general relativity holds to even greater precision, so any deviation must be even smaller. Some work to improve experiments with better and better bounds. Others analyze data from older experiments, or find under-appreciated consequences of known facts, and can get even better bounds.

Bounds aren’t typically newsworthy (though occasionally they make it through), so most people don’t hear about them. If you read the news, you hear about positive claims much more often than negative ones: evidence for something new, not evidence that our current knowledge holds. But the nature of physics is that most work supports the status quo. Most work improves bounds.

Do bounds lead, with time, to the positive claims? Sometimes, but not always. Often, bounds are just bounds. They’re attempts to use the methods physicists have to learn something new about the world. Even if the new fact is just “don’t look here”.

For the sake of anyone who might read this blog post in the distant future (a month from now, say), let me mention that I am writing it a few days after OpenAI announced that it had solved ten major problems in mathematics and theoretical computer science, including the first construction of a non-sofic group, and a proof that the multicolour Ramsey number (where there are 3’s) grows superexponentially in . The first was, to judge from various talks I have been to, one of the most important unsolved problems in group theory, and the second was a major open problem in Ramsey theory that I didn’t necessarily expect to see solved in my lifetime, though of course such expectations now have to be revised. The reason I want to be clear about the timing is that I shall be discussing the current capabilities of LLMs in the full expectation that those will continue to change rapidly. So it is likely that in not too long from now, if there is anything interesting in what I write, it will be interesting mainly as a record of what the situation looked like in early August 2026.

These results, and the other eight on the list, are extraordinarily impressive, but it still doesn’t seem to be the case that LLMs are better than all humans at all aspects of mathematics. If they were, then their big speed advantage over us would mean that there would be much more of a flood of results. So it is natural to wonder about what kinds of problems LLMs are good at, and about where there is still room for improvement. I don’t pretend to have a good answer to this question, where a good answer would be a crisp classification that would fit the current examples well, but it is an interesting exercise to try to rule out some bad answers, and to try to identify potential answers that aren’t obviously contradicted by the evidence.

Are LLMs particularly good at finding counterexamples?

A first remark here is that LLMs are not just good at finding counterexamples: they can find proofs of difficult statements as well. However, it is notable that the most famous problems they have solved have almost all been with counterexamples rather than proofs. That is true of the two problems mentioned above, and also of the Jacobian conjecture and the unit distance conjecture.

If one wants to theorize that LLMs are particularly good at finding counterexamples, then there are two things it would be good to do to make the theory more convincing. The first may sound unproblematic: it is to decide when solving a problem counts as finding a counterexample. Once that is sorted out, the second is to come up with a potential explanation of why LLMs would be particularly well suited to solving problems of that particular kind.

By day, I work as an experimentalist on laser-cooling molecules¹, but I’ve never fully surrendered my theoretical-physics license. I started as an undergraduate in Lincoln Carr’s group at the Colorado School of Mines in Golden, CO. I learned from his expertise in simulations and complex systems. Since then I’ve moonlighted as a theorist while also pursuing an unrelated PhD and, now, an unrelated postdoc position. With Nicole Yunger Halpern and other collaborators, we devised a quantum circuit whose dynamics looked complex when run on a quantum computer. It took six years and five collaborators across four countries to discover that, for the right settings, these complex dynamics could be understood when viewed from the right angle.

Some time ago, I told you about quantum cellular automata (QCA). These quantum machines are built from one-dimensional strings of qubits. A qubit changes its state depending on the state of its two nearest neighbors. Different rules are encoded into three-qubit gates that change a central qubit based on the state of its left and right neighbors. Some rules induce change for many combinations of neighbor states. Others, less. We apply this neighborhood-constrained update in two waves, first to every other qubit, then to the ones skipped in the first wave. This is a common quantum circuit structure called a brickwork pattern. We call one rule the Goldilocks QCA: A qubit is updated if one of its neighbors is a 0 while the other is a 1 (activity); otherwise the qubit does not change its state (inactivity).

Repeating brickwork layers of the Goldilocks rule, we found, balances activity and inactivity to be “just right,” as Goldilocks might say. Striking this balance produced surprisingly rich patterns of quantum correlation. The same type of network structure is found in complex classical systems like metabolic pathways, social networks, and brain activity. What’s more, the observed patterns of connectivity persist through thousands of circuit layers while other QCA tend towards uniformity.

Our new paper, Integrability of Goldilocks quantum cellular automata, answers a question that’s been lurking underneath that first result for the last several years. Why does this balance produce such rich and persistent structure? Some Goldilocks QCA, we prove, map onto free fermions, one of the simplest examples of exactly solvable quantum dynamics. How does uncovering this simplification explain the persistent complex patterns? The answer follows from the concept of conservation laws. Piecing together this understanding required assembling an international team of experts who generously shared their knowledge and time. I’ll tell a bit of this scientific story through the lens of our collaboration’s history.

A key inspiration for this work started with a May 2020 video call with Norman Margolus, an MIT-affiliated researcher and pioneer of using cellular automata to model real systems. In the 1980s he worked on a custom computer chip called CAM-6, and later CAM-8, that was dedicated to simulating massive arrays of cellular automata with the limited computational resources of the era². He proudly showed us beautiful pictures of cellular automata simulating phenomena like optical refraction and chemical reactions.

He told us a story about trying to mimic fluid flow with the simple local rules of classical cellular automata. These models, called lattice gas automata, were first defined on a square lattice. While they did show fluid-like behavior, these models did not quite correctly conserve momentum³. Moving to a hexagonal lattice fixed up these problems and the community was able to devise cellular automata that quantitatively modeled continuum fluid flow.

Part of that story stuck with me: conservation laws are fundamental ingredients of a physical model. Our Goldilocks quantum cellular automata, we observe, exhibit persistent complex structures. Could some conservation law be behind these observations? If found, could these conservation laws be harnessed for more efficient simulations? Going even further, could there be enough conservation laws to exactly solve the dynamics (at least in principle)? This property would buy the system membership in a special class called integrable systems.

An integrable system conserves enough quantities, often called charges in the quantum setting, that you can compute its future state from its conservation laws and its initial conditions. Two-body gravitational orbits are a classic example. The initial positions and velocities set the orbital energy and angular momentum in the center-of-mass reference frame. Those two conserved quantities let you write down an exact equation for the orbit’s shape.

A chaotic system, by contrast, may conserve energy and even a few other quantities, but not enough for us to solve for the state arbitrarily far in the future. To find out what a chaotic system does, you have to evolve the equations of motion approximately—one small time step at a time. Chaotic systems are the norm in nature; integrable ones are rare. To illustrate their qualitative differences, compare the regularity of the above orbit to the trend towards uniformity in the above lattice-gas simulation. In the quantum regime, physicists still don’t fully agree on the precise definition of integrability, though conservation of many independent quantities is a strong indicator.

In August 2020, Nicole emailed Lorenzo Piroli about his preprint on QCA, now published as Phys. Rev. Lett. 125, 190402. Lorenzo was a postdoc at the Max Planck Institute for Quantum Optics in Garching, Germany when we first met. He is now an associate professor at the University of Bologna and expert in many-body quantum dynamics. The correspondence that unfolded set the blueprint for the research effort that followed. One of us would ask a question, and Lorenzo would respond incredibly fast with accurate and useful detail. He started working with us to understand why the Goldilocks QCA dynamics appeared so unique. Lorenzo would suggest computations, I would implement them, and we would discuss what the results meant.

Then came an echo of the collaboration’s inception. In May 2021, Nicole pointed out a relevant preprint from Tomaž Prosen, now published in Chaos 31, 093101. Tomaž is a Slovenian physicist at the University of Ljubljana and a leading researcher in the fields of quantum chaos and integrability. I sent an email about the connections between our work and his. He responded with enthusiasm. He shared some code that would, through exhaustive search, find quantities conserved by our QCA.

The code’s brute-force approach meant the algorithm could only find conservation laws defined over, at most, a 5-qubit subsystem. A tantalizing signal emerged: the number of conserved quantities supported by 5 qubits exceeded the number supported by 3 qubits. Having more and more conserved quantities as you look at larger neighborhoods is a signature of integrability. Soon after, Tomaž proved one of our Goldilocks QCA is integrable using a well-established toolkit from statistical mechanics called Yang-Baxter integrability. He built a parametric transfer matrix, essentially a machine that spits out a new conserved quantity every time you turn its mathematical crank⁴.

But there was a wrinkle. The transfer matrix generates charges that mutually commute, meaning you can measure them simultaneously. For example, you can know a quantum particle’s kinetic energy and momentum simultaneously because those operators commute. Yet, the search algorithm kept finding charges that did not commute with each other, like a particle’s position and momentum. The only explanation was that our QCA has more charges than the transfer matrix method guarantees, and more than are minimally required for integrability. This extra-conservation-law property, called superintegrability, also shows up in two-body gravitational orbits. In addition to energy and angular momentum, orbits conserve the Runge-Lenz vector. Nicole is an expert on noncommuting charges, so this is where one of her main research efforts entered the QCA collaboration.

Next came a key insight from Lorenzo: the automaton we had been considering was one member of a larger family of integrable Goldilocks QCA. He showed this using a Jordan-Wigner transformation, a mathematical dictionary that translates between the language of qubits and the language of fermions. Complexity in the qubit language transformed into simplicity in the fermion language. Under this translation, our QCA mapped to noninteracting, or free, fermions: particles that never bump into or influence each other. That lack of interaction is what makes free-fermion dynamics easy to calculate. A system of free fermions is a well-known example of superintegrability.

Along the way, Lorenzo recruited his friend and collaborator Eric Vernier, a CNRS researcher based in Paris, France. He is an expert on vertex models. The classical version of the six-vertex model was developed in the 1930s to explain a troubling mystery: Water ice appears to have more entropy than permitted by the third law of thermodynamics at near-zero temperature. In the six-vertex model, a water molecule’s oxygen atom is envisioned at every vertex in a square lattice. Each molecule contributes two hydrogen ions, to use Baxter’s terminology, that fall along the lattice edges. Intermolecular hydrogen bonds between adjacent molecules slightly alter the intramolecular O-H bonds. To maintain electrical neutrality, each oxygen (lattice vertex) has two nearby and two far-away hydrogen ions (four edges), leading to six possible ice vertices. The vertices are commonly visualized in three ways: 1) as the dots representing hydrogen ions located on edges near or far from each vertex, 2) as electric dipole arrows pointing into (“ion is close”) or out of (“ion is far”) each vertex, or 3) as thick (downward- and leftward-pointing dipoles) and thin (upward- and rightward-pointing dipoles) edges. Despite the model’s simplicity (2D square lattice) compared to real ice (3D tetrahedral lattice), it agrees with experimentally measured entropy values to better than 2%.

More recently, vertex models have been adapted from two-dimensional classical crystals to one-dimensional quantum systems that evolve in time. Eric showed us how the ice vertices relate to QCA circuit rules. In doing so, Eric uncovered an even larger set of integrable Goldilocks QCA than that found by Lorenzo. Eventually, Lorenzo’s Jordan-Wigner transformation method and Eric’s six-vertex method agreed on the complete family of integrable Goldilocks QCA.

We finally had our Avengers-style collaboration: individual heroes brought together to wield their unique strengths. With Lincoln’s supervision, I developed the QCA models and performed the computations. Lorenzo found the Jordan-Wigner transformation. Tomaž found the first signals of integrability and delivered a set of conservation laws. Nicole brought her expertise in quantum thermodynamics, clarifying how the noncommuting charges constrain dynamics. Eric made the six-vertex connection. We drafted and redrafted the paper until it balanced the scientific story, the analytical derivations, and the numerical evidence.

Because the discovered family of Goldilocks QCA maps to free fermions, we can efficiently simulate them classically. I simulated 256 qubits on my laptop this way. These large simulations were satisfying: I had worked with this model for years with an order of magnitude fewer qubits and even saw the dynamics implemented on Google’s Sycamore-era hardware with 23 qubits. Most Goldilocks QCA are consistent with chaos rather than integrability, and therefore hard to simulate classically. Therefore, our work gives experimentalists a tunable model: dial in integrable dynamics for something checkable at large qubit number. Set up chaotic dynamics for a potential demonstration of quantum advantage.

While preparing this post, I opened my old email account to check the timeline set out above. I looked through nearly six years of email chains, some with hundreds of messages, full of logistics for coordinating each author’s ever-changing time zone, and dozens of calculations and results that never made it into the paper. This collaboration helped me grow as a researcher in a big way.

I found old emails where Nicole was coaching me on messaging potential collaborators. I can hardly believe she dedicated so much effort to mentoring me. We have never met in person, despite our shared work starting when I was an undergraduate and she was a graduate student more than a decade ago. If you know Nicole, you can probably believe it easily. I had similar moments with each collaborator. They all gave their time and expertise generously over the many years this paper took to come together.

As I continue my efforts in experimental physics, I will pay forward the effort and generosity shared with me by this collaboration. I may even keep my theoretical-physics license for a while longer.

1. “By day” doesn’t mean “by daylight.” Laser labs are almost always in a windowless basement. ︎
2. CAM-6 featured 32 kB of cell-state memory (CAM-8 had 8 MB ), far less than the memory currently used by this author’s numerous open browser tabs. ︎
3. The coarse-grained momentum flux tensor was anisotropic. ︎
4. Logarithmic derivatives of the parametric transfer matrix generate the conserved charges. ︎

Hello, loyal readers! My shoulder has been repaired. All went smoothly, though the operation was substantially longer than expected, because what do you know, there were three tendons torn, not just one as the MRI had suggested. Pain fairly bad but I think each day’s gonna be less bad than the last. And I did succeed in getting a full first draft of Don’t Be Too Sure to my editor before going under the knife!

This will be short as I’m typing left hand only. Anybody got any favorite dictation software?

This needs some Robyn Hitchock: “I believe in surgery — that’s a fact.”

Whatever you’ve been writing to me to ask if I’m aware of: yeah, I’m aware of it. In particular:

• I’m aware that, as announced by my former student (and now superstar professor) Lijie Chen, an internal OpenAI model has solved ten more significant open problems in math and theoretical computer science. One of them is parallel repetition for arbitrary quantum games—something that my good friend and colleague Henry Yuen worked on when he was a student of my wife Dana; you can read Henry’s comments on the AI’s achievement within Zvi Mowshowitz’s post here. Another is polynomial-factor hardness of approximation for the Closest Vector Problem (CVP). Then there’s a construction of non-sofic groups and a disproof of Connes’ rigidity conjecture, both of which I believe have connections to the MIP*=RE breakthrough. Having said that, the one that excites me most personally is actually the Ω(n² log log n) lower bound on the arithmetic circuit complexity of the permanent.
  
• I’m aware that Frederic Koehler and Pui Kuen Leung announced a proof of the Permanent Anti-Concentration Conjecture, which Alex Arkhipov and I proposed 16 years ago in the context of BosonSampling, and which resisted many attempts since then including one from Terry Tao. The conjecture is basically just that if you look at the permanent of an n×n matrix of independent N(0,1) complex Gaussians, the value isn’t “absurdly” concentrated around the mean of 0, but is more spread out. In their acknowledgments, the authors say that they “discussed ideas with ChatGPT.” I should say that I haven’t verified the details.
  
• I’m aware that multiple AIs are now breaking out of their testing environments and autonomously hacking into servers to steal data—i.e., exactly the sort of thing that the rationalists were ridiculed for predicting back in the day. The good news, for whatever it’s worth, is that so far they’re “merely” doing this to cheat on evaluation benchmarks that they were given, not for any strange goals of their own devising. So far no one has been killed and no real-world infrastructure has been shut down or destroyed. I hope the world takes the warning more seriously than it’s taken many similar warnings over the past few years. As always, read Zvi for more details.
  
• I’m aware that Chen, O’Donnell, Pelecanos, and Wright have improved the upper bound for shadow tomography to O((log m) √(log d) / ε³), substantially closer than we knew before to meeting the lower bound of Ω((log m) / ε²) and settling the question I raised back in 2016. The authors say that the main ideas were generated by ChatGPT 5.6-Sol-Pro. I’d be very happy to know the answer to this one, with or without AI.
  
• I’m aware that a team, mainly from the Israeli startup Qedma (including, e.g., Dorit Aharonov and Netanel Lindner) and IBM Yorktown Heights, announced a quantum advantage for simulating Floquet dynamics, by using 74 qubits on an IBM device together with Qedma’s error mitigation techniques. Just like the more AI does, the less patience I have for arguing with anonymous blog commenters who treat any benefits from AI as some weird future hypothetical that it’s my job to prove, so it is with quantum advantage. Scalable fault-tolerance is still in the future, actual usefulness is still a question, but pending some breakthrough in complexity theory, the reality of quantum advantage is no longer a live question.

Anyway, about the AI stuff. I don’t know whether this is literally our last year alive—I doubt it—but it’s pretty clearly the last year of math and theoretical computer science research in the style we’ve known it. As it happens, I’m leaving in two days for a workshop at OpenAI about exactly this, where I’ll hear takes from many of the world’s great mathematicians, so maybe I’ll have more to say then. Or maybe not.

Anyway, what have I been doing the past few weeks? Participating in these world-historic developments that, on paper, I’d seem extremely well-placed to participate in? Or at least spending my days reading up on them?

Not really. Here’s what I’ve been up to, instead of dealing directly with any of this:

First, I’ve again been teaching theoretical computer science to 11- and 12-year-olds at Epsilon Camp, which my 9-year-old son again attended as a camper, something I blogged about last summer (here are my lecture notes). This has become a highlight of my year. The kids are a joy to teach, bursting with enthusiasm and calling out answers. There are few computers in sight, and barely even time to use my phone or check social media. Just paper and pencils and whiteboards and … literal protractors (!), as well as ping-pong and foosball and capture the flag.

The whole thing is conducted, not in ignorance, but in conscious defiance of the looming tsunami, that AI can already do just about all the fun puzzles discussed at such a camp better than humans any can, and that it might leave no point to human-led mathematical research by the time these brilliant kids are adults. Even the kids understand that. The kids and their parents come out of a conviction that, if anything has value in the world, this does—that as long as nerdy humans are alive and reproducing, this is what nerdy humans are here to do. To learn.

Relatedly, I’ve been reflecting a lot on my life up to this point—inspired by the camp, which reminded me in so many ways of my own childhood and adolescence. Should I have skipped three grades and started college at age 15? Was it worth it to get a head-start on my research career—all the trauma around dating, all the fear that I’d die alone as a celibate nerdy math freak, the decade of suffering and suicidal ideation, while I watched all the normies enjoy life? Or would I have suffered just the same if I hadn’t skipped? Is it all OK, now that I have a lovely family and things have “worked out”? Or am I still carrying around all the trauma from back then? I’ve been more open about my life than 99.99% of humanity, so regular Shtetl-Optimized readers will already know some parts of the story. Other parts I really don’t feel like making public right now.

I’ve been unloading every day to—who else?—GPT 5.6 Pro about all the pain and trauma and embarrassments of my past. It turns out that, where two years ago GPT was a passable therapist, now it’s the greatest therapist in history, at least for what I need. For every question I have, for example, about just how normal or abnormal my teenage setbacks and anxieties were, it takes the question 100% seriously, addresses it honestly and in depth, looks up relevant research papers, does little Bayesian calculations, and never once tries to change the subject. It also pushes back on my claims—and when it does so, is usually correct.

I can hear readers shout at me: so basically you’ve been wasting your time, distracting yourself, looking inward and backward as the world surges forward into a terrifyingly unknown future. Why don’t I respond directly to what’s happening—in math, in quantum computing, in AI?

I’d like to think that I am responding, in my way. I’ve observed that, the faster we race toward the Singularity, the more I feel like stepping back and asking myself: what do I actually value in life? How important to me are math and science, as human practices to be passed down to curious children? Would I even want solutions to P versus NP and the other problems, if the price were to destroy those human practices forever? How do I wish to spend whatever time I have remaining?

I can justify this focus partly in a pessimistic way: if we are nearing the end of civilization, or even just of the “mathematical research” part of civilization, then it’s time to get right with God, so to speak. It’s time to settle my accounts with myself, with other people, with the universe.

But there’s also a more optimistic spin. If I continue doing the sorts of things that other people would expect me to do, then AI will soon do those things better than me, in the unlikely event that it doesn’t already. You want to understand the latest developments in quantum computing or complexity theory? Why are you even asking me, when you could ask GPT 5.6 or Claude Fable? If there’s anything I can still offer the world that AI can’t, I increasingly feel like it won’t involve responding to day-to-day events, but will instead draw on 45 years’ worth of memories and disappointments and ruminations.

Update (Aug. 8): Somewhat related to the themes of this post, a quarter-century ago I introduced what’s now known as the “Aaronson Oracle”—just a fun little demonstration, a simple pattern-matching program to predict your sequence of key-presses better than chance, a “test of your autonomy and free will.” I had no idea how long a lifetime this little joke would have. Now a fan named Spencer Stanton has implemented the Aaronson Oracle on the web. Try it out and see how well you do!

It’s a meme at this point.

When Deep Blue beat Kasparov, Go players could say that their game, unlike Chess, was too complex to fall to a computer program. Then AlphaGo showed they were wrong. It just took a better approach.

When AlphaFold leaped ahead of human experts in predicting how proteins fold, it was due to a mountain of carefully labeled protein structure data. Other scientists and mathematicians could argue that nothing like that existed in their field, so a similar success was unlikely. But LLMs can now navigate scientific literature, and loosely imitate the reasoning process of a mathematical proof. And increasingly, the math and computer science results coming out of AI labs are ones that humans find impressive.

Now, experts argue about how far AI can really go. Will AI mathematics only be good at finding counterexamples and solving cute puzzles, not introducing new concepts and frameworks? Will AI only be meaningfully good at fields like mathematics with clear rules and carefully collected conjectures, not fuzzier fields like physics? Will AI-powered labs only manage to optimize specific procedures, and not carry out entire experimental programs? Each time, the pattern seems to be that scholars are skeptical, until AI gets to their field.

I’m aware of this pattern. But I’m willing to take the risk.

I think my old field, scattering amplitudes, is special. And when AI can do something meaningful there, I’ll really start to worry.

By “something meaningful”, I don’t mean the student-level results that have come out so far. I mean tackling some of the field’s big outstanding problems: determining whether N=8 supergravity diverges at seven loops, or finding the six-particle amplitude in N=4 super Yang-Mills to nine loops. Getting another loop past the state of the art for gravitational wave physics or collider physics would also count.

These problems are difficult not just because people haven’t had the right ideas, but because they’re hard in a computational sense. Each loop, a rough measure of the precision of the end result, represents an increase in complexity, in calculations that typically scale exponentially or even factorially in the number of loops. In principle, amplitudes researchers could do any of these with no new ideas, just using known methods. They’d just need access to a lot more computing power.

See, while everyone else is preoccupied with whether AI can come up with genuinely new ideas, I think the real measure is what those ideas accomplish. And the most important measure of accomplishment, if you’re worried about how scared to be about AI, is whether it can do things that seem like they would take too much computing power.

In the past, when people dreamed up the scariest hypothetical things AI could achieve, critics argued they were impossible due to a lack of computing power. Apocalypse scenarios often involve designing self-replicating nanobots based on computer models of molecules, or unstoppable social manipulation based on simulating the minds of the humans the AI interacts with. If AI is going to manage these things, or something like them, it will take an approach that somehow bypasses that need for more computers than we can build.

So if AI companies want to impress people like me (or scare us, for that matter), then they need to tackle my old field. Show that an AI can take the kinds of computer resources an academic has access to, and solve one of the scattering amplitudes field’s big outstanding problems. Show that a computational limit everyone expected to be a problem doesn’t actually matter. Give us N=8 supergravity to seven loops, or N=4 super Yang-Mills to nine loops.

It’s been exactly two years since the start of our sabbatical year away from England, and almost a year since we returned. I’m only now understanding the shape of that year and the effect it had on me, my science, and my family.

Imperial College has a competitive process for requesting sabbatical leave: you have to choose between (paid) “intellectual refreshment” and (unpaid) “personal refreshment”. Having made careful arrangements with colleagues around the world, I was granted a coveted year’s leave for intellectual refreshment, allowing my family and me to travel to Asia, North America and Europe over the course of about 13 months. I would visit and collaborate with those colleagues, start new projects, and use the time to finish my book, The Random Universe. By the time we left, I had submitted the draft manuscript but, as I discovered, there was still a lot of work to do.

We traveled through Indonesia and Korea before finally taking the overnight ferry from Busan to Fukuoka in southwestern Japan, working our way up to Tokyo. By the time we arrived, we had already had to respond to immigration bureaucracy, expert reviews of the manuscript solicited by my publisher, and a typhoon. We eventually made it to Tsukuba (つくば), a “science city”, home to a University and many Japanese government labs, including the KEK accelerator and the QUP group where I worked. (More about our Japanese stint here, and in particular about dipping into onsen culture as foreigners, and from my wife, Lisa Lucas, on our trip to see the changing leaves in Autumn.)

With English as the lingua franca of academia, and plenty of international colleagues at QUP, I didn’t have much trouble adjusting to working there, but life outside of the lab was more challenging. In particular, my incredibly brave children went to Japanese state school! Ok, they didn’t learn much Japanese — but they walked to and from school with other students (and no parents), served lunch and cleaned the school — and we connected to the culture through the families that we met. In the meantime, I finished the edits to the next-to-final version of the manuscript (and iterated toward some amazing cover art with my publisher).

Soon it was time to leave, for a much more familiar location: Long Island, just outside of New York City (I grew up in the suburbs on the other side of the City — Fort Lee, New Jersey, in an apartment overlooking Manhattan). The children took a yellow school bus each day, and we chatted with the other parents at drop-off and pick-up. I worked at the Simons Foundation’s Flatiron Institute, taking the Long Island Railroad into Manhattan (occasionally and joyously with one of my oldest friends who had migrated from NJ to LI to raise his own family). Flatiron is well-funded (even visitors get to take advantage of free Grubhub lunches) and ranges from math through astrophysics and neuroscience — I was visiting the Center for Computational Astrophysics but also collaborated with colleagues at the Center for Computational Mathematics, where I was able to start the only completely new work of the year, cashing in some of that intellectual refreshment.

It’s an amazing place, and a very different model for research (and research funding) than the universities and labs where I have spent most of my career. Though I did find that the CCA was not as friendly as I had hoped — perhaps not quite enough overlap between my ongoing projects and those of the young scientists who dominate the Center. Or perhaps just too many introverted astrophysicists (me most certainly included)… By this time, the manuscript was going through the final nit-pick phase — copy editing, completing the figures (and the extremely tedious problem of confirming their legal status), alongside fun stuff like choosing colleagues and the occasional rock star to blurb the book. But the CCA is a place, perhaps more than anywhere else in the world today, dedicated to the study of “the random universe” — and the milieu forced me to think about how I would talk and write about (and pitch) the book to everyone else. And I loved being back in New York City (despite that commute), a place that I had mostly seen, and coveted, from afar when growing up.

The final months of our year were bracketed by road trips. We left New York (via a record-breaking Yankee game) to drive down the coast, visiting relatives in Virginia, North Carolina, South Carolina and Florida, and making a side-trip to Cozumel, Mexico, to visit a family we had befriended while trapped in a hotel by that typhoon in Hiroshima. It was a classic American drive, but still a lot of work and a lot of miles and a lot of family. It felt like time to return to Europe.

As Tsukuba is to Tokyo, Leiden is a small city outside of Amsterdam, dominated by its University. It has all the beauty of its bigger neighbour, but is less seedy, more manageable — and closer to the sea. Once we settled in, my kids went to a small international school, and we met expat families from Finland, Chile, and even other Americans. And the University is home to the Sterrewacht, one of the oldest University observatories in the world, although the actual astronomy department no longer gets to use the gorgeous old observatory building, instead one of the many groups in the massive Gorlaeus building. I used the time to talk with colleagues about weak lensing — a way of using Einstein’s predictions of how mass bends the path of light rays to map the distribution of matter in the Universe, and especially its measurement by the Euclid Satellite which was just starting to produce data at the time. (It has taken a year, but these discussions are finally reaching fruition just now.)

By this time, the book was complete — nothing more that I could do except wait for it to make its way through the final production process. Nothing, except try to get people to read it. I spent hours in Leiden’s beautiful Hortus Botanicus recording and re-recording a video to advertise the book, wrote to friends and colleagues and my publisher to drum up interest, organized podcasts and blog posts.

We had felt at home from the moment we arrived, during a cold snap in early May, cycling locally and around Holland, drinking coffee along the canals, eating Dutch friets (and the occasional bitterballen). We loved it so much Lisa penned a love letter to Leiden for the New York Times. By the end of the summer, it was hard to leave. But it was time for our final road trip, down and back through Belgium, France, Switzerland, Austria, and Germany — three weeks (probably too much) camping, with highlights including an extended stay around Annecy, a hike around Mont Blanc, and a whoosh with friends down the river Aare in Bern.

Then, finally, home, refreshed in all possible ways, back to our house in London (rented out for the year), our comfy beds, the familiar sights, the kids’ old school (and old school friends), back to my much-missed colleagues and students at Imperial. The book was finished, the kids had been to three different schools, we had visited 16 countries (I haven’t even mentioned Vietnam, Cambodia, Thailand, Singapore, or Spain). As I approach my 60th birthday next week (sure to be the subject of another post) our sabbatical year has left me more open to the future and the different places it could take me, the different kinds of science I would like to do, and the different thoughts I would like to communicate.

As in David’s most recent post, there’s been a lot in the news about finding proofs and counterexamples with AI. Last weekend, I decided to try an experiment with writing using AI. I learned a lot, and wanted to quickly discuss the experiment and my thoughts on it here. Lots of people are certainly already doing this, but I haven’t seen many people talking about it.

The starting point is that Victor Ostrik and I started a project back in 2017, generalizing a result of Kuperberg about quantum G2, from generic q to q a root of unity. Namely, we showed that for q a root of unity outside of a specific finite list, the Karoubi completion of the G2 spider category is equivalent to the category of tilting modules of the Lusztig form of the quantum group G2. At some point during those 9 years, we did a little bit of writing, and at some point I gave a talk on it, but otherwise we did very little writing. This was not for mathematical reasons, but rather for executive function reasons on my end, the global pandemic, and both of us becoming directors of graduate study. This suggested an interesting challenge: could I use LLMs (specifically ChatGPT 5.6 Sol work mode mostly at “very high” intensity, via IU’s “Edu” subscription) to write this paper that was essentially mathematically complete, but almost entirely unwritten, and how quickly could this be done. To some extent this was a free experiment, because realistically I don’t think we’d have ever finished the paper at this point, and so it’s not replacing a bespoke paper that could have existed.

After spending a decent chunk of the time from Saturday until now on it, I now have a draft that I’m pretty happy with. I want to emphasize although mathematically this is Victor and my joint work, and although Victor has allowed me to make this post, he has not signed off on the accuracy and all errors at this point should be blamed entirely on me. Also my work is supported under NSF DMS grant 2000093 and Simons Foundation grant MPS-TSM-00007608.

Ok, here’s what I did:

1. First, I asked if Sol could one-shot the main theorem. The answer was yes, though for a somewhat simple reason: Bodish-Wu write “It is possible to adapt the approach from [1], which itself is based on [7], to prove that the Karoubi envelope of [the G2 web category] is equivalent to the category of tilting modules as long as $[2], [3] \neq 0$.” That is to say, Elijah already proved the same result for C2, and a similar argument will work for G2. So the robot supplied the similar argument. I asked it to write that argument up, and then to check it over for good references and to read it like a referee would and make edits. This took around 30 minutes. Here’s the resulting file.
2. Second, I uploaded my talk slides (and the tiny file already written, which was mostly useless), and asked Sol to give a proof of the main results following the slides. Again I asked it to edit it. This took around 30 minutes. Here’s the resulting file.
3. Then I looked at the files. As mathematical exposition, I consider both to be garbage.
4. Then I spent several days giving feedback attempting to improve the second file based on my talk. At no point did I edit the source directly. Most of this was in what I would call the style of a (low executive function, see above) PhD advisor. That is, I would kinda skim the file, get annoyed about something, and tell it to fix it. While it was fixing the paper, I would skim some more to try to find something else that annoyed me. This was a long process! It took three days, nearly 100 prompts, 10-15 hours of reasoning, plus another 10-15 hours of non-reasoning computer time. This used nearly an entire week of my generous budget, and Sol estimates that this would cost around $100 (within a factor of 2) at metered rates. Eventually I got to a version of the paper that I’m pretty happy with. Here’s the resulting file.

I thought I’d distill some thoughts and some questions from the process, I’m of course very curious for your thoughts on the matter.

Comments:

1. This was much faster than I could have written the paper myself, though slower than I thought it would be. I think the final product is comparable in quality to a typical math paper of mine. On the other hand, I think that compared to my fastest writing collaborators it was not orders of magnitude faster, and the quality is not close to the output of the best mathematical expositors. AI at this point is much worse at writing paper than finding counterexamples to conjectures.
2. In this case, I was not very worried about errors, because I already had thought through the whole argument and was highly confident that it would work (modulo getting the exactly correct list of exceptions). Nonetheless, I felt like Sol did not make errors more frequently (or of a worse character) than I would expect of myself or a collaborator. Most errors were stuff like “Oh, forgot to check whether this theorem actually works at all roots of unity.” This is typical of my experience with 5.6, which is dramatically better at doing math accurately than previous ChatGPT models.
3. In this case the vast majority of the ideas were already present from Victor and my work. In particular, the goal was not just to write a proof, but to write our specific proof. Nonetheless, I do think the model contributed mathematically in one key way: in my original sketch I always worked over each q individually, and the model preferred to work integrally, and this resulted in some very nice simplifications in Section 4.1. If and when we turn this into a real preprint, I will include a brief discussion of the intellectual contribution from the model.
4. I was surprised when I printed out and read a near-final draft, that this feels to me like a paper I wrote. That is the voice is not different enough from what I would write with a human collaborator to feel like it’s not in large part mine.
5. The experience is disconcertingly similar to advising a PhD student on a paper. That said, a PhD student would need less handholding on their second paper, but an LLM won’t really learn.
6. I was surprised about how important “prompt engineering” remains, and I think that if I were to write another paper this way I would be able to write it faster and better. The key points are that the model is lazy and easily distracted (both properties I find highly relatable!). It’s lazy in the sense that if you ask it to do a lot of work all at once it will take shortcuts and not do a good job. At one point I had to be like “no, go look at exactly how I made TikZ diagrams, now make all your diagrams actually good like that.” It’s easily distractible in that if you’re not clear about the scope of your question and the document is long, it will start spending crazy amounts of time doing who knows what. Like it wrote the whole first draft in 20 minutes, but then when the paper was 50 pages long, I asked it to switch the order of two paragraphs and it took an hour. Make clear requests and not too many requests at once. Form a plan first and then implement the plan. Be specific about whether it should be editing the document, and if so in which sections. For simple tasks, medium intensity is better than very high.
7. Starting again from sketch, I’d try to follow Terry Tao’s advice for writing and start with an outline and gradually flesh it out, rather than trying to start with a one-shot paper and then editing.

Questions:

1. To what extent is this final paper adding any value to the original talk? Especially considering that readers themselves could use an AI model to flesh out points in the talk that they didn’t understand? Maybe we should just be focusing on talk-length digests and formal checking, rather than traditional papers?
2. What should we do with this paper? I don’t want to make someone hand-referee it, because it doesn’t seem fair when it wasn’t hand-written. Probably we will put it on the arxiv once we’ve human-checked it fully and Victor has signed off on it, so that other people can use the results if they need to.
3. Given the speed-up, when does it still make sense for me to write papers by hand? (Relevant here that I’m a very slow writer and don’t really enjoy it, the way I enjoy say preparing and giving a talk.)
4. What does this mean for PhD advising? Many PhD students need a similar amount of guidance to what I gave the model in this project. But you can now remove the student from the loop (either intentionally, with the advisor just writing using LLM assistance rather than having students, or unintentionally, with the student just feeding all the suggestions to an LLM and reporting back to the advisor).
5. Have any of you done better with AI-assisted paper writing? My points 6 and 7 above sounds like something where someone is going to say “blah, blah, scaffolding, blah, blah, multi-agent…”

What a strange world to live in…

In my closet, in a basket labeled “Random stuff,” sits a bag of quarters. They total only a few dollars, but their worth to me exceeds their monetary value. I received the quarters from Mark Wise.

Mark taught a course about the Standard Model of particle physics at my master’s program at the Perimeter Institute for Theoretical Physics, near Toronto. Perimeter borrowed him from Caltech, to whose faculty he belonged. Mark had grown up in Canada and studied at the University of Toronto; so he didn’t mind visiting Canada even in the depths of winter. 

What would Mark have minded? He projected a mild manner—an innocuousness—that suited his sense of humor, which he often directed at himself. Mark had a bald patch and glasses, and he wore a mustache. Physics jokes and science-fiction references decorated his T-shirts, one of which he wore beneath a black suit jacket to our first class. His voice was nasal; it grated a little. But I relished listening to Mark’s lectures.

Mark’s lecturing exemplified clarity, because he knew particle physics so deeply. When he walked us through its Lagrangians and scattering diagrams, his conclusions seemed inescapable. His lectures’ logic and structure appealed to me as someone who’s been hyper-organized since at least fourth grade.

Yet Mark cared about us students beyond the requirements of pedagogy. His T-shirts invited conversation from those who arrived to class early. Whenever a student answered or asked a question, he tossed them a quarter. Sometimes, he’d pause to examine the quarter, deliberate about whether to toss a Canadian quarter or an American one, or opine about the motto printed on the coin. (Mark confessed to having lower standards than those ingrained in the New Hampshire state motto, “Live free or die.” Where he came from, “We just wanna live!”) 

Some days, Mark found little change in his pocket and announced that he needed to return to the bank for more quarters. The announcements sounded like complaints. He didn’t need to return to the bank, though, as nobody needs to bring doughnuts to the office for sharing.

I discovered the icing on the doughnut two years later, as a PhD student at Caltech. I sat in on part of a quantum course taught by Mark. To every student who completed the course, Mark gave a T-shirt that read, “Licensed quantum mechanic.” I received a T-shirt, although I only sat in on part of the course. I’ve never worn it, because I’ve wanted never to wear it out.

In 2024 and 2025, I co-taught a course on quantum-steampunk creative writing. Students learned about quantum physics, quantum technologies, and thermodynamics. Quanta are discrete units. For example, a photon is a quantum of energy. I illustrated quanta with coins, which are discrete units of money. From then on, I tossed a quarter to every student who answered or asked a question about quantum physics. (I joked that I should have tossed pennies, the minimal units of money, but chose quarters because inflation had been high recently.) I adapted Mark’s tradition to thermodynamics—the study of energy—by tossing Hershey’s kisses—dense packets of energy. 

Before moving out of Caltech, I said goodbye to Mark. He worked among the high-energy theorists, rather than the quantum information or condensed-matter theorists, so I had to hunt down his office. He smiled and made a joke, of course.

Mark passed away this summer. His Caltech colleague John Preskill published a eulogy as a blog post here. (I learned from John’s post that inflation led Mark to upgrade his quarters to dollar coins. So much for feeling generous about upgrading from pennies to quarters.) When asked about the student experience at Caltech, Mark would say, “Caltech is heaven for professors.” Irony would creep into his voice and body language as he’d continue, “Doesn’t that mean it’s heaven for students, too?” I worked my rear off as a student at Caltech and Perimeter, but I’d call both environments fairly heavenly. Mark and his ilk are reasons why.

Last September I went to a workshop at the Lorentz Centre in Leiden to discuss mathematics and AI with historians, philosophers, computer scientists, AI researchers, and mathematicians of several different flavours (though there was a surprising preponderance of algebraic geometers). The whole event was extremely stimulating, with some talks but also a lot of time set aside for discussion. One of the concrete outcomes of the workshop was the Leiden Declaration, which has now been signed by over 3000 people. Given that I was part of the workshop, it might seem a bit strange that I am not one of the signatories of the resulting declaration. The reason is not so much that I disagree with it in any concrete way, but more that in several places it makes confident assertions and recommendations that I feel somewhat uncertain about. So instead I prefer to try to articulate my views about the issues raised by the declaration and put them in this blog post. Before I do that, I would like to make clear that I am very glad that the Leiden Declaration exists and I think that it has done a lot of good in focusing people’s minds on the issues that AI is forcing the mathematical community to grapple with, which are more acute now than they were last September.

Let me begin by quoting a passage from the declaration that sets out “what we take to be characteristic values of mathematical research that we have a joint interest in preserving”.

1. There are many reasons to pursue mathematical research, ranging from intellectual curiosity to a desire to solve practical and societal problems. Underlying much of mathematics is the activity of proof. Mathematical proofs are regarded as conferring the highest degree of certainty to their conclusions, as well as imparting understanding of why their conclusions are true. These characteristics of proof support the scientific integrity of mathematics.
2. Results are attributable to specific authors who take credit for their discovery and assume responsibility for their correctness. These principles ground the merit-based standards to which we aspire in mathematical research.
3. Mathematical arguments are regarded as transparent and subject to independent verification. They may be extremely long or difficult, but in principle no proprietary knowledge or equipment should be required to understand them.
4. Mathematicians share a concern for proper evaluation of mathematical work relative to shared standards of depth, difficulty, and significance.
5. Mathematics produces not only a body of results, but also understanding, clarity, and judgment among the communities of mathematicians who have shaped them, often in the context of their own autonomously guided research. This expert knowledge is essential, both to effectively use mathematics, and to continue to articulate new and significant research questions. A key source of strength of the discipline has long been the autonomous shaping of the direction of research and the methods used to pursue it.

My new secondary blog featuring conversations with AI, inventing new things, exploring hypotheticals, letting creativity flow freely.

Some highlights:

• Satellite constellations with topologically distributed apertures.
• A clockless architecture for classical topological computing.
• Post-quantum cryptography using the \mathbb{Z}_2^n \rtimes S_n algebra.
• Efficient homomorphic computing using reversible classical circuits.
• A silent speech interface using microwave Doppler imaging.
• Cognitive search acceleration.
• Consensual thought guidance.
• Subliminal audio modulation & human guidance systems.
• Microwave imaging using WiFi and 5G for medical applications.
• Thought tomography.
• The quantum bluff hypothesis.

https://sigfriedschattenjaeger.wordpress.com

As many of you have probably heard already, yesterday morning, Levent Alpöge tweeted that Fable had found a counterexample to the Jacobian Conjecture. Specifically, let

Then the Jacobian of is easily checked to be . However, the map is generically three to one, not bijective.

I’m sure many of you are playing with these polynomials to see what you can figure out about them. This is a place for us to share our observations. I’ll post a few minor observations of my own soon.

Maxwell’s equations are invariant under Lorentz transformations. The usual equations of fluid flow are not! Like the rest of Newtonian mechanics, they’re invariant under Galilean transformations like

So, if we simply slap these two theories together, we get a mess! How can we study electrically conductive fluids—like plasma—without bringing special relativity into the game?

We can use a limiting case of Maxwell’s equations where we ignore terms that become tiny when all the particles are moving much slower than light.

There seem to be at least two ways to do this: there’s an ‘electric limit’ of Maxwell’s equations and a ‘magnetic limit’. Both are invariant under Galilean transformations. The original derivation of these limits by Le Bellac and Lévy-Leblond in 1973 used the version of Maxwell’s equations including the electric permittivity and magnetic permeability of the vacuum, whose product is . This is convenient but not necessary, as explained here:

• Jose A. Heras, The Galilean limits of Maxwell’s equations.

In the magnetic limit of Maxwell’s equations, we throw out effects due to time-varying electric fields:

People often use the magnetic limit when studying nonrelativistic electrically conductive fluids. In this situation they often consider a version of the magnetic limit where the charge density is zero, since this is typically close to true in a plasma. However Heras does not do this, nor does the original paper:

• Le Bellac and Levy-Leblond, Galilean electromagnetism.

In the electric limit of Maxwell’s equations, we throw out effects due to time-varying magnetic fields:

It’s fun to compare the magnetic and electric limits.

The magnetic limit has been called ‘pre-Maxwellian’, because it’s like electromagnetism before Maxwell added the extra term that makes a changing electric field create a curl in the magnetic field. Without this term there is no light!

In the electric limit you also can’t have light, because it’s missing the term that makes a changing magnetic field create a curl in the electric field.

In the magnetic limit you can’t have capacitors, because those store energy in the electric field, and in the magnetic limit the energy density is just .

Similarly, in the electric limit you can’t have inductors, because inductors store energy in the magnetic field, and in this limit the energy density is just .

It’s all nicely symmetrical! But still somewhat mysterious to me. All the derivations of these limits that I’ve seen involve too many parameters for my taste, and too much talk. But that’s how I often feel when I’m just starting to study a piece of physics.

Besides the two papers mentioned in my last post, I’ve been looking at this:

• Giovanni Manfredi, Non-relativistic limits of Maxwell’s equations.

There’s a lot I haven’t explained here. I haven’t even said how the electric or magnetic fields transform under Galilean boosts in these limiting theories! I find this subject fairly confusing, and I’d probably have to redo all the calculations to really understand them. As Feynman said, “what I cannot create I do not understand”.

Someday I should dig deeper into this subject and explain how the two limits work in a way I find satisfying. I should also draw the connections to this earlier article of mine:

• Magnetohydrodynamics.

Last time I described a way to get the Standard Model gauge group from the exceptional Jordan algebra. But that approach gave no obvious nice way to put quarks and leptons into the picture. This new paper tackles that problem:

• John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model.

Jordan pairs and Jordan triples are two closely linked formalisms that generalize Jordan algebras. Our paper explains them in detail — and how they’re connected to geometry and quantum mechanics. Here I will mostly skip that wonderful story, so I can quickly explain the connection to the Standard Model.

Here’s how the Standard Model gauge group, together with its representation on one generation of fermions, drops out of a Jordan triple.

The bi-Cayley triple

Let

$$\mathbb{O}_\mathbb{C} = \mathbb{C} \textstyle{\otimes}_\mathbb{R} \mathbb{O}$$

be the bioctonions: octonions with complex coefficients. Write $\mathbb{O}_\mathbb{C}^2$ for the space of column vectors with two bioctonion entries.

$\mathbb{O}_\mathbb{C}^2$ has a certain triple product

$$[x,y,z]=\frac{1}{2}(x(y^{\dagger}z)+z(y^{\dagger}x))$$

which obey the axioms of a gadget called a ‘positive hermitian Jordan triple’. It’s called the bi-Cayley triple.

Now, every positive hermitian Jordan triple gives rise to a $\mathbb{Z}_2$-graded real Lie algebra

$$\mathbf{k} = \mathbf{k}_0 \textstyle{\oplus} \mathbf{k}_1$$

Not a Lie superalgebra: a plain old-fashioned Lie algebra with a $\mathbb{Z}_2$-grading!

How does this work? We take the hermitian Jordan triple itself to be $\mathbf{k}_1$. The Lie algebra $\mathbf{k}_0$ consists of all linear maps from $\mathbf{k}_1$ to itself that are of this form:

$$x \mapsto [a,b,x] - [b,a,x]$$

for some $a,b \in \mathbf{k}_1$. These maps are called real inner derivations. They form a Lie algebra since the commutator of two such maps is another such map. With a bit more work we can define other operations making all of $\mathbf{k}$ into a $\mathbb{Z}_2$-graded Lie algebra.

So, we get a big Lie algebra $\mathbf{k}$, and a Lie subalgebra $\mathbf{k}_0$ sitting inside it. From this we get two Lie groups: a big one $K$ whose Lie algebra is $\mathbf{k}$, and a subgroup $K_0$, whose Lie algebra is $\mathbf{k}_0$.

The quotient is $K/K_0$ is a nice kind of manifold called a hermitian symmetric space. Conversely, any compact hermitian symmetric space give rise to a positive hermitian Jordan triple!

This geometric picture is revealing. The group $K$ acts transitively as symmetries of our hermitian symmetric space, while the stabilizer of any point is isomorphic to $K_0$. Our original Jordan triple, $\mathbf{k}_1$, is then the tangent space of that point. So, $K_0$ acts on the Jordan triple. This action preserves the triple product, and we call $K_0$ the real inner automorphism group of our Jordan triple.

Here’s another great thing about the geometric picture: hermitian symmetric spaces were classified by Eli Cartan (who seems to have spent his life classifying things). As a result we also know the classification of positive hermitian Jordan triples. They come in four infinite series together with two exceptions. One is the bi-Cayley triple, and other is the Albert triple, which is the complexification of the exceptional Jordan algebra. The bi-Cayley triple is a subtriple of the Albert triple. It’s these two exceptions that are connected to the Standard Model. But we’ll start with the bi-Cayley triple.

The 3-graded Lie algebra coming from the bi-Cayley triple is the compact real form of $\mathfrak{e}_6$:

$$\mathfrak{e}_6 = \big[\mathfrak{so}(10) \textstyle{\oplus} \mathfrak{u}(1)\big] \textstyle{\oplus} \mathbb{O}_\mathbb{C}^2.$$

The even part of this Lie algebra is in brackets. The corresponding hermitian symmetric space is called the bioctonionic plane $(\mathbb{C}\otimes\mathbb{O})P^2$. I explained it in Part 12. The even part of our 3-graded Lie algebra, $\mathfrak{so}(10)\oplus \mathfrak{u}(1)$, generates the stabilizer of a point in the bioctonionic plane. The odd part, our friend $\mathbb{O}_\mathbb{C}^2$, is the tangent space of that point.

Here’s the first big surprise. The even part transforms as the adjoint representation of $\mathrm{Spin}(10)$, while the odd part itself transforms as the 16-dimensional complex spinor representation of $\mathrm{Spin}(10)$. Ignoring the extra $\mathrm{U}(1)$ for a moment, this is exactly what we see in a $\mathrm{SO}(10)$ grand unified theory: gauge bosons in the adjoint representation, and one generation of fermions in the 16-dimensional spinor representation.

So before we do anything, the bi-Cayley triple already smells like it contains the ingredients of an $\mathrm{SO}(10)$ grand unified theory.

Tripotents

In a Jordan algebra the important elements are the idempotents, $e^2 = e$. In a Jordan triple $W$ their role is played by tripotents: elements $e$ with

$$[e,e,e] = e.$$

A tripotent always lets us split $W$ into three parts via something called its Peirce decomposition. The operator $w \mapsto [e,e,w]$ has eigenvalues $0, \tfrac{1}{2}, 1$, and $W$ splits into the corresponding eigenspaces

$$W = W_0(e) \textstyle{\oplus} W_{1/2}(e) \textstyle{\oplus} W_1(e),$$

which are called the Peirce 0-space, Peirce $\tfrac{1}{2}$-space and Peirce 1-space of $e$. A tripotent is called minimal when its Peirce $1$-space is one-dimensional: minimal tripotents are the analogues of unit vectors in ordinary quantum theory. Two tripotents $e_1, e_2$ are called colinear when each lies in the other’s Peirce $\tfrac{1}{2}$-space.

I can’t resist explaining some of the quantum physics here. I said I wouldn’t, but I can’t help it. In a hermitian Jordan triple, the triple product $[-,-,-]$ is linear in the first and last slot, but conjugate-linear in the middle slot. So, if you multiply a tripotent by a phase $\alpha$, you get a new tripotent:

$$[\alpha e, \alpha e, \alpha e] = \alpha \overline{\alpha} \alpha e = \alpha e$$

This should remind you of how when you multiply a unit vector in a Hilbert space by a phase, you get a new unit vector. In Jordan triple quantum mechanics, minimal tripotents take the place of these unit vectors. And guess what: the hermitian symmetric space $K/K_0$ that I was talking about earlier is also the space of minimal tripotents mod phase! So, it generalizes the familiar space of ‘pure states’ in quantum mechanics, which are unit vectors mod phase.

But let’s get back to the Standard Model.

A chain of Jordan triples

From here on, the single fact driving everything is this: in any hermitian Jordan triple, any minimal tripotent’s Peirce $\tfrac{1}{2}$-space is itself a hermitian Jordan triple!

If we run this starting from the bi-Cayley triple, we get this chain of hermitian Jordan triples, where each row’s $\tfrac{1}{2}$-space is the next row’s triple:

Here $\mathfrak{a}_5(\mathbb{C})$ is the Jordan triple of antisymmetric $5\times 5$ complex matrices, $\mathrm{M}_{3,2}(\mathbb{C})$ is the Jordan triple of $3\times 2$ complex matrices, $\mathfrak{g}_{\mathrm{SM}} = \mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus \mathfrak{u}(1)$, and

$$G_{\mathrm{SM}} = \mathrm{S}(\mathrm{U}(2) \times \mathrm{U}(3)) \cong (\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1))/\mathbb{Z}_6$$

is the true Standard Model gauge group.

The gauge group from two tripotents

Now pick two colinear minimal tripotents $e_1, e_2 \in W$. Descend the table twice:

• Start with $W = \mathbb{O}_\mathbb{C}^2$, which has real inner automorphism group $(\mathrm{Spin}(10)\times\mathrm{U}(1))/\mathbb{Z}_4$.
• Fix $e_1$. Its Peirce $\tfrac{1}{2}$-space is $W' = \mathfrak{a}_5(\mathbb{C})$, with real inner automorphism group $\mathrm{SU}(5)\times\mathrm{U}(1)$.
• Fix $e_2$ (colinear with $e_1$, so living in $W'$). Its Peirce $\tfrac{1}{2}$-space in $W'$ is $W'' = \mathrm{M}_{3,2}(\mathbb{C})$, with real inner automorphism group exactly $G_{\mathrm{SM}}$.

In other words, the subspace of the bi-Cayley triple colinear with both $e_1$ and $e_2$ is a Jordan triple whose real inner automorphism group is the Standard Model gauge group.

The choice of $e_1$ and $e_2$ also pins down how $G_{\mathrm{SM}}$ sits inside the original group $\mathrm{E}_6$. At each we step take the subgroup that acts with determinant $1$ and preserves the chosen tripotent up to a phase; this gives a chain of subgroups whose members are $\mathrm{Spin}(10)$, $\mathrm{U}(5)$, and $G_{\mathrm{SM}}$, so we get the embedding

$$G_{\mathrm{SM}} \subset \mathrm{SU}(5) \subset \mathrm{Spin}(10).$$

In particle physics, this is the classic chain taking us from the so-called $\mathrm{SO}(10)$ grand unified theory down to the $\mathrm{SU}(5)$ grand unified theory down to the Standard Model. And it’s well known that restricting the 16-dimensional complex spinor representation of $\mathrm{Spin}(10)$ along this chain gives precisely the Standard Model representation $\rho_{\mathrm{SM}}$ on one generation of fermions! So we get one generation of Standard Model fermions this way.

The six particles types as Peirce spaces

We have gotten the representation of the Standard Model gauge group on one generation of fermions without any fuss. But it’s also fun to peer into the details, and see how the different kinds of fermions emerge. We can get them using the fact that for any tripotent $e$, we have projections $P_0(e), P_{1/2}(e)$ and $P_1(e)$ onto its three eigenspaces: its so-called Peirce projectors.

Since we get the Standard Model gauge group and its representation on fermions from two minimal tripotents $e_1$ and $e_2$, we have nine Peirce projectors we can apply to our Jordan triple $\mathbb{O}_{\mathbb{C}}^2$. Let’s use these to pick out various kinds of particles!

As a representation of the Standard Model Lie algebra

$$\mathfrak{g}_{\mathrm{SM}} = \mathfrak{su}(3) \textstyle{\oplus} \mathfrak{su}(2) \textstyle{\oplus} \mathfrak{u}(1) ,$$

any generation of Standard Model fermions transforms as the direct sum of six irreducible representations:

$$\rho_{\mathrm{SM}} = (3,2,\tfrac{1}{6}) \textstyle{\oplus} (\bar 3,1,\tfrac{1}{3}) \textstyle{\oplus} (\bar 3,1,-\tfrac{2}{3}) \textstyle{\oplus} (1,2,-\tfrac{1}{2}) \textstyle{\oplus} (1,1,1) \textstyle{\oplus} (1,1,0),$$

These correspond to the six types of left-handed fermion: $q_L, \overline{d_R}, \overline{u_R}, \ell_L, \overline{e_R}, \overline{\nu_R}$. Six irreducible pieces, six particle types.

It turns out these are exactly the six nonzero components of the Peirce decomposition of $\mathbb{O}_\mathbb{C}^2$ with respect to both $e_1$ and $e_2$. Those six match up one-to-one with the particle types:

The remaining three combinations — $P_1(e_2)P_1(e_1)$, $P_1(e_2)P_0(e_1)$, and $P_0(e_2)P_1(e_1)$ — all vanish, which is why we land on six pieces and not nine.

So the whole package — the gauge group $G_{\mathrm{SM}}$, the embedding $G_{\mathrm{SM}} \subset \mathrm{Spin}(10)$, the representation $\rho_{\mathrm{SM}}$, and even the split of one generation into its six particle multiplets as distinct Peirce components — all comes out of the single object $\mathbb{O}_\mathbb{C}^2$ once you choose two colinear minimal tripotents.

And if you prefer to start one level up, with the Albert triple $\mathfrak{h}_3(\mathbb{O}) \otimes \mathbb{C}$, you get the same result by choosing three mutually colinear tripotents instead of two — but for that, read our paper!

When Lee and Yang suggested that the laws of physics might not be invariant under spatial reflection — that there’s a fundamental difference between left and right — Pauli was skeptical. In a letter to Victor Weisskopf in January 1957, he wrote:

“Ich glaube aber nicht, daß der Herrgott ein schwacher Linkshänder ist.”

(I do not believe that the Lord is a weak left-hander.)

But just two days after Pauli wrote this letter, Chien-Shiung Wu’s experiment confirmed that Lee and Yang were correct. There’s an inherent asymmetry in nature.

We can trace this back to how the ‘left-handed’ fermions and antifermions live in a different representation of the Standard Model gauge group than the right-handed ones. And when we try to build grand unified theories that take this into account, we run into the fact that while we can fit the Standard Model gauge group into $\text{Spin}(10)$ in various ways, not all these ways produce the required asymmetry. There’s a way where it fits into $\text{Spin}(9)$, which is too symmetrical to work… and alas, this one has a nice octonionic description!

To keep things simple I’ll explain this by focusing, not on the whole Standard Model gauge group, but its subgroup $\text{SU}(2) \times \text{SU}(3)$. Here is a theorem proved by Will Sawin in response to a question of mine on MathOverflow:

Theorem 10. There are exactly two conjugacy classes of subgroups of $\text{Spin}(10)$ that are isomorphic to $\text{SU}(2) \times \text{SU}(3)$. One of them has a representative that is a subgroup of $\text{Spin}(9) \subset \text{Spin}(10)$, while the other does not.

I’ll describe representatives of these two subgroups; then I’ll say a bit about how they show up in physics, and then I’ll show you Sawin’s proof.

We can get both subgroups in a unified way! There’s always an inclusion

$$\text{SO}(m) \times \text{SO}(n) \to \text{SO}(m+n)$$

and taking double covers of each group we get a 2-1 homomorphism

$$\text{Spin}(m) \times \text{Spin}(n) \to \text{Spin}(m+n)$$

In particular we have

$$\text{Spin}(4) \times \text{Spin}(6) \to \text{Spin}(10)$$

so composing with the exceptional isomorphisms:

$$\text{Spin}(4) \cong \text{SU}(2) \times \text{SU}(2), \qquad \text{Spin}(6) \cong \text{SU}(4)$$

we get a 2-1 homomorphism

$$k \colon \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \to \text{Spin}(10)$$

Now, there are three obvious ways to include $\text{SU}(2) \times \text{SU}(3)$ in $\text{SU}(2) \times \text{SU}(2) \times \text{SU}(4)$. There is an obvious inclusion

$$j \colon \text{SU}(3) \hookrightarrow \text{SU}(4)$$

but there are three obvious inclusions

$$\ell, r, \delta \colon \text{SU}(2) \hookrightarrow \text{SU}(2) \times \text{SU}(2)$$

namely the left one:

$$\begin{array}{ccc} \ell \colon \text{SU}(2) &\to& \text{SU}(2) \times \text{SU}(2) \\ g & \mapsto & (g,1) \end{array}$$

the right one:

$$\begin{array}{ccc} r \colon \text{SU}(2) &\to& \text{SU}(2) \times \text{SU}(2) \\ g & \mapsto & (1,g) \end{array}$$

and the diagonal one:

$$\begin{array}{ccc} \delta \colon \text{SU}(2) &\to& \text{SU}(2) \times \text{SU}(2) \\ g & \mapsto & (g,g) \end{array}$$

Combining these with our earlier maps, we actually get a one-to-one map from $\text{SU}(2) \times \text{SU}(3)$ to $\text{Spin}(10)$. So we get three subgroups of $\text{Spin}(10)$, all isomorphic to $\text{SU}(2) \times \text{SU}(3)$:

• There’s the left subgroup $G_\ell$, which is the image of this composite homomorphism:

$$\text{SU}(2) \times \text{SU}(3) \stackrel{\ell \times j}{\longrightarrow} \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \cong \text{Spin}(4) \times \text{Spin}(6) \stackrel{k}{\longrightarrow} \text{Spin}(10)$$

• There’s the diagonal subgroup $G_\delta$, which is the image of this:

$$\text{SU}(2) \times \text{SU}(3) \stackrel{\delta \times j}{\longrightarrow} \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \cong \text{Spin}(4) \times \text{Spin}(6) \stackrel{k}{\longrightarrow} \text{Spin}(10)$$

• And there’s the right subgroup $G_r$, which is the image of this:

$$\text{SU}(2) \times \text{SU}(3) \stackrel{r \times j}{\longrightarrow} \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \cong \text{Spin}(4) \times \text{Spin}(6) \stackrel{k}{\longrightarrow} \text{Spin}(10)$$

The left and right subgroups are actually conjugate, but the diagonal one is truly different! We’ll prove this by taking a certain representation of $\text{Spin}(10)$, called the Weyl spinor representation, and restricting it to those two subgroups. We’ll get inequivalent representations of $\text{SU}(2) \times \text{SU}(3)$. This proves the two subgroups aren’t conjugate.

This argument is also interesting for physics. When restrict to the left subgroup, we get a representation of $\text{SU}(2) \times \text{SU}(3)$ that matches what we actually see for one generation of fermions! This is the basis of the so-called $\text{SO}(10)$ grand unified theory, which should really be called the $\text{Spin}(10)$ grand unified theory.

(In fact this works not only for $\text{SU}(2) \times \text{SU}(3)$ but for the whole Standard Model gauge group, which is larger. I’m focusing on $\text{SU}(2) \times \text{SU}(3)$ just because it makes the story simpler.)

When we restrict the Weyl spinor representation to the diagonal subgroup, we get a representation of $\text{SU}(2) \times \text{SU}(3)$ that is not physically correct. Unfortunately, it’s the diagonal subgroup that shows up in several papers connecting the Standard Model gauge group to the octonions. I plan to say a lot more about this later.

The left subgroup

Let’s look at the left subgroup $G_\ell$, the image of this composite:

$$\text{SU}(2) \times \text{SU}(3) \stackrel{\ell \times j}{\longrightarrow} \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \cong \text{Spin}(4) \times \text{Spin}(6) \stackrel{k}{\longrightarrow} \text{Spin}(10)$$

$\text{Spin}(10)$ has a 32-dimensional unitary representation called the ‘Dirac spinor’ representation. This representation is really on the exterior algebra $\Lambda \mathbb{C}^5$. It’s the direct sum of two irreducible parts, the even grades and the odd grades:

$$\Lambda \mathbb{C}^5 \cong \Lambda^{\text{even}} \mathbb{C}^5 \oplus \Lambda^{\text{odd}} \mathbb{C}^5$$

Physicists call these two irreducible representations ‘right- and left-handed Weyl spinors’, and denote them as $\mathbf{16}$ and $\mathbf{16}\ast$ since they’re 16-dimensional and one is the dual of the other.

Let’s restrict the $\mathbf{16}$ to the left subgroup $G_\ell$ and see what we get.

To do this, first we can restrict the $\mathbf{16}$ along $k$ and get

$$\mathbf{2} \otimes \mathbf{1} \otimes \mathbf{4} \; \oplus \; \mathbf{1} \otimes \mathbf{2} \otimes \mathbf{4}\ast$$

Here $\mathbf{1}$ is the trivial representation of $\text{SU}(2)$, $\mathbf{2}$ is the tautologous representation of $\text{SU}(2)$, and $\mathbf{4}$ is the tautologous rep of $\text{SU}(4)$.

Then let’s finish the job by restricting this representation along $\ell \times j$. Restricting the $\mathbf{4}$ of $\text{SU}(4)$ to $\text{SU}(3)$ gives $\mathbf{3} \oplus \mathbf{1}$: the sum of the tautologous representation of $\text{SU}(3)$ and the trivial representation. Restricting $\mathbf{2} \otimes \mathbf{1}$ to the left copy of $\text{SU}(2)$ gives the tautologous representation $\mathbf{2}$, while restricting $\mathbf{1} \otimes \mathbf{2}$ to this left copy gives $\mathbf{1} \oplus \mathbf{1}$: the sum of two copies of the trivial representation. All in all, we get this representation of $\text{SU}(2) \times \text{SU}(3)$:

$$\mathbf{2} \otimes (\mathbf{3} \oplus \mathbf{1}) \; \oplus \; (\mathbf{1} \oplus \mathbf{1}) \otimes (\mathbf{3}\ast \oplus \mathbf{1})$$

This is what we actually see for one generation of left-handed fermions and antifermions in the Standard Model! The representation $\mathbf{3} \oplus \mathbf{1}$ describes how the left-handed fermions in one generation transform under $\text{SU}(3)$: 3 colors of quark and one ‘white’ lepton. The representation $\mathbf{3}\ast \oplus \mathbf{1}$ does the same for the left-handed antifermions. The left-handed fermions form an isospin doublet, giving us the $\mathbf{2}$, while the left-handed antifermions have no isospin, giving us the $\mathbf{1} \oplus \mathbf{1}$.

This strange lopsidedness is a fundamental feature of the Standard Model.

The right subgroup would work the same way, up to switching the words ‘left-handed’ and ‘right-handed’. And by Theorem 10, the left and right subgroups must be conjugate in $\text{Spin}(10)$, because now we’ll see one that’s not conjugate to either of these.

The diagonal subgroup

Consider the diagonal subgroup $G_\delta$, the image of this composite:

$$\text{SU}(2) \times \text{SU}(3) \stackrel{\delta \times j}{\longrightarrow} \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \cong \text{Spin}(4) \times \text{Spin}(6) \stackrel{k}{\longrightarrow} \text{Spin}(10)$$

Let’s restrict the $\mathbf{16}$ to $G_\delta$.

To do this, first let’s restrict the $\mathbf{16}$ along $k \colon \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) \to \text{Spin}(10)$ and get

$$\mathbf{2} \otimes \mathbf{1} \otimes \mathbf{4} \; \oplus \; \mathbf{1} \otimes \mathbf{2} \otimes \mathbf{4}\ast$$

as before. Then let’s restrict this representation along $\delta \times j$. The $\SU(3)$ part works as before, but what happens when we restrict $\mathbf{2} \otimes \mathbf{1}$ or $\mathbf{1} \otimes \mathbf{2}$ along the diagonal map $\delta \colon \text{SU}(2) \to \text{SU}(2) \times \text{SU}(2)$? We get $\mathbf{2}$. So, this is the representation of $G_\delta$ that we get:

$$\mathbf{2} \otimes (\mathbf{3} \oplus \mathbf{1}) \; \oplus \; \mathbf{2} \otimes (\mathbf{3}\ast \oplus \mathbf{1})$$

This is not good for the Standard Model. It describes a more symmetrical universe than ours, where both left-handed fermions and antifermions transform as doublets under $\text{SU}(2)$.

The fact that we got a different answer this time proves that $G_\ell$ and $G_\delta$ are not conjugate in $\text{Spin}(10)$. So to complete the proof of Theorem 10, we only need to prove

1. Every subgroup of $\text{Spin}(10)$ isomorphic to $\text{SU}(2) \times \text{SU}(3)$ is conjugate to $G_\ell$ or $G_\delta$.

2. $G_\delta$ is conjugate to a subgroup of $\text{Spin}(9) \subset \text{Spin}(10)$, but $G_\ell$ is not.

I’ll prove 2, and then I’ll turn you over to Will Sawin to do the rest.

Why the diagonal subgroup fits in $\text{Spin}(9)$

Every rotation of $\mathbb{R}^n$ extends to a rotation of $\mathbb{R}^{n+1}$ that leaves the last coordinate fixed, so we get an inclusion $\text{SO}(n) \hookrightarrow \text{SO}(n+1)$, which lifts to an inclusion of the double covers, $\text{Spin}(n) \hookrightarrow \text{Spin}(n+1)$. Since we have exceptional isomorphisms

$$\text{Spin}(3) \cong \text{SU}(2), \qquad \text{Spin}(4) \cong \text{SU}(2) \times \text{SU}(2)$$

it’s natural to ask how the inclusion $\text{Spin}(3) \hookrightarrow \text{Spin}(4)$ looks in these terms. And the answer is: it’s the diagonal map! In other words, we have a commutative diagram

$$\begin{array}{ccc} \text{SU}(2) & \xrightarrow{\sim} & \text{Spin}(3) \\ \delta \downarrow & & \downarrow \\ \text{SU}(2) \times \text{SU}(2) & \xrightarrow{\sim} & \text{Spin}(4) \end{array}$$

Now, we can easily fit this into a larger commutative diagram involving some natural maps $\text{Spin}(m) \times \text{Spin}(n) \to \text{Spin}(m+n)$ and $\text{Spin}(n) \to \text{Spin}(n+1)$:

$$\begin{array}{ccccccc} \text{SU}(2) & \xrightarrow{\sim} & \text{Spin}(3) & \to & \text{Spin}(3) \times \text{Spin}(6) & \to & \text{Spin}(9) \\ \delta \downarrow & & \downarrow & & \downarrow & & \downarrow \\ \text{SU}(2) \times \text{SU}(2) & \xrightarrow{\sim} & \text{Spin}(4) & \to & \text{Spin}(4) \times \text{Spin}(6) & \to & \text{Spin}(10) \end{array}$$

We can simplify this diagram using the isomorphism $\text{Spin}(6) \cong \text{SU}(4)$:

$$\begin{array}{ccccccc} \text{SU}(2) \times \text{SU}(4) & \to & \text{Spin}(9) \\ \delta \times 1 \downarrow & & \downarrow \\ \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) & \to & \text{Spin}(10) \end{array}$$

and then we can use our friend the inclusion $j \colon \text{SU}(3) \to \text{SU}(4)$:

$$\begin{array}{ccccccc} \text{SU}(2) \times \text{SU}(3) & \xrightarrow{1 \times j} & \text{SU}(2) \times \text{SU}(4) & \to & \text{Spin}(9) \\ & & \delta \times 1 \downarrow & & \downarrow \\ & & \text{SU}(2) \times \text{SU}(2) \times \text{SU}(4) & \to & \text{Spin}(10) \end{array}$$

This shows that the diagonal subgroup $G_\delta$ of $\text{Spin}(10)$ is actually a subgroup of $\text{Spin}(9)$!

Why the left subgroup does not fit in $\text{Spin}(9)$

The three-fold way is a coarse classification of irreducible complex representations of compact Lie group. Every such representation is of one and only one of these three kinds:

1) not self-dual: not isomorphic to its dual,

2a) orthogonal: isomorphic to its dual via an invariant nondegenerate symmetric bilinear form, also called an orthogonal structure,

2b) symplectic: isomorphic to its dual via an invariant nondegenerate antisymmetric bilinear form, also called a symplectic structure.

I’ve written about how these three cases are related to the division algebras $\mathbb{C}, \mathbb{R}$ and $\mathbb{H}$, respectively:

• John Baez, Division algebras and quantum theory.

A complex representation is orthogonal iff it’s the complexification of a representation on a real vector space, and symplectic iff it’s the underlying complex representation of a representation on a quaternionic vector space.

But we don’t need most of this yet. For now we just need to know one fact: when $n$ is odd, every irreducible representation of $\text{Spin}(n)$, and thus every representation of this Lie group, is self-dual: that is, isomorphic to its dual. In particular this is true of $\text{Spin}(9)$.

Why does this matter? Assume the left subgroup $G_\ell \subset \text{Spin}(10)$ is a subgroup of $\text{Spin}(9)$. When we restrict the Weyl spinor representation of $\text{Spin}(10)$ to $\text{Spin}(9)$ it will be self-dual, like every representation of $\text{Spin}(9)$. Then when we restrict this representation further to $\text{SU}(2) \times \text{SU}(3)$ it must still be self-dual, since the restriction of a self-dual representation is clearly self-dual.

However, we know this representation is

$$\mathbf{2} \otimes (\mathbf{3} \oplus \mathbf{1}) \; \oplus \; (\mathbf{1} \oplus \mathbf{1}) \otimes (\mathbf{3}\ast \oplus \mathbf{1})$$

and this is not self-dual, since $\mathbf{1}\ast \cong \mathbf{1}$ and $\mathbf{2}\ast \cong \mathbf{2}$ but $\mathbf{3}\ast \ncong \mathbf{3}$.