Planet Musings

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.

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.

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?

In case you weren't aware, in addition to the OMB proposed rule changes which just concluded their public comment period, the NSF has put forward proposed changes in their proposal procedures for public comment, and the last day for feedback is August 24.  The proposal and award policies and procedures guide (PAPPG) is to be replaced with the revised document called the "Guide for Financial Assistance" (GFA).

The proposed revised document is here: https://www.regulations.gov/document/NSF-2026-OTR-0001-0003 

The place to file a comment is here: https://www.regulations.gov/docket/NSF-2026-OTR-0001/document?withinCommentPeriod=true

The place read already-filed comments is here: https://www.regulations.gov/docket/NSF-2026-OTR-0001/comments

The Federation of American Societies for Experimental Biology has a brief summary of the proposed changes here.  There are a number of (IMO) problematic issues with the would-be revisions, not the least of which is they presumptively assume that the OMB changes are going into effect, including the very controversial ones that would allow termination of awards at any time, for any reason (including not aligning with presidential priorities), with no appeal.  It's a good idea to be aware of what's being proposed and to take advantage of the opportunity to provide feedback.  

Update:  The APS has also put together a guide to the proposed changes.

Speaking of feedback, the American Physical Society is doing a broad survey (beyond just physicists) asking for input about up to three things that people think the NSF does well and up to three things that the NSF should do differently.  It's open to everyone, and the response spaces are brief (500 char ea.). Again, if this is something that matters to you, I encourage you to contribute - it's quick.  

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.

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.

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”.

This post concerns the following conjecture of Sendov, as well as its strengthening by Phelps–Rodriguez:

Conjecture 1 (Sendov’s conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with .

Conjecture 2 (Phelps–Rodriguez conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with , unless is on the unit circle and is a scalar multiple of .

By applying a rotation around the origin, we can normalize to be a real number with .

From the work of Rubinstein, both conjectures were already established in the case, so one can restrict to the case. Both of these conjectures then follow from

Conjecture 3 (Sendov’s conjecture in interior) Let . Let be a degree polynomial with all zeroes in the unit disk. Then if is a zero of , there exists a critical point of with .

All three of these conjectures were established for (in a sequence of papers culminating in this paper of Brown and Xiang) and for sufficiently large (in a paper of myself, which in turn built upon several partial results in this setting). This left the case of intermediate to be settled. My arguments used some qualitative ingredients (most notably analytic continuation) and as such did not easily lend themselves to quantifying the threshold of above which the argument was valid.

Recently, Lech Mazur was able to use an AI tool to resolve Sendov’s conjecture for all , with the proof verified in Lean. However, the AI-generated proof was not human-digested to be in the form of a publication-ready preprint; and it has taken me several days (with heavy AI assistance) to perform such a digestion, to place the proof in proper context with previous literature and to simplify and streamline the argument to highlight the main ideas. (Note: the above chat log only represents a portion of the digestion work: the rest was performed with pen and paper, or using some further AI agents.) The same arguments also give a new proof of Rubinstein’s theorem, which I also give below the fold.

One consequence of this digestion is that the argument in fact demonstrates Conjecture 3, and thus resolves both the Sendov conjecture and the Phelps–Rodriguez conjecture in full generality.

The proof ends up being remarkably elementary. No complex analysis is used other than the fundamental theorem of algebra (and very basic facts about Möbius transformations); and the deepest inequality used as input is the Maclaurin inequality (and we only need a special case of that inequality which can be derived from the arithmetic mean-harmonic mean inequality and an induction argument).

Using an AI agent, I have been able to formalize the entire argument in Lean, extended to by some minor modifications to the proof. This formalization is more streamlined than the original formalization (it has about 15,000 lines of code, compared with around 90,000 for the original proof).

We now prove Conjecture 3. The cases have long been known but need to be treated separately; a short proof using the machinery developed here is provided at the end of the post. Suppose now that we have a counterexample for some , thus one can find a degree polynomial with zeroes

To capture the fact that the critical points lie at a distance at least from , we write these critical points as

Example 4 If and , then are the non-trivial roots of unity, while the are all equal to . Strictly speaking this is not actually a counterexample to Conjecture 3, because is not strictly less than one; nevertheless this is an important motivating near-counterexample for the arguments below.

Example 5 A generalization of the previous example was studied in Section 4 of my paper. Here one took

where was an asymptotic parameter going to infinity,

was a low-degree polynomial for some ,

and were constants. This polynomial has a zero at , critical points at , and additional critical points near . If all the critical points were at distance at least one from , one would have

and

while if all the zeroes were in the unit disk, the calculations in my paper showed that

Here denotes a quantity that goes to zero as . If one ignores the errors, one can show that these conditions are only simultaneously feasible if and all the vanish, but the argument was somewhat subtle (I had to proceed by inspecting the second Fourier coefficient of (1)). This illustrates the fact that the regime is particularly delicate.

We now have two sets of points in the closed unit disk: and . They “communicate” with each other through the polynomial and its first derivative , both of which can be expressed in terms of either set of points (as well as and ). Indeed, if we normalize to be monic, then we can factor in terms of the zeroes as

In a similar vein, can be factored

It is convenient to rule out the easy case right away. In this case we see from (3), (4) that

By inspecting or at various natural locations, we can thus obtain a number of identities relating the to the . We record the ones that we actually need here:

Lemma 6 (Communication identities) Let denote the function

• (i) (Centroid identity) We have

  

  That is to say, the centroid of the zeroes equals the centroid of the critical values.
• (ii) (Polar identity) We have

  

• (iii) (First origin identity) We have

  

• (iv) (Second origin identity) We have

  

  (Again, we are using the convention of removing singularities to deal with the case where some of the vanish.)

Proof: For (i), we inspect the behavior of as . From (2) we have

For (ii), we consider the expression . On the one hand, from (2), (3) one has

For (iii), we evaluate . From (2) we have

For (iv), we similarly evaluate . From (3) we have

Remarkably, the polynomial will play no further role in the argument: the identities in (i)-(iv), together with the hypotheses that and lie in the closed unit disk, will be sufficient by themselves to obtain a contradiction.

Example 7 Continuing the example in Example 4, in (i) both sides vanish. In (ii), both sides are equal to one. For (iii) and (iv), we have , with both sides of (iii) equal to one, and both sides of (iv) equal to zero.

Remark 8 The centroid identity is extremely classical, going back to this 1948 paper of Popoviciu. The comparison of the polynomial at a location and at the polar inversion of that location across the closed unit disk is a familiar trick in the literature; see, e.g., Lemma 5 and Theorem 8 of Dégot. The specific form of the polar identity is implicit in the first part of Section 5 of Mazur’s AI-generated proof, while the origin identities are extracted from equation (6.3) of that proof. The first origin identity is also very close to Theorem 6 of Dégot, while the second origin identity is similar to some identities appearing in the proof of Lemma 6 of Dégot, as well as the work of Mir–Nazir–Wani and (in the case) Rubinstein. The work of Meir–Sharma and Mir–Nazir–Wani also contain several further identities relating the to the ; see in particular Lemma 15 below. Variants of (5) also appear in Proposition 10 of Miller.

Remark 9 The first origin identity (9) is already strong enough to handle asymptotically all examples of the form in Example 5, except in the endpoint case where vanish and the are all . Indeed, as the are in the closed unit disk, (9) implies that

On the other hand, routine calculations (omitted here) show that

leading asymptotically to the constraint

But all terms here are non-negative (since ), so this forces a contradiction unless (and hence also ) and the all vanish.

As mentioned in Example 5, the most delicate regime occurs when . It is convenient to introduce the normalized version

A key role in the argument will be played by the mean

It will be convenient to work with the quadratic polynomial

The arguments will revolve around the relationship between and . Specifically, we will establish the following two inequalities below the fold. The first inequality, which we call the “polar inequality”, comes in three forms:

Proposition 10 (Polar inequality)

• (i) (Raw polar inequality) We have

  

• (ii) (Polar inequality in , form) We have

  

• (iii) (Simplified polar inequality) We have

  

  In particular, since , one has

  

It will be the inequality (18) that we use in practice, but it will be derived from (17), which in turn is a consequence of (16), which will follow from the polar identity (8) together with the fact that the and lie in the unit disk. The bound (18) is only slightly weaker than (17); see the (Gemini-generated) image below.

I was not able to find an exact duplicate of the above polar inequalities in past literature, but the paper of Dégot contains several similar inequalities. The inequality (16) was extracted from (5.1) of Mazur’s AI-generated proof; the subsequent bounds (17), (18) arose from my attempts to simplify the arguments after that point.

The second inequality, which is more difficult, also will come in several forms:

Proposition 11 (Origin inequality) Let .

• (i) (Raw origin inequality) We have

  

• (ii) ( bound) We have

  

• (iii) (Origin inequality in , form) We have

  

Part (i) (which was extracted with some effort from Section 6 of the original AI-generated argument) will be deduced from the first and second origin identities (9), (10), as well as the centroid identity (7). Part (ii) will follow from (i) and the polar inequality (18), while part (iii) is an elementary consequence of (i).

As it turns out, the last three terms in (21) are asymptotically negligible as . Dropping those terms gives a competing feasibility region for and which is disjoint from the one coming from the polar inequality (17) (or (18)):

This already suggests that one can use this approach to recover my previous result on Sendov’s conjecture holding for all sufficiently large . In fact, even with the three error terms in (21) added, there is enough room between the two inequalities (18), (21) to obtain a contradiction for all (using the additional bound to control these errors), although showing this for medium-sized (such as ) requires a certain amount of computer assistance.

For fixed , the right-hand side of (21) is monotone increasing in (or equivalently, monotone decreasing in ). In view of (18), we can thus replace by in this inequality, so that is replaced by

We can now eliminate all large degrees, e.g., , as follows. The quadratic attains its minimum at . For we have

Dropping some and terms, we conclude that

The remaining range to handle is when

I have also verified this bound in Lean.

— 1. The polar inequality —

We begin with a proof of Proposition 10.

As is well known, the Möbius transform maps the closed unit disk to itself. In particular, we have

Bounding by and using the quantities from (11), (15), we observe that

On evaluating the integral in (17), we obtain

Both sides of (27) vanish at . Taking derivatives, it suffices to show that

Remark 12 As the image in the introduction suggests, the bound (18) is only slightly weaker than (17). For small , one can perform Taylor approximation on the latter bound to obtain

while the former bound is

Note that is slightly smaller than .

Relating to this, the constant in (27) cannot be improved.

— 2. The origin inequality —

Now we turn to the proof of Proposition 11, which is more difficult and revolves around an analysis of the function defined in (6). We begin with a heuristic analysis. Inserting the approximation for small into (6) and using (12), we are led to the approximation

As lie in the closed unit disk, the product does also. However, past experience with the Sendov conjecture has taught us that the worst cases tend to be when lie very close to the boundary of the disk, so that is close to one. For instance, in Example 4 all the and lie on the unit circle, and . See Remark 3 of Dégot or Theorem 1.10(ii) of my own paper for other places where this heuristic is noted. To simplify the discussion, let us assume for now that is exactly one, so that all lie on the unit circle. This leads in particular to the inversion identities

We turn to the details. To make the approximation (30) more precise, we note that , and hence by the fundamental theorem of calculus

When faced with a similar expression in (24), we used the arithmetic mean-geometric mean inequality. Here, the analogous tool is Maclaurin’s inequality, which gives

The next step is thus to estimate . When , then all the were on the unit circle and we could use (31) (and the centroid identity) to proceed. Now, we are no longer assuming to equal , but we can still adapt the previous arguments with a loss proportional to . The key lemma is

Lemma 13 (Defect lemma) Let be some points in the closed unit disk. Then

Proof: By a limiting argument we may assume that none of the vanish. If we write for some , then we can calculate that

We remark that the lemma can also be proven by direct induction, without an appeal to hyperbolic trigonometry.

From taking complex conjugates of the centroid identity (7) and performing some algebra, we have

Next we eliminate the role of the imaginary term . Observe for any complex number with positive real part that

— 2.1. Upper bound on —

Now we can prove (20). Suppose for contradiction that ; since , this implies that . Crudely discarding the term in (19) and bounding by , we have

— 2.2. A simplified estimate —

Now we show (21). Note from (11) that

— 3. Rubinstein’s theorem —

We now adapt the arguments to give a proof of Rubinstein’s theorem that the Phelps–Rodriguez conjecture holds in the case, i.e.,

Theorem 14 (Rubinstein’s theorem) Let , and let be a degree polynomial with all zeroes in the unit disk. If , then there exists a critical point of with , unless is a scalar multiple of .

The argument here is essentially in Remark 5.1 of this paper of Tang and Zhang.

Taking contrapositives, we may assume that the critical points are of the form for some in the closed unit disk, and normalize to be monic; our task is to show that .

The polar identity (8), based on calculating degenerates to a triviality when , but we have the following usable substitute, valid for any choice of , first observed in equation (3.2) of Meir–Sharma:

Lemma 15 (Meir–Sharma identity) If and the critical points are of the form then all the zeroes are not equal to , and

Proof: By hypothesis, is not a critical point of , so and for all . Instead of computing , we instead consider the expression . On the one hand, from (4) we have

Now take . Since lie in the closed unit disk, has real part at least , while is at most . Thus, the only way that the above identity can hold is if for all , hence for all . Thus all critical points are at the origin, which forces for some . Since , we conclude that , giving the claim.

— 4. The cases —

We now prove the cases of Conjecture 3. The starting point is (23). Using the triangle inequality and , this implies that

Remark 16 The same argument also works for , but breaks down for higher .

— 5. Further directions —

The Sendov and Phelps–Rodriguez conjectures are now resolved, but several related conjectures remain open. The following strengthening of Sendov’s conjecture, by Borcea, is open for any :

Conjecture 17 (Borcea conjecture) Let and , and let be a degree polynomial with zeroes satisfying . Then for every zero of , there exists a critical point of with .

Sendov’s conjecture is the limiting case of this conjecture. There has been relatively little progress on this conjecture: the cases were established by Khavinson, Pereira, Putinar, Saff, and Shimorin, and in this previous paper we reported the negative result that AlphaEvolve failed to find a counterexample to the conjecture. The proof methods here do not seem to extend easily; all the identities relating zeroes and critical points continue to hold, but now that the are only constrained to the unit disk in an averaged moment sense, all of the inequalities developed above now fail.

Another strengthening of Sendov’s conjecture that remains open is Schmeisser’s conjecture:

Conjecture 18 (Schmeisser’s conjecture) Let , and let be a degree polynomial with all zeroes in the closed unit disk. Then for any in the convex hull of the zeroes of , there exists a critical point of with .

Schmeisser proved several special cases of this conjecture, and AlphaEvolve again failed to find a counterexample, but there has not been much further progress. Here, the are now back in the closed unit disk, but we no longer have , again rendering most of the previous identities invalid. But perhaps some modification of the arguments here can make some progress on this conjecture.

A common generalization of the Borcea and Schmeisser conjectures was proposed in Conjecture 2.4 of this paper of Zhang. A slightly different strengthening was also proposed in Conjecture 1.10 of Tang and Zhang:

Conjecture 19 (Tang–Zhang conjecture) Let , and let be a degree polynomial with all zeroes in the closed unit disk and critical points . Then for any , one has .

Sendov’s conjecture is the limiting case . By Hölder’s inequality, the case is the strongest form of the conjecture.

Another well known variant of Sendov’s conjecture is Smale’s problem:

Conjecture 20 (Smale’s problem) Let , and let be a degree polynomial. Then for any zero of , there exists a critical point of with .

The constant is best possible, as can be seen by the example and . Using the Koebe one-quarter theorem, Smale proved this conjecture with replaced by . Some slight improvements of this bound have been obtained over the years; for instance for , the improved bound of was obtained by Crane. Again, AlphaEvolve failed to find a counterexample to this conjecture. This problem does not seem to have a direct relationship with Sendov’s conjecture, and there is no useful normalization of the zeroes and critical points that is confined to the unit disk. Nevertheless there may be some hope of making progress on this conjecture, perhaps working first in the asymptotic regime .

Needless to say, I did try some desultory attempts to use AI tools to attack these questions, but without much notable success.

One potential way forward is to find further proofs of Sendov’s conjecture that utilize other techniques that might be more broadly applicable to this larger family of problems. The proof here is remarkable in that the zeroes and critical points are treated almost as independent mathematical objects, communicating with each other only very narrowly through four identities in which one only inspects the underlying polynomial (and its derivative) at a small number of points. It could be that an approach focusing on more global features of the polynomial may lead to new proofs of Sendov’s conjecture, and perhaps also of its generalizations.

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.

A function of one variable can be translated in space by a spatial shift to obtain a new function

Some functions obey finite linear relations between their time-frequency shifts. For instance, a sinusoid obeys the relation

Conjecture 1 (HRT conjecture) If is non-zero, then there is no relation of the form

for some distinct time-frequency shifts and some coefficients , not all zero.

A special case of the HRT conjecture, which was also open, makes the additional assumption that was Schwartz.

Many positive results towards this conjecture were known. I will mention only a few here. A simple case is when we only have frequency shifts rather than spatial shifts:

What about the non-collinear case? Suppose first that all the lie in the lattice , thus for some integers . Here, the phase shift in (1) disappears, and all the time-frequency shifts commute with each other. This suggests that it should be possible to diagonalize the situation with a suitable transform to convert (2) to a pointwise equation similar to (3). To find this diagonalization, observe that if one restricts the function to a coset of the integers, then just multiplies the function by the scalar , while shifts the function on this coset by . The latter translation operation can also be converted to pointwise multiplication by performing the Fourier transform on the integers. Thus, if one introduces the Zak transform

More generally, there is a result of Linnell that the conjecture is true if lie in a translate of a discrete subgroup of ; this (together with the argument handling the collinear case) establishes all cases where , and several partial results involving the cases are also known. The conjecture is also known if is decays at a suitably super-exponential rate, by work of Bownik and Speegle.

I was aware of this conjecture through various talks and conversations with colleagues, and even briefly tried my hand at it for a while, though not with particularly serious effort (or progress). It was thus a nice surprise to see that it has just been resolved by Faulhuber, Petersen, van Velthoven, and Voigtlaender, even in the Schwartz case:

Theorem 2 There exist complex numbers , not all zero, distinct points , and a non-zero Schwartz function such that

It is perhaps unsurprising that this result is AI-assisted. However, I think the authors have disclosed their AI use responsibly, with the final arguments written by hand with a readable overview of the argument, as well as proper discussion of methods, relation to past literature, and other independent numerical checks on the result.

The negative result lies only a little beyond the positive results: is now increased to , and all but one of the points lie in (a translate of) a discrete subgroup of (in fact the explicit subgroup is used). The functions constructed are smooth and rapidly decaying, but not analytic or super-exponentially decaying, which would start being in conflict with the known positive results.

In addition to AI being used to come up with the initial proof strategy, a more traditional numerical computation was used to verify one step of the argument.

I have not had the time to do a full digestion of the result, but (after reading the introduction, and using a little AI assistance of my own) I was able to understand the main ideas at a high level. The first few reductions are relatively standard. Setting and , one can view the problem as one of solving an eigenvalue problem

How to solve this equation? The motivating scenario here is if the matrix function was replaced by a rank one function

The main remaining obstacle is that the “eigenvalue function” is varying in the parameter rather than constant. (This issue was, by the way, anticipated to some extent in previous work of Demeter, who observed that eigenfunctions of the almost Matthieu discrete Schrödinger operator gave a near-miss counterexample to the HRT conjecture, but with an eigenvalue that depended on an auxiliary phase shift parameter rather than constant.) However, if one was able to solve the scalar cocycle equation

Yesterday I attended the 40th annual summer research colloquium of the Smalley-Curl Institute at Rice, a fun internal conference that provides a great opportunity for undergrads (including visitors), graduate students, and a few postdocs to present their work.  The keynote speaker was our EVPR, Prof. David Sholl, who gave a very informative talk about reproducibility in the chemical engineering/materials literature.  We hear a lot these days about crises of reproducibility in scientific research, and Prof. Sholl rightly points out that in some fields the expectation of reproducible results is high - no one would spend $1B on a chemical engineering plant if they weren't very sure that the catalytic processes were going to work as expected at scale.  Keys to reproducibility include, unsurprisingly, repeated results and independent replication.  One metaresult that was interesting is this paper, looking at the literature on metal-organic frameworks and how often there are published replications of syntheses; not as often as you would think or want!  

A truly surprising (to me, anyway) result is this one.  The Brunauer–Emmett–Teller (BET) (yes, that Teller) method is a long-established technique that uses gas adsorption measurements to infer the surface area of porous materials.  Many research groups were given identical raw adsorption isotherms and asked to calculate the specific surface areas, resulting in a surprisingly large spread of results (Fig 1 of the paper).  Clearly not everyone had the same analysis procedures even for a technique developed in the 1930s!

Some take-away lessons from this are encapsulated here, in an article titled "Five easy ways to make your research more reproducible".  Good stuff.  The talk raised a number of questions relevant to our present era of huge enthusiasm about AI-based materials research and "self-driving" labs.  If the AI models are all trained on the literature, and the literature is not representative of complete and reproducible procedures, that's a problem.  

One personal anecdote about reproducibility and its challenges in materials synthesis.  Twenty years ago (!), I was working with a colleague who had a postdoc who was synthesizing Fe3O4 (magnetite) nanoparticles via wet chemistry methods (see here). We did some fun electronic transport experiments bridging very closely spaced electrodes with such nanoparticles, and we saw some very dramatic hysteretic response kick in as \(T\) was reduced below about 120 K.  That's the temperature of the Verwey transition in magnetite, where the material enters a more insulating low temperature phase.  Basically all of the devices we made with that batch of nanoparticles showed this phenomenon.  Then the postdoc took up a faculty position and a senior grad student came in and took over the synthesis, and for several months, subsequent batches of nanoparticles just didn't seem to show the effect.  The key issue is oxygen stoichiometry.  Get a little oxygen rich, and you form nanoparticles that include some \(\gamma\)-Fe2O3, which doesn't have the Verwey physics and in nanoparticle form looks really similar in x-ray diffraction to the desired magnetite.  Anyway, we started working with a collaborator who could grow epitaxial Fe3O4 films, and in those devices the electronic effect was there all the time.  All this led to this publication and subsequent papers, and I still think it's a cool set of results about a nonequilibrium transition in a correlated material.  In the end, after several months the chemistry grad student did get back to making nanoparticle batches that showed the transition. It turns out that at some point he had changed the length of a piece of tubing in the gas manifold, and unexpectedly that had altered the reaction kinetics just a little.  Changing it back got the synthesis to be reliable again.  This is an example of how finicky materials synthesis can be!

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.

At least it wasn’t the Yankees.

Yes, Adley Rutchman, an Oriole yesterday, is a Red Sock today. Not much to say about it that hasn’t been said over the last 20 hours. I understand that he probably wasn’t going to be an Oriole beyond 2027. It still feels bad. Together with yesterday’s trade of Dean Kremer, it feels like an acknowledgment of what we sort of already knew: the plan failed. There is not going to be a championship Orioles team built on the Rutschman-Henderson-Cowser-Westburg-Kjerstad-Holliday foundation that in 2023 looked like, well, the foundation of a championship team. It has not developed that way. I hate to say that I was right. I was, though. That was our chance.

I am aware that, on paper, the group of prospects we got back from the Red Sox has more value than the next season and a half of Adley Rutschman. But we’ve had top pitching prospects before. That didn’t work out either. And who will be the first pitcher the new-look Orioles face as we go up against the Angels tonight? Who could it be but Grayson Rodriguez? Grayson will take the mound and look at the Orioles, and the Orioles will look at him, and both will think, this isn’t how it was supposed to turn out.

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.

Dean Kremer, the last Oriole among those acquired in the Manny Machado trade, is now a Twin. Almost the last link to the dark days of the rebuild. Ryan Mountcastle is still, tenuously here, and OK, there’s Keegan Akin, who I’d forgotten has been on this team since 2020.
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Kremer was never a star for the Orioles. But he was a guy who could more or less be counted on to show up and be a legitimate major league pitcher, so that every time he took the hill he was taking a start away from a guy of whom that could not be said. Guys like that have a lot of value. The Orioles seem to feel that, with Chris Bassitt coming back, and Cade Povich, Trey Gibson, and Nestor German ready to go, they have enough legitimate starting pitching. Does anybody have enough legitimate starting pitching? As for Trey Gibson, how did I only notice now that he has the first name of one recent Oriole and the surname of another? We should sign a Kyle Mancini.

The last time I saw Dean Kremer appear for the Orioles, he wasn’t pitching. He was catching the ceremonial first pitch, from Baltimore deli owner Marc Attman. The guy I was with said “Why’s Kremer out there? Usually a bullpen catcher does this.” Because he’s the only Jew on the team, of course! Attman’s Deli had a location in Cabin John Mall in the 2010s, my home mall, the place where as a kid I used to go to the Bagel Den, later the Delly Den. I’d always get their special, which was two mini-sandwiches, one corned beef, one pastrami. And play Space Invaders; they were the only restaurant my parents liked that had a machine, and it was tabletop style, true Space Invaders as far as I was concerned.

The game was this lousy affair, which the Orioles had many chances to win. Trevor Rogers had a sterling start but the Braves defense was equally sterling, keeping the Orioles from putting more than two runs on the board. Rico Garcia came in and Madison West High School’s own Drake Baldwin tied it up with the game’s first home run. Bottom of the 9th, the Braves intentionally walk Gunnar Henderson with two outs to load the bases — that’s a stupid move you should get punished for! Especially with Taylor Ward, who’s leading the league in walks, on deck, and a reliever who’s shaky finding the plate. 3-0 to Ward and it looks like we’re about to win. But Ward swings through one, then fouls one off, then grounds out on 3-2 and we go to extra innings. Everything falls apart, Matt Olson things happen, and we’re down 7-3. But then the bats finally wake up, or at least unexpected sensation Christian Encarnacion-Strand wakes up and hits a three-run homer (violating nominative determinism, as Tom points out, by consistently failing to strand baserunners.) But of course that’s it. Orioles lose 7-6. Weird team this year, playing roughly .500 ball but not seeming like a mediocre team at all, more like a team that’s great and terrible at once. O’s the Great and Terrible! Anyway, I got an XL Orioles Hawaiian shirt that I’m going to be wearing a lot. Anyway, I’m glad I made it to OPACY this year. Anyway, I wish there was a good place to get a pastrami sandwich in Madison, WI. Anyway, I’ll miss Dean Kremer.

The trade deadline for ancient civilizations is approaching. I am traded for Aeneas.

I had two pieces up this month, one in New Scientist and one in Quanta. I figured I’d give a bit of “bonus info” for both here.

The New Scientist piece covered a paper arguing that, according to the supernova evidence, there is no need for dark energy, because the universe isn’t speeding up after all. That’s a pretty dramatic claim, and it’s one the majority of cosmologists don’t agree with. But a small group has been hammering at the consensus.

Long-time followers of this blog have already heard of these folks. I criticized news coverage around a paper with one of the authors, Subir Sarkar, back in 2016, when he was only arguing that the supernova evidence was insufficient to establish dark energy, not that it was literally the wrong way around. He and his co-authors wanted an opportunity to state their case, so I posted their response a few weeks later. Later, I got to know Rameez, another author on the new paper. He ended up doing a guest post in 2019, on the next step in the story.

That story went from a statistical critique to an active proposal for something outside the consensus, in the form of inhomogeneous cosmology, the idea that the universe may be lumpier than typically assumed, with flowing currents that explain things otherwise attributed to exotic physics. That’s another topic I’ve covered before, but in this article I didn’t have the space to go into it.

Sarkar and Rameez are claiming something much more radical than most proponents of inhomogeneous cosmology, arguing against dark energy as a whole, not just more subtle effects. They told me a lot about their reasoning, most of which also didn’t make it into the piece.

It’s largely not going to make it here, either. Not due to space limitations: I’ve been pushing the blog post lengths you folks tolerate for a long time, no reason to stop now. Rather, it’s because I genuinely don’t feel qualified to judge this.

The issue is, cosmology is messy.

How do you figure out how the universe is expanding? You can look at supernovae, and use how dim they are to estimate how far away they are. The issue is, supernovae vary, even famous “standard candles”: they move in different ways, they shine differently depending on different things. Simulations aren’t nearly advanced enough to tell you all this from first principles, and so there are a wash of different corrections based largely on empirical observations, taking this or that correlation seriously and factoring them out of the measurement, while disregarding other correlations as spurious. Which of these combinations of corrections you trust determines whether the universe seems to be accelerating or decelerating.

It makes me glad I went into particle physics instead of cosmology, where you can mostly model everything from first-principles, and compare to your models.

Of course, even that can get you into trouble.

Take that Quanta piece. I wrote an update of the muon g-2 measurement. In some ways, this is an issue that many think of as already solved, by precisely that kind of first-principles modeling. Muons seemed to be violating the Standard Model, according to a calculation based in part on empirical formulas. Lattice QCD researchers figured out how to do that part of the prediction via a first-principles simulation instead, with enough accuracy that it could substitute for the empirical formulas. And lo and behold, the new prediction agreed with the experiment. Eventually, the experts accepted it, and all was well.

Except, as I commented at the time, not really. Because the empirical formulas were based on other experiments, colliding electrons and positrons. And as I now understand in much more detail, the disagreement between those electron-positron experiments is worryingly large.

So nobody is resting on their laurels. The lattice folks improved their calculation, with a result published in Nature this year that prompted Quanta to ask me to report on it. The people using the empirical method are still trying to sort out what happened. And the experimentalists are busy scrutinizing how they analyze their data, and collecting and analyzing more.

A few details that didn’t make it into the piece:

First: it goes beyond the level I was aiming for in the piece, but I really do want to emphasize the role of error estimates. Ultimately, every step in the story was not just one where experiments and predictions disagree, but one where they disagree past their reported error. The problem with disagreement between the new experiments isn’t that they disagree, it’s that they disagree so badly that they’re straining the statistical methods the people using the empirical method use to combine results together, so badly that if taken seriously, those methods would have to throw away ten years of progress of increasing precision. I don’t expect it, but I really hope for a postmortem in which we learn how to better estimate the kinds of experimental errors that can cause this. I haven’t seen that kind of postmortem for other results, so I don’t expect one here. But I have to believe that in the background someone is learning something, and getting better at this.

Second: one interesting question my editors raised was whether it was possible to do a first-principles calculation to compare with the electron-positron experiments. The answer is yes, but with some caveats. One will be recognizable to physicists: lattice QCD can only compute energy-integrated cross-sections, not measurements at particular energies like experiments find. But they can compute integrals with a modulating function…such as one strongly peaked at a particular energy. It’s a familiar type of trick for a quantum field theorist, though in this case it’s trickier than it sounds. I actually misunderstood, and thought that the same people I was talking to about the muon g-2 calculation had done that, and found it agreed with one of the electron-positron experiments over another. That’s not true: their comparisons were more indirect, while other groups attempting more direct comparisons haven’t quite gotten something good enough to do more than gesture at an agreement. So while the direction was correct, there are indeed lattice-based suggestions that favor one experiment over the others, the piece phrased things much too definitely. There’ll be a correction fixing this.

The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows.

Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse).

The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be derived from the Weierstrass preparation theorem, but is omitted here; see also Lemma 5 of this previous blog post.) Also, from the fundamental theorem of algebra, once the Jacobian polynomial is non-zero, it must be constant. So the hypothesis “Jacobian is a non-zero constant” can be replaced with “ is locally invertible”. So the Jacobian conjecture can be viewed as an assertion that local invertibility implies global invertibility. The complex numbers can be easily replaced with other fields of characteristic zero by the Lefschetz principle, but I prefer to work in the concrete setting of the complex numbers.

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well):

Theorem 2 (Counterexample to conjecture) There exists a polynomial which has non-zero constant Jacobian, but is not invertible.

The example can be stated completely explicitly: one can take

The example has since been retroactively explained in more geometric terms. As a “digestion” exercise to myself, I sought to write this explanation with relatively little use of algebraic geometry, in a manner that minimizes the amount of “miracles” required, although there are still a few places where some remarkable phenomena occur.

It is convenient to use the local injectivity formulation, and to generalize the domain to an equivalent affine variety. Namely, we will show

Theorem 3 (Counterexample, reformulated) There exists an affine variety that is isomorphic to by polynomial changes of variable, and a polynomial map which is locally injective, but not globally injective.

Clearly one can get from Theorem 3 to Theorem 2 by composing with the isomorphism and using the previously mentioned fact that local injectivity implies non-zero constant Jacobian. Our objective is now to find data , that obeys three separate properties:

• (a) is locally injective on .
• (b) is not globally injective on .
• (c) is isomorphic to by polynomial changes of variable.

(A pedantic remark: strictly speaking, in the arguments below, we not only replace the domain of by an equivalent variety , but also replace the range of by an equivalent variety . But the equivalence between and is a boring linear isomorphism ( will just be a hyperplane in a four-dimensional vector space ), so we do not highlight this aspect of the construction.)

It turns out that and can be built out of the operation of multiplication of low degree polynomials. Namely, consider the following three simple affine spaces:

• The space of linear homogeneous polynomials of two complex variables .
• The space of quadratic homogeneous polynomials of two complex variables .
• The space of cubic homogeneous polynomials of two complex variables .

The map , essentially a map from to , is clearly polynomial; it is given explicitly in coordinates as

• If one applies a scaling for some non-zero complex numbers , then the product is scaled by : .
• If one applies a change of variables for some invertible linear transformation , then the product is transformed by : .

The five-dimensional domain is of course larger than the four-dimensional range , so the map clearly cannot be injective. This can already be seen from the scaling symmetry, as the specific scalings

It will be convenient to “spend” the scaling symmetry to obtain a useful normalization. If is a linear polynomial and is a quadratic polynomial, the (homogeneous) resultant can be defined by the determinant

We now have a restricted multiplication map (which by abuse of notation we will continue to call ) from the four-dimensional variety

But we now also have property (a)! Suppose we want to show the local injectivity of in the neighborhood of a pair with . As the resultant is non-vanishing, the root of (which exists in the Riemann sphere, or projective line if you prefer) is distinct from the two roots of (though the latter two roots could be equal to each other). Applying the action (which performs Möbius transforms on the roots), one can assume without loss of generality that is the point at infinity (or equivalently ), thus for some complex number and for some complex numbers , with the resultant condition (7) simplifies to (so in particular are also non-zero). It is then clear that if one perturbs and by a small amount (say, modifying each coefficient by ), then the root of will perturb to something large (), while the roots of stay bounded. Thus, just from knowledge of the product , one can reconstruct which of the three roots of this cubic polynomial will be the perturbed root of , and which two will be the perturbed roots of ; from this and (6), (7) we can also reconstruct the leading coefficient of , and this completely determines both and . This establishes the local injectivity property (a). (In fact it is étale, but we will not need the machinery of étale maps here.)

Unfortunately, (the four-dimensional analogue of) condition (c) fails: the quadric hypersurface (8) is not isomorphic to the affine space . But we can try to get around this by passing to a three-dimensional slice. Let be some three-dimensional affine plane of (which we will take to avoid the origin for technical reasons), then we can restrict as a map from the set

Let’s see how. The affine hyperplanes in avoiding the origin are parameterized by the dual space of avoiding the origin, which one can think of as the non-zero third order homogeneous differential operators in two variables. Indeed, every such operator generates an affine hyperplane that avoids the origin, and conversely by duality every affine hyperplane avoiding the origin arises in this form uniquely. Just as the cubic polynomials in can be factored into three linear polynomials, the differential operators in the dual space can also be factored into three linear differential operators, e.g.,

• Operators where the three roots are all distinct, thus for independent first-order operators .
• Operators where two roots coincide and one is distinct, thus for independent first-order operators .
• Operators where all three roots coincide, thus for some first-order operator .

It turns out that the affine miracle for (9) occurs precisely in the second case, when has two identical roots. I do not have a completely satisfactory geometric explanation for this miracle, but one can verify it by the following coordinate computation.

By applying the action, we can normalize so that , thus is now the affine hyperplane of cubic polynomials with . Using (2) and (5), the variety (9) can now be described explicitly in coordinates as

So we just need to glue back in the fiber. Indeed, from (10) we see that the fiber at is just

The standard way to proceed here is to manipulate various tangent spaces using the modern machinery of algebraic geometry and commutative algebra, but given my own background, I prefer to adopt the language of analysis, and in particular big-O notation (in place of the ideals used in algebraic geometry), in order to investigate the limit by hand. On the variety (10), let us use to denote any multiple of by a polynomial expression in . Thus, for instance, the equation implies that

We can get some more precise asymptotics by also taking advantage of (15). Substituting into (15), we obtain after some algebra

Expanding the error term in (16) as , and doing a little more algebra, we thus have a polynomial change of variables

The previous computations, when expanded out, also gives polynomial inverse maps:

AI disclosure: I used an AI chatbot to discuss various aspects of this problem and to confirm several of the calculations made here.

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 higher 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?)

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.

First, some science, clearing out a number of papers and articles that I've been collecting for some time in my far-too-numerous browser tabs:

• I've written before (here) about dimensional analysis and similarity, techniques commonly used in the engineering world that can seem quasi-miraculous at times.  This paper gets into why this approach works, and different categories of physical similarity.  I'd mentioned it when the book came out, but this kind of thinking is also a key component of Anthony Zee's Fly by Night Physics.
• This review article is about fundamental limits in photonics and electromagnetics.  This is a very handy review that I'm going to point my students toward.
• On a lighter note, a colleague pointed me to this collection of (AI-generated) songs related to thermodynamics.    
• Speaking of thermodynamics, here is a recent paper about the thermodynamic description of wealth inequality (treating the flow of money in a physics formalism - see here for a prior discussion.).  Edging closer....  
• There was a nice post on substack about Wojciech Zurek's approach to decoherence in quantum mechanics (quantum darwinism).   Cleanly written.  
• Along these lines, this paper is a survey/guide to issues in quantum foundations and interpretations of quantum mechanics.
• Lastly, Jim Freericks has a new quantum mechanics textbook out (for free!), with the challenging idea of making the subject accessible without calculus or differential equations.  (I'm jealous.  My textbook's UK publisher would not let me use Steve Martin's quote for a chapter epigraph, but Prof. Freericks was able to do it - well played.)

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.

Just in case you’re up for a short talk at the top of Mount Wilson followed by an evening of observing through the historic telescopes on Saturday 25th July… this might be for you! Go to Mount Wilson Observatory’s website for more. –cvj

The post On top of the Mountain again appeared first on Asymptotia.

As an atheist, I started out frustrated by how little interest religious people had in debating their beliefs. Much of that was probably to do with how obnoxious it was to be “debated” by a socially awkward ten-year-old. But as I appreciate now, defending religious beliefs is just not a core activity for most religious people, even the experts. While there are many theologians who study the doctrines of their various religions, only a few engage in apologetics: arguments designed to convince people on the outside. The rest work within a particular religion, working out its implications.

There’s a similar, less-often-noticed behavior when it comes to interpretations of quantum mechanics.

Much like religions, there are many different interpretations of quantum mechanics, from people who envision a fundamentally undetermined world to those who picture a vast multiverse of all possibilities, to people who think quantum mechanics needs to be supplemented with faster-than-light signals, deterministic rules, or even consciousness. And as there isn’t yet any broad consensus for any of these options, you’d be forgiven for assuming that these people are trying their hardest to convince others that their interpretation is right.

But most of the work these people do is “quantum theology”, not “quantum apologetics”. The average paper connected to a quantum interpretation isn’t designed to convince people with different interpretations. It starts out with an interpretation in place, and tackles a more detailed question of what the interpretation should actually mean in practice. This work can be quite valuable and impressive…provided you’re already convinced. But if you’re not, and you see someone glowingly praise a paper on say the many-worlds interpretation, you might mistakenly think they’re on the cusp of closing the question of which interpretation is right, and not merely solving a technical issue within many-worlds.

Are there other areas of science with this pattern?

Let’s talk about string theory.

Physicists really started getting excited about string theory around forty years ago. Some people hear that number and wonder what happened. Have string theorists been looking for evidence for forty years and not found any? Isn’t that a huge waste of time?

That would be “string apologetics”. And these days, string apologetics is not actually that common. It’s a priority for some, to be sure, but most string theorists aren’t working on proving string theory. It’s clearly not an easy thing to do, and as a result, most people don’t spend their time on it.

Realizing that, some people assume that string theorists are actually practicing “string theology”, and get mad all over again. If string theorists are just assuming string theory and spending their time figuring out the “string versions” of known facts, then many would also deride their work as pointless.

But actually, string theology is also not very common. There are certainly some people who work to figure out the “string version” of this or that, or do research that only makes sense assuming string theory. But most of the string theory community doesn’t do that either.

Instead, they do something that, to continue the analogy, you might call “string pastoral care”.

Most theologians aren’t apologists, but similarly, most priests don’t spend their time doing theology. They use their training to advise their congregations on how to live their lives, solving day-to-day problems with some religious inspiration.

Similarly, most people these days who call themselves string theorists are working on more general questions about the types of theories used in particle physics. They’re doing this making use of their background in string theory, as an inspiration for solutions, a source of mathematical tools to solve problems, and a motivation for which questions are the most interesting. But if string theory turns out to be false, most of these peoples’ work will still be useful. It’s “string pastoral care”, used to solve problems for the people around them, not “string theology”.

Do you know any other fields that this applies to? Let me know in the comments!

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!

A week ago, a philosopher named Amit Hagar put out a preprint entitled The NISQ Trap: Eight Years of Demonstrations the Hardware was Built to Lose. Here’s the abstract:

With a single clear exception, every NISQ-era flagship demonstration of ‘quantum advantage’ has, within eighteen months of its announcement, been classically reproduced, shown to rest on classically tractable structure, or closed by a simulability theorem. Six theoretical results from 2024 through April 2026 explain the pattern: the regions of circuit-space NISQ hardware can run with sufficient fidelity coincide with the regions classical algorithms compress efficiently, because the features that admit one (low effective depth, strong algebraic structure, geometric locality) are the features that admit the other. This reading dates the NISQ programme from its 2018 articulation as an interim retreat from the unmet conditions of the 1996 threshold theorems, characterises the eight years that followed as a closed loop in which the demonstrations the hardware could run were drawn from regions classical methods could already reach, and locates the exit from the loop where the threshold theorems originally located it: in fault tolerance. The empirical pattern could in principle break with a demonstration that escapes the current simulability results. After eight years and more than thirty advantage-class announcements, the burden of producing such a demonstration falls to the defenders of NISQ.

You can also read some debates about the paper on SciRate here. I think it’s fair to say that the paper is purely polemical, without new ideas, and Pangram agrees with my suspicion (and that of a SciRate commenter) that significant portions of it are AI-generated.

Nevertheless, the basic thesis—that quantum supremacy in the NISQ (Noisy Intermediate Scale Quantum computing) era has been a failure, or even an example of pathological science—seems surprisingly widely shared, along with the opposite thesis that quantum computing already gives oodles of useful advantages for optimization and finance.

So it seems worth stating for the record that I have an extremely different view. I would say:

1. Sampling-based quantum supremacy experiments, including those based on Random Circuit Sampling and BosonSampling, passed the point about two years ago where, absent a breakthrough in classical algorithms, they quite clearly are beating what can easily be simulated on any existing classical computer. Hagar seems to claim that these experiments have been killed by the October 2025 paper Classical simulation of noisy random circuits from exponential decay of correlation, but he ignores that the algorithm from that paper still needs time that’s exponential in the circuit depth (see Theorem 2).
2. Indeed, simulating deep ~100-qubit random circuits, like those that Google and Quantinuum have now demonstrated experimentally, still seems pretty hopeless with any current classical method. This is particularly true for Quantinuum’s experiments, which had high enough gate fidelity to maintain a Linear Cross-Entropy score of order 1 (i.e., they’re no longer all that “noisy”). The central drawback of these experiments is no longer lack of confidence about quantum advantage; rather, it’s just that we only get samples as output, and directly verifying the quality of the samples seems just as intractable for a classical computer as spoofing the samples.
3. As of this past year, however, we have some strong candidates for verifiable quantum advantage. One is the Google OTOC experiment, as even Hagar himself acknowledges (that’s his “single clear exception”). A second is the simulations of the 2D Fermi-Hubbard model on Quantinuum and Google machines, like this one. The 1D Fermi-Hubbard model can be classically simulated pretty easily (see here for example), but the 2D one still presents challenges, meaning that in some regimes, the best available estimates of certain observables apparently now come from quantum computers. I wish I could write about other examples that will be public shortly.
4. Yes, the “real” goal remains, as it’s been since the 1990s, to build a scalable fault-tolerant quantum computer—and I’m glad that Hagar (unlike, say, Gil Kalai) never suggests that we’ve learned anything to rule that goal out. In the meantime, an intermediate goal would be to use NISQ devices to do physics and chemistry simulations that are commercially useful, or that help solve important scientific problems. The point of quantum supremacy experiments, you might say, is that by demonstrating the reality of quantum speedup about as clearly as it can be demonstrated with current hardware, they let us cleanly turn our attention to those more ambitious goals.

Anyway, my son and I need to catch a plane to Utah now, for the next iteration of the wonderful Epsilon Camp, where I’ll again be teaching theoretical computer science to 11- and 12-year-olds. But feel free to discuss in the comments! Nothing about world affairs in this thread please, just quantum supremacy.

Update (July 19): Not unrelated to the subject of this post, here’s a podcast I did with Gill Eapen of “Scientific Sense” about the current situation in quantum computing including recent experimental victories.

Mark Wise, the John A. McCone Professor of High Energy Physics at Caltech, passed away on July 10 at age 72. At a recent memorial service, John Preskill made these remarks.

I’m John Preskill, Mark’s colleague on the Caltech physics faculty for more than four decades. Our friendship goes back even farther. My wife Roberta and I met Mark and Jackie not long after they arrived at Harvard in 1980. We’ve been friends since then. We attended the bris for both Barry and Jonathan during those Harvard days. Mark and Jackie have two boys and we have two girls who are a few years younger, who were thrilled to connect with Barry and Jonathan when the families would get together for occasions like Passover or Hanukkah or Thanksgiving. When the kids were little, Mark and I would sometimes muse about the potential for forging even closer family ties if those relationships blossomed.

That didn’t happen. But Mark would preside at each Seder with a light hand, sprinkling the occasion with corny jokes as was his style, and Jackie would be determined to make it to the end of that customized family-friendly Haggadah she had meticulously prepared. The children, meanwhile, would be wondering when they’d be able to continue their game of sock baseball.

Many of you know that Mark was deeply dedicated to his family and friends. I’ll make some brief remarks about three facets of Mark I know especially well: Mark the scientist, Mark the teacher and mentor, and Mark the colleague and friend.

Because of his self-deprecating manner, those of you who are not scientists may not appreciate Mark’s stature as a physicist. He was one of the most influential figures in theoretical particle physics of his generation. It was not obvious things would turn out that way. Growing up in Toronto, Mark was an indifferent student, and his poor grades reflected that lack of interest. As a 9^th grader, though, it struck him that he better change his ways and figure out how to make something of his life. He liked sports — the possibility of being a professional athlete was briefly considered, but discarded. Somehow he decided that science would be a better fit. I’m not sure why — he had recently failed math. But he worked hard and had inspiring teachers, so by the time he finished high school Mark was an excellent student, and he sailed into the University of Toronto well prepared to major in physics,

At U of T, Mark came under the influence of a young professor, Nathan Isgur, who would later become his close research collaborator. Under Nathan’s guidance, Mark sought admission to the PhD programs of the most prestigious US research universities, intent on a career devoted to deep exploration of the fundamental laws of physics. He was rejected everywhere he applied. He should have been discouraged. But he wasn’t. Mark shrugged and said: “It’s okay. I’ll stay another year in Toronto, I’ll get a master’s degree, I’ll apply again and I’ll get in somewhere.” And that’s what happened. He went to Stanford, where, under the kind tutelage of Fred Gilman, Mark took off like a rocket. Hired to the Caltech faculty in 1982, he was a tenured full professor three years later at the age of 31, and appointed as the John A. McCone Professor of High Energy Physics while still in his 30s.

Mark liked action movies, such as those starring Arnold Schwarzenegger or Clint Eastwood. In serious moments, we would sometimes ponder together why we’re successful at what we do, and Mark would always quote Clint Eastwood as Harry Callahan in Magnum Force: “A man’s got to know his limitations.” We would both laugh, but those were words of wisdom. Mark understood what he did well as a research scientist and what he was less good at. Finding problems he could solve that would have interesting consequences for experiments that had been done or could be done was where he excelled – he did it again and again. Mark never lost his zest for calculating things, often by hand with pen and paper, his head resting on one arm with his glasses pushed up onto his forehead as he scribbled. Getting to an answer that was experimentally relevant never stopped giving him a thrill.

Mark also never lost his sense of appreciation for the teachers and mentors who had inspired and helped him. Perhaps that’s why he became such a dedicated teacher and mentor himself. It’s hard to impress Caltech students, but Mark’s lectures where extremely popular, not just for their pedagogical value but also for the humanity and humor he displayed. Students had to pay attention because otherwise one might miss the jokes, which inevitably became known as “Wisecracks.” There is even an account on X with the handle @MarkWiseSays, curated by students who want to preserve Mark’s pithy lessons in physics and in life.

For example, Mark might say: “If you really get depressed, I recommend diagonalizing a 2×2 matrix.” For physics students, this is both funny and sage advice. Or he might say. “This calculation will knock your socks off.” A cliché you might hear from anyone. But who besides Mark would then proceed to remove his shoes, rip off his socks, hurl them at the blackboard, put his shoes back on and resume lecturing?

Most famously, Mark would come to class with an ample supply of coins. He would ask the class questions, sometimes about physics and sometimes random trivia, rewarding a student who gave an answer Mark approved of by tossing a coin. At first the coins were quarters. But Mark, who had a scholarly interest in finance as well as physics, eventually felt that due to inflation he needed to upgrade to dollar coins. These are harder to come by, so it took frequent visits to the bank to make sure he wouldn’t run out. His antics in class made Mark human and approachable, and students responded. Mark felt that many Caltech students don’t fully realize how smart they are. He saw part of his job as building their self-confidence and relieving their stress.

As a colleague and mentor to graduate students and postdoctoral scholars, Mark was highly collaborative. He believed that interactions with others sparked his creativity. He was never at all pompous. I know this started early. When we were in the Harvard Society of Fellows we were obligated to have dinner with the Senior Fellows on Monday nights. It was a rather stuffy occasion. And, though I don’t think they do this anymore, after a sumptuous meal we would literally retire for brandy and cigars. Once, while puffing on his cigar after dinner, Mark had an inspiration. He gathered up a few junior fellows and led them to a theater for a movie he thought everyone should see right away. The movie was Conan the Barbarian. And everyone had a blast. That was a perfect Mark moment.

As the news about Mark has spread, accolades have poured in from physicists all over the world. He was admired not just for his scientific brilliance, but almost as much for his quirky sense of humor and his kindness. Mark was a wonderful friend to many of us. When you were with him, you were sure to laugh and feel good. He touched the lives of countless colleagues, students and friends. We miss him terribly but there are so many memories that we’ll cherish. We are all so very fortunate to have known and loved Professor Mark Wise.

I don’t think people who argue about reductionism are really arguing about reductionism.

Reductionism is the idea that the behavior of big, complicated things (people, economies, ecosystems) boils down to the behavior of their smallest constituents (molecules, atoms, subatomic particles). It’s often contrasted with emergence, the idea that new rules emerge in those big, complicated systems that are more than just the rules that govern the smallest scales.

Emergence can be divided into two kinds: weak and strong. In strong emergence, the big, complicated things have their own causal powers that aren’t due to smaller things at all. This tends to get mystical, with ideas like “lifeforce” and “consciousness”. Weak emergence is much milder, and while the big complicated things are best described by their own laws, in weak emergence they are in principle still caused by laws on smaller scales.

In practice, basically no-one believes in strong emergence: it seems way too much like magic for most scientists. And basically everyone believes in weak emergence: it would be nuts to insist that economists and biologists aren’t discovering important rules that would be almost impossible to find for someone who just had physics and chemistry to work with.

So if everyone agrees, what do people argue about?

Arguments about reductionism are really arguments about attitudes. If you position yourself as a reductionist, or an emergentist, you’re defending a particular way of thinking about the world, one that privileges one scale over another. When people argue for emergence, what they really seem to be doing is opposing a kind of “reductionist chauvinism”, where people like physicists insist that their perspective is the most valuable one.

And that’s understandable, because physicists can definitely be jerks sometimes. Let it be known I am no fan of jerks.

But I think it’s worth defending reductionism, not as a philosophy, but as an attitude or perspective. Worth arguing not about whether things in practice reduce or not, but about whether reduction is a good goal, about whether science that reduces more successfully is healthier science.

Because I think it is. And the reason boils down to agreement.

Simpler systems are less controversial systems. When we write down the laws that govern subatomic particles, they’re more definite: less heuristic, more precisely specified, with fewer exceptions. That isn’t to say there’s zero controversy in these subjects, there’s even controversy between mathematicians. But the more thoroughly you can boil something down to simple rules, the easier time you have of convincing others you’re right.

In contrast, the laws of the largest scales, like psychology and biology, are deeply heuristic. They thrive on exceptions and guidelines, general tendencies without clearly defined limits. And the problem with such laws is that they can lead to intractable arguments. Different schools of thought in psychology may simply never be able to convince each other, and may just have to wait for one to die off, the aesthetic feel of one set of ideas falling out of fashion as people become preoccupied with different sorts of problems.

Every time you can reduce, you avoid those insoluble disagreements. The simpler a system you can invoke, the more you can cooperate and build off each other’s work, the less time you have to waste disagreeing, the more you can accept people with different aesthetic and philosophical preferences as just different perspectives on what are ultimately the same facts. Reductionism is a technology for peace, and one of the most powerful we have.

So yeah, I’m a bit of a chauvinist about reductionism. That’s typical of an ex-physicist, sure. But it comes from a place of concern. I want a peaceful world, where we can learn from each other and find ways to agree. And reductionism is how I get there.

I believe that the creation of visualization apps to illustrate mathematical or scientific concepts is a particularly favorable use case for modern coding agents, as many of the downside risks attached to other LLM use cases are limited:

1. Not mission-critical. As such apps are not authorative sources of truth and only used for secondary purposes, a small positive error rate in the output can be acceptable.
2. Stand-alone. As the applets are not destined to be incorporated into a larger codebase or literature, the technical debt incurred by delegating all the coding to an LLM agent is bounded.
3. End product is deterministic (and sandboxed). As the applets run on a deterministic language (Javascript), are sandboxed against file or internet access, and do not make any LLM calls at run-time, security and privacy concerns are minimal, and the applet can be maintained without continued premium LLM access or resource-intensive compute.
4. Not replacing primary skills. While deskilling is the tradeoff one accepts when relying on these tools to accelerate output, I am perfectly willing to forego the opportunity to keep my Javascript skills at a high level, as this is a tertiary skill for me at best in my chosen profession. (I continue to manually program in Lean and in Python to keep in practice with programming in general.)
5. Not competing with humans. To my knowledge, there is no existing human effort that is being duplicated by these applets (the activity in this direction appears to have peaked two decades ago).

I would however caution against unrestricted LLM use when one or more of the above five favorable situations is not in effect.

With these points in mind, I have used such an agent to create two further apps. The first app illustrates the “zeta process” that was introduced in my recent paper with Alexeev, Barreto, Li, Lichtman, Price, Shah, and Tang, though it was first discovered by an AI. For each , the zeta distribution is a random natural number with distribution

It has long been known that this distribution has good number-theoretic properties: for instance, the number of times a given prime divides has a geometric distribution of mean . However, the new observation is that these random variables can be chained together into a single stochastic process, which we call the “zeta process”, which is an infinite divisibility chain. I used an agent to create an app to visualize this process:

The underlying process is generated by several exponential random variables at each prime: in the above instantiation of the process, two such variables are visible at the prime , and one variable at the primes . At a given choice of , is formed by collecting all the variables below this threshold (and for which all predecessors also lie below the threshold); in the above illustration, this amounts to one variable at each of the primes , leading to in this case. Additional visualizations in the app display the distribution of each , as well as the distribution of the hitting probability , which among other things can be used to give a quick solution to Erdős problem #1196.

The second app is rather different in nature, and is a somewhat whimsical attempt to display the motion of the heavens, both at “human” scales of space and time, and at more “astronomical” scales (in which the motion of the planets in particular are more apparent). It is very loosely inspired by the game “Katamari Damacy“, in which one absorbs both terrestrial and celestial objects of many different scales. Here is how the app typically looks at a human scale:

And here is how it looks when one’s perspective leaves the Earth’s atmosphere:

(As I did not want to render an entire explorable world in this app, the observer in the app is only limited to changing his or her size, from a human to a creature of comparable size to the Earth itself; they cannot move horizontally on the planet.) At the largest scales of space and time, the classic orrery diagram appears:

After lengthy conversations with the agent, I was able to implement many astronomical phenomena, including phases of the Moon, the effect of Earth’s rotation against the fixed stars (though one can also stabilize one’s view against those stars to see the Earth’s rotation more directly), and so forth.

Here at the National Academy of Sciences, it seems that my first job is to serve on the selection committee for the prestigious Michael and Sheila Held Prize in combinatorial and discrete optimization and related areas. The committee chair, my former MIT colleague Madhu Sudan (now at Harvard), invited me to share the following message here on Shtetl-Optimized. (I’d add: put in the effort to nominate someone, and you can actually influence how things go!)

Dear Colleagues

I am writing to seek nominations for the 2027 Michael and Sheila Held Prize. The scope of the prize and nomination needs are described below. If you intend to nominate someone I would appreciate a heads up by email to madhu@cs.harvard.edu one month before the deadline (so email by Sept 8, 2026) to let me know your nomination is coming. (We may also reach out to you in response to coordinate multiple/overlapping nominations.)

The Held prize honors outstanding, innovative, creative, and influential research in the areas of combinatorial and discrete optimization, or related parts of computer science, such as the design and analysis of algorithms and complexity theory. This $100,000 prize is intended to recognize recent work (defined as published within the last eight years, i.e., on or after October 6, 2018).

All nominations must be submitted online by Monday, October 5, 2026 and include:

1. Nomination letter describing the candidate’s work and why he or she should be selected for the award. No more than three (3) pages.

2. Curriculum vitae. No more than two (2) pages.

3. Bibliography listing no more than twelve (12) of the nominee’s most significant publications.

4. Suggested citation. A 50-word summary stating why the nominee should be considered for this award.

5. Two letters of support. No more than one letter of support can be written by someone of the same primary work institution as the nominee.

The Held Prize is given to a person or a set of persons, as supported by a paper or a body of work. Unless otherwise stated, preference will be given to scientists who may be earlier in their careers or those whose work has not been recognized by other prizes or awards. Nomination restrictions can be found here. Joint nominations will only be considered when nominees have collaborated closely on the paper to be recognized by the award. If nominating multiple individuals for a paper with additional authors, please clearly explain the reason for nominating those chosen, as well as the reason for excluding other collaborators, if applicable. 

Please feel free to circulate this call further within your department

Best
Madhu Sudan, on behalf of The Michael and Sheila Held Prize Selection Committee

And while I have your attention, a second CS theory announcement: David Soloveichik, my wonderful friend and colleague in UT Austin’s Electrical and Computer Engineering Department, has funding for a postdoc for 1-2 years, to work on the thermodynamics of computation here at UT. This is a topic that I’ve been trying to learn more about as well, so I might get involved too! David writes, “the big picture is to think of thermodynamics (energy dissipation / entropy production) as CS complexity measures like time and space usage.” If you’re on the postdoc market and this sounds potentially up your alley, email David to learn more.

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 $\text{Spin}(n) \to \text{Spin}(n+1)$:

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