On December 10, I gave a keynote address at the Q2B 2025 Conference in Silicon Valley. This is a transcript of my remarks. The slides I presented are here.

The first century

We are nearing the end of the International Year of Quantum Science and Technology, so designated to commemorate the 100^th anniversary of the discovery of quantum mechanics in 1925. The story goes that 23-year-old Werner Heisenberg, seeking relief from severe hay fever, sailed to the remote North Sea Island of Helgoland, where a crucial insight led to his first, and notoriously obscure, paper describing the framework of quantum mechanics.

In the years following, that framework was clarified and extended by Heisenberg and others. Notably among them was Paul Dirac, who emphasized that we have a theory of almost everything that matters in everyday life. It’s the Schrödinger equation, which captures the quantum behavior of many electrons interacting electromagnetically with one another and with atomic nuclei. That describes everything in chemistry and materials science and all that is built on those foundations. But, as Dirac lamented, in general the equation is too complicated to solve for more than a few electrons.

Somehow, over 50 years passed before Richard Feynman proposed that if we want a machine to help us solve quantum problems, it should be a quantum machine, not a classical machine. The quest for such a machine, he observed, is “a wonderful problem because it doesn’t look so easy,” a statement that still rings true.

I was drawn into that quest about 30 years ago. It was an exciting time. Efficient quantum algorithms for the factoring and discrete log problems were discovered, followed rapidly by the first quantum error-correcting codes and the foundations of fault-tolerant quantum computing. By late 1996, it was firmly established that a noisy quantum computer could simulate an ideal quantum computer efficiently if the noise is not too strong or strongly correlated. Many of us were then convinced that powerful fault-tolerant quantum computers could eventually be built and operated.

Three decades later, as we enter the second century of quantum mechanics, how far have we come? Today’s quantum devices can perform some tasks beyond the reach of the most powerful existing conventional supercomputers. Error correction had for decades been a playground for theorists; now informative demonstrations are achievable on quantum platforms. And the world is investing heavily in advancing the technology further.

Current NISQ machines can perform quantum computations with thousands of two-qubit gates, enabling early explorations of highly entangled quantum matter, but still with limited commercial value. To unlock a wide variety of scientific and commercial applications, we need machines capable of performing billions or trillions of two-qubit gates. Quantum error correction is the way to get there.

I’ll highlight some notable developments over the past year—among many others I won’t have time to discuss. (1) We’re seeing intriguing quantum simulations of quantum dynamics in regimes that are arguably beyond the reach of classical simulations. (2) Atomic processors, both ion traps and neutral atoms in optical tweezers, are advancing impressively. (3) We’re acquiring a deeper appreciation of the advantages of nonlocal connectivity in fault-tolerant protocols. (4) And resource estimates for cryptanalytically relevant quantum algorithms have dropped sharply.

Quantum machines for science

A few years ago, I was not particularly excited about running applications on the quantum platforms that were then available; now I’m more interested. We have superconducting devices from IBM and Google with over 100 qubits and two-qubit error rates approaching 10^{-3}. The Quantinuum ion trap device has even better fidelity as well as higher connectivity. Neutral-atom processors have many qubits; they lag behind now in fidelity, but are improving.

Users face tradeoffs: The high connectivity and fidelity of ion traps is an advantage, but their clock speeds are orders of magnitude slower than for superconducting processors. That limits the number of times you can run a given circuit, and therefore the attainable statistical accuracy when estimating expectations of observables.

Verifiable quantum advantage

Much attention has been paid to sampling from the output of random quantum circuits, because this task is provably hard classically under reasonable assumptions. The trouble is that, in the high-complexity regime where a quantum computer can reach far beyond what classical computers can do, the accuracy of the quantum computation cannot be checked efficiently. Therefore, attention is now shifting toward verifiable quantum advantage — tasks where the answer can be checked. If we solved a factoring or discrete log problem, we could easily check the quantum computer’s output with a classical computation, but we’re not yet able to run these quantum algorithms in the classically hard regime. We might settle instead for quantum verification, meaning that we check the result by comparing two quantum computations and verifying the consistency of the results.

A type of classical verification of a quantum circuit was demonstrated recently by BlueQubit on a Quantinuum processor. In this scheme, a designer builds a family of so-called “peaked” quantum circuits such that, for each such circuit and for a specific input, one output string occurs with unusually high probability. An agent with a quantum computer who knows the circuit and the right input can easily identify the preferred output string by running the circuit a few times. But the quantum circuits are cleverly designed to hide the peaked output from a classical agent — one may argue heuristically that the classical agent, who has a description of the circuit and the right input, will find it hard to predict the preferred output. Thus quantum agents, but not classical agents, can convince the circuit designer that they have reliable quantum computers. This observation provides a convenient way to benchmark quantum computers that operate in the classically hard regime.

The notion of quantum verification was explored by the Google team using Willow. One can execute a quantum circuit acting on a specified input, and then measure a specified observable in the output. By repeating the procedure sufficiently many times, one obtains an accurate estimate of the expectation value of that output observable. This value can be checked by any other sufficiently capable quantum computer that runs the same circuit. If the circuit is strategically chosen, then the output value may be very sensitive to many-qubit interference phenomena, in which case one may argue heuristically that accurate estimation of that output observable is a hard task for classical computers. These experiments, too, provide a tool for validating quantum processors in the classical hard regime. The Google team even suggests that such experiments may have practical utility for inferring molecular structure from nuclear magnetic resonance data.

Correlated fermions in two dimensions

Quantum simulations of fermionic systems are especially compelling, since electronic structure underlies chemistry and materials science. These systems can be hard to simulate in more than one dimension, particularly in parameter regimes where fermions are strongly correlated, or in other words profoundly entangled. The two-dimensional Fermi-Hubbard model is a simplified caricature of two-dimensional materials that exhibit high-temperature superconductivity and hence has been much studied in recent decades. Large-scale tensor-network simulations are reasonably successful at capturing static properties of this model, but the dynamical properties are more elusive.

Dynamics in the Fermi-Hubbard model has been simulated recently on both Quantinuum (here and here) and Google processors. Only a 6 x 6 lattice of electrons was simulated, but this is already well beyond the scope of exact classical simulation. Comparing (error-mitigated) quantum circuits with over 4000 two-qubit gates to heuristic classical tensor-network and Majorana path methods, discrepancies were noted, and the Phasecraft team argues that the quantum simulation results are more trustworthy. The Harvard group also simulated models of fermionic dynamics, but were limited to relatively low circuit depths due to atom loss. It’s encouraging that today’s quantum processors have reached this interesting two-dimensional strongly correlated regime, and with improved gate fidelity and noise mitigation we can go somewhat further, but expanding system size substantially in digital quantum simulation will require moving toward fault-tolerant implementations. We should also note that there are analog Fermi-Hubbard simulators with thousands of lattice sites, but digital simulators provide greater flexibility in the initial states we can prepare, the observables we can access, and the Hamiltonians we can reach.

When it comes to many-particle quantum simulation, a nagging question is: “Will AI eat quantum’s lunch?” There is surging interest in using classical artificial intelligence to solve quantum problems, and that seems promising. How will AI impact our quest for quantum advantage in this problem space? This question is part of a broader issue: classical methods for quantum chemistry and materials have been improving rapidly, largely because of better algorithms, not just greater processing power. But for now classical AI applied to strongly correlated matter is hampered by a paucity of training data.  Data from quantum experiments and simulations will likely enhance the power of classical AI to predict properties of new molecules and materials. The practical impact of that predictive power is hard to clearly foresee.

The need for fundamental research

Today is December 10^th, the anniversary of Alfred Nobel’s death. The Nobel Prize award ceremony in Stockholm concluded about an hour ago, and the Laureates are about to sit down for a well-deserved sumptuous banquet. That’s a fitting coda to this International Year of Quantum. It’s useful to be reminded that the foundations for today’s superconducting quantum processors were established by fundamental research 40 years ago into macroscopic quantum phenomena. No doubt fundamental curiosity-driven quantum research will continue to uncover unforeseen technological opportunities in the future, just as it has in the past.

I have emphasized superconducting, ion-trap, and neutral atom processors because those are most advanced today, but it’s vital to continue to pursue alternatives that could suddenly leap forward, and to be open to new hardware modalities that are not top-of-mind at present. It is striking that programmable, gate-based quantum circuits in neutral-atom optical-tweezer arrays were first demonstrated only a few years ago, yet that platform now appears especially promising for advancing fault-tolerant quantum computing. Policy makers should take note!

The joy of nonlocal connectivity

As the fault-tolerant era dawns, we increasingly recognize the potential advantages of the nonlocal connectivity resulting from atomic movement in ion traps and tweezer arrays, compared to geometrically local two-dimensional processing in solid-state devices. Over the past few years, many contributions from both industry and academia have clarified how this connectivity can reduce the overhead of fault-tolerant protocols.

Even when using the standard surface code, the ability to implement two-qubit logical gates transversally—rather than through lattice surgery—significantly reduces the number of syndrome-measurement rounds needed for reliable decoding, thereby lowering the time overhead of fault tolerance. Moreover, the global control and flexible qubit layout in tweezer arrays increase the parallelism available to logical circuits.

Nonlocal connectivity also enables the use of quantum low-density parity-check (qLDPC) codes with higher encoding rates, reducing the number of physical qubits needed per logical qubit for a target logical error rate. These codes now have acceptably high accuracy thresholds, practical decoders, and—thanks to rapid theoretical progress this year—emerging constructions for implementing universal logical gate sets. (See for example here, here, here, here.)

A serious drawback of tweezer arrays is their comparatively slow clock speed, limited by the timescales for atom transport and qubit readout. A millisecond-scale syndrome-measurement cycle is a major disadvantage relative to microsecond-scale cycles in some solid-state platforms. Nevertheless, the reductions in logical-gate overhead afforded by atomic movement can partially compensate for this limitation, and neutral-atom arrays with thousands of physical qubits already exist.

To realize the full potential of neutral-atom processors, further improvements are needed in gate fidelity and continuous atom loading to maintain large arrays during deep circuits. Encouragingly, active efforts on both fronts are making steady progress.

Approaching cryptanalytic relevance

Another noteworthy development this year was a significant improvement in the physical qubit count required to run a cryptanalytically relevant quantum algorithm, reduced by Gidney to less than 1 million physical qubits from the 20 million Gidney and Ekerå had estimated earlier. This applies under standard assumptions: a two-qubit error rate of 10^{-3} and 2D geometrically local processing. The improvement was achieved using three main tricks. One was using approximate residue arithmetic to reduce the number of logical qubits. (This also suppresses the success probability and therefore lengthens the time to solution by a factor of a few.) Another was using a more efficient scheme to reduce the number of physical qubits for each logical qubit in cold storage. And the third was a recently formulated scheme for reducing the spacetime cost of non-Clifford gates. Further cost reductions seem possible using advanced fault-tolerant constructions, highlighting the urgency of accelerating migration from vulnerable cryptosystems to post-quantum cryptography.

Looking forward

Over the next 5 years, we anticipate dramatic progress toward scalable fault-tolerant quantum computing, and scientific insights enabled by programmable quantum devices arriving at an accelerated pace. Looking further ahead, what might the future hold? I was intrigued by a 1945 letter from John von Neumann concerning the potential applications of fast electronic computers. After delineating some possible applications, von Neumann added: “Uses which are not, or not easily, predictable now, are likely to be the most important ones … they will … constitute the most surprising extension of our present sphere of action.” Not even a genius like von Neumann could foresee the digital revolution that lay ahead. Predicting the future course of quantum technology is even more hopeless because quantum information processing entails an even larger step beyond past experience.

As we contemplate the long-term trajectory of quantum science and technology, we are hampered by our limited imaginations. But one way to loosely characterize the difference between the past and the future of quantum science is this: For the first hundred years of quantum mechanics, we achieved great success at understanding the behavior of weakly correlated many-particle systems, leading for example to transformative semiconductor and laser technologies. The grand challenge and opportunity we face in the second quantum century is acquiring comparable insight into the complex behavior of highly entangled states of many particles, behavior well beyond the scope of current theory or computation. The wonders we encounter in the second century of quantum mechanics, and their implications for human civilization, may far surpass those of the first century. So we should gratefully acknowledge the quantum pioneers of the past century, and wish good fortune to the quantum explorers of the future.

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Individual physicists don’t ask for a lot for Newtonmas. Big collaborations ask for more.

This year, CERN got its Newtonmas gift early: a one billion dollar pledge from a group of philanthropists and foundations, to be spent on their proposed new particle collider.

That may sound like a lot of money (and of course it is), but it’s only a fraction of the 15 billion euros that the collider is estimated to cost. That makes this less a case of private donors saving the project, and more of a nudge, showing governments they can get results for a bit cheaper than they expected.

I do wonder if the donation has also made CERN more bold about their plans, since it was announced shortly after a report from the update process for the European Strategy for Particle Physics, in which the European Strategy Group recommended a backup plan for the collider that is just the same collider with 15% budget cuts. Naturally people started making fun of this immediately.

There were more serious objections from groups that had proposed more specific backup plans earlier in the process, who are frustrated that their ideas were rejected in favor of a 15% tweak that was not even discussed and seems not to really have been evaluated.

I don’t have any special information about what’s going on behind the scenes, or where this is headed. But I’m amused, and having fun with the parallels this season. I remember writing lists as a kid, trying to take advantage of the once-a-year opportunity to get what seemed almost like a genie’s wish. Whatever my incantations, the unreasonable requests were never fulfilled. Still, I had enough new toys to fill my time, and whet my appetite for the next year.

We’ll see what CERN’s Newtonmas gift brings.

The traditional celebration is Chinese food and a movie, so in that spirit: AB and I tried a new kind of dumpling on our recent trip to New York, sheng jian bao. These are soup dumplings, but with a thicker, crispier wrapper than the xiao long bao I’m more familiar with. Excellent!

As for a movie, we all re-watched Trading Places last night. Except it turns out Dr. Mrs. Q has never actually seen Trading Places! Arguably the very greatest Christmas movie.

Merry Christmas, everyone! Ho³!

Here’s my beloved daughter baking chocolate chip cookies, which she’ll deliver tomorrow morning with our synagogue to firemen, EMTs, and others who need to work on Christmas Day. My role was limited to taste-testing.

While (I hope you’re sitting down for this) the Aaronson-Moshkovitzes are more of a latke/dreidel family, I grew up surrounded by Christmas and am a lifelong enjoyer of the decorations, the songs and movies (well, some of them), the message of universal goodwill, and even gingerbread and fruitcake.

Therefore, as a Christmas gift to my readers, I hereby present what I now regard as one of the great serendipitous “discoveries” in my career, alongside students like Paul Christiano and Ewin Tang who later became superstars.

Ever since I was a pimply teen, I dreamed of becoming the prophet who’d finally bring the glories of theoretical computer science to the masses—who’d do for that systematically under-sung field what Martin Gardner did for math, Carl Sagan for astronomy, Richard Dawkins for evolutionary biology, Douglas Hofstadter for consciousness and Gödel. Now, with my life half over, I’ve done … well, some in that direction, but vastly less than I’d dreamed.

A month ago, I learned that maybe I can rest easier. For a young man named Aaron Gostein is doing the work I wish I’d done—and he’s doing it using tools I don’t have, and so brilliantly that I could barely improve a pixel.

Aaron recently graduated from Carnegie Mellon, majoring in CS. He’s now moved back to Austin, TX, where he grew up, and where of course I now live as well. (Before anyone confuses our names: mine is Scott Aaronson, even though I’ve gotten hundreds of emails over the years calling me “Aaron.”)

Anyway, here in Austin, Aaron is producing a YouTube channel called PurpleMind. In starting this channel, Aaron was directly inspired by Grant Sanderson’s 3Blue1Brown—a math YouTube channel that I’ve also praised to the skies on this blog—but Aaron has chosen to focus on theoretical computer science.

I first encountered Aaron a month ago, when he emailed asking to interview me about … which topic will it be this time, quantum computing and Bitcoin? quantum computing and AI? AI and watermarking? no, diagonalization as a unifying idea in mathematical logic. That got my attention.

So Aaron came to my office and we talked for 45 minutes. I didn’t expect much to come of it, but then Aaron quickly put out this video, in which I have a few unimportant cameos:

After I watched this, I brought Dana and the kids and even my parents to watch it too. The kids, whose attention spans normally leave much to be desired, were sufficiently engaged that they made me pause every 15 seconds to ask questions (“what would go wrong if you diagonalized a list of all whole numbers, where we know there are only ℵ₀ of them?” “aren’t there other strategies that would work just as well as going down the diagonal?”).

Seeing this, I sat the kids down to watch more PurpleMind. Here’s the video on the P versus NP problem:

Here’s one on the famous Karatsuba algorithm, which reduced the number of steps needed to multiply two n-digit numbers from ~n² to only ~n^1.585, and thereby helped inaugurate the entire field of algorithms:

Here’s one on RSA encryption:

Here’s one on how computers quickly generate the huge random prime numbers that RSA and other modern encryption methods need:

These are the only ones we’ve watched so far. Each one strikes me as close to perfection. There are many others (for example, on Diffie-Hellman encryption, the Bernstein-Vazirani quantum algorithm, and calculating pi) that I’m guessing will be equally superb.

In my view, what makes these videos so good is their concreteness, achieved without loss of correctness. When, for example, Aaron talks about Gödel mailing a letter to the dying von Neumann posing what we now know as P vs. NP, or any other historical event, he always shows you an animated reconstruction. When he talks about an algorithm, he always shows you his own Python code, and what happened when he ran the code, and then he invites you to experiment with it too.

I might even say that the results singlehandedly justify the existence of YouTube, as the ten righteous men would’ve saved Sodom—with every crystal-clear animation of a CS concept canceling out a thousand unboxing videos or screamingly-narrated Minecraft play-throughs in the eyes of God.

Strangely, the comments below Aaron’s YouTube videos attack him relentlessly for his use of AI to help generate the animations. To me, it seems clear that AI is the only thing that could let one person, with no production budget to speak of, create animations of this quality and quantity. If people want so badly for the artwork to be 100% human-generated, let them volunteer to create it themselves.

Even as I admire the PurpleMind videos, or the 3Blue1Brown videos before them, a small part of me feels melancholic. From now until death, I expect that I’ll have only the same pedagogical tools that I acquired as a young’un: talking; waving my arms around; quizzing the audience; opening the floor to Q&A; cracking jokes; drawing crude diagrams on a blackboard or whiteboard until the chalk or the markers give out; typing English or LaTeX; the occasional PowerPoint graphic that might (if I’m feeling ambitious) fade in and out or fly across the screen.

Today there are vastly better tools, both human and AI, that make it feasible to create spectacular animations for each and every mathematical concept, as if transferring the imagery directly from mind to mind. In the hands of a master explainer like Grant Sanderson or Aaron Gostein, these tools are tractors to my ox-drawn plow. I’ll be unable to compete in the long term.

But then I reflect that at least I can help this new generation of math and CS popularizers, by continuing to feed them raw material. I can do cameos in their YouTube productions. Or if nothing else, I can bring their jewels to my community’s attention, as I’m doing right now.

Peace on Earth, and to all a good night.

For those in the US:  Here is a link for submitting a public comment regarding OSTP's request for information about "Accelerating the American Scientific Enterprise".   The US government does this at some rate, requesting public feedback and input on policies under consideration or development.  Unfortunately, this public comment period ends on December 26.  

Here is the slightly longer version of what I submitted (I had to trim it back b/c of character limits).  It doesn't touch on everything in the RFI, but it's a start.  Many readers will likely disagree with what I wrote, and that's fine.  Submit your own comments.  Better that OSTP hear from a variety of perspectives.

Update:  I’ve had some ask me what the point of this is, since the comments will likely not be read in a meaningful way.  I assume they will feed the comments through an AI tool and have staffers spot check ones that are flagged as interesting (by whatever criteria). I know I’m old fashioned, but I still feel like silence is acquiescence edging toward complicity. Saying something at least helped me organize my ideas a bit.

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I appreciate the desire and need to improve the American scientific enterprise. Other rival nations are making enormous investments in basic and applied scientific research, and it is critical for US competitiveness that the US not flag or falter.

I have two overarching suggestions before delving into the specific questions posed in the RFI.  

First, it is of vital importance that the US scientific enterprise solicit advice and input from practicing scientists and engineers through advisory panels and committees.  In the last year, the US government research agencies have dissolved or disbanded a huge fraction of advisory bodies.  I think this is a mistake.  While there are always concerns about whether experts are too entrenched to be bold, across-the-board devaluing expertise is not the way to go.

Second, there needs to be coherence between priorities and financial allocations.  Complaining that the US is falling behind in critical research areas (e.g. advanced materials) while simultaneously slashing federal support of those research areas is both self-defeating and internally inconsistent.  There have been large scale cuts across many agencies in critical scientific areas, seemingly at odds with the stated policies about innovation and competitiveness.  

Regarding specific points from the RFI:

Encouraging public-private collaboration runs into the problem that existing economic incentives (the constant need for short-term returns) disfavor companies investing in research that doesn't have clear, nearly-immediate payoffs.  Norm Augustine has spoken publicly about how opening a research laboratory caused Lockheed Martin's stock to fall. (see here:  https://www.aaas.org/news/basic-research-needs-sustained-federal-investment-says-norman-augustine)  Moreover, since many very wealthy corporations (e.g. Tesla, 3M) already pay almost no federal taxes, it is challenging to encourage public-private partnerships through offering companies tax incentives.  The issue is less of a problem for small and startup companies, in part thanks to programs like SBIR and STTR.  Programs like ARPA-E and ARPA-H also are important to consider and expand.

The most financially well-off sectors, all of which depend critically on a spectrum of research from basic to the applied and near to long term, are the financial industry, energy, biomedical/pharma, and tech/semiconductors (especially tied into AI).  In the past, research consortia like International Sematech were very effective at guiding research of tremendous economic benefit.  Some state governments have had interesting and impactful programs (e.g., the Cancer Prevention Research Institute in Texas, supported by state bonds).  The US could encourage the development of industrial consortia aligned to specific aspects of research; these could be supported financially by joint investment of the constituting companies and special bond issues expressly for these topics, with the proviso that funds be used to support use-inspired basic and early-stage applied research.  It is critical that any such plan have actual oversight, to make sure that this does not become a giant give-away to the companies that have the best lobbyists.

In terms of supporting models for research that complement traditional university structures and enable projects that require vast resources, interdisciplinary coordination, or extended timelines:  There are already large-scale center/institute programs that have been deployed in the past by various federal agencies.  While not perfect, these are not a bad model - in the last year, these have suffered greatly due to budgetary uncertainties and outright cancellations.  You can't complain about failing to do research requiring extended timelines if funding is annualized and perpetually at risk.  One approach to dealing with this would be to endow such structures at the outset for multiyear support. Congress would have to be willing to do this, and given the level of impasse that seems very difficult, but with executive branch support it could be possible.  The short version: Don't complain about lack of long-term stickiness if the US government cannot be considered a reliable partner year over year.

In terms of identifying and developing scientific talent across the country:  the RFI mentions "leveraging digital tools", which sounds a lot like trying to use AI to identify people.  This is frought with risk.  Programs such as the NSF graduate research fellowship, the NDSEG, the SCGSR, and others are essential at developing talent, as are many research experience for undergraduate programs and internship programs at national laboratories.  The recent cuts to these programs are short-sighted and counterproductive.  If we want the US to develop top technical talent and have people choose technical paths over "easier" trajectories, then we need to make it financially worthwhile for students to pursue these aims.  Federally supported undergraduate scholarship, graduate fellowships, policies that reward companies for providing internships, policies that, e.g. reduce endowment taxes if university resources are spent providing such support, etc. are all worth considering.  Again, an advisory task force could likely provide quantitative information about what strategies are likely to work best.

I could go on, but these are a starting point for a discussion.  I would be happy to talk further with interested parties and engage in discussions going forward, if that would be helpful.

Problem 1026 on the Erdős problem web site recently got solved through an interesting combination of existing literature, online collaboration, and AI tools. The purpose of this blog post is to try to tell the story of this collaboration, and also to supply a complete proof.

The original problem of Erdős, posed in 1975, is rather ambiguous. Erdős starts by recalling his famous theorem with Szekeres that says that given a sequence of distinct real numbers, one can find a subsequence of length which is either increasing or decreasing; and that one cannot improve the to , by considering for instance a sequence of blocks of length , with the numbers in each block decreasing, but the blocks themselves increasing. He also noted a result of Hanani that every sequence of length can be decomposed into the union of monotone sequences. He then wrote “As far as I know the following question is not yet settled. Let be a sequence of distinct numbers, determine

This problem was added to the Erdős problem site on September 12, 2025, with a note that the problem was rather ambiguous. For any fixed , this is an explicit piecewise linear function of the variables that could be computed by a simple brute force algorithm, but Erdős was presumably seeking optimal bounds for this quantity under some natural constraint on the . The day the problem was posted, Desmond Weisenberg proposed studying the quantity , defined as the largest constant such that

Though not stated on the web site, one can formulate this problem in game theoretic terms. Suppose that Alice has a stack of coins for some large . She divides the coins into piles of consisting of coins each, so that . She then passes the piles to Bob, who is allowed to select a monotone subsequence of the piles (in the weak sense) and keep all the coins in those piles. What is the largest fraction of the coins that Bob can guarantee to keep, regardless of how Alice divides up the coins? (One can work with either a discrete version of this problem where the are integers, or a continuous one where the coins can be split fractionally, but in the limit the problems can easily be seen to be equivalent.)

AI-generated images continue to be problematic for a number of reasons, but here is one such image that somewhat manages at least to convey the idea of the game:

For small , one can work out by hand. For , clearly : Alice has to put all the coins into one pile, which Bob simply takes. Similarly : regardless of how Alice divides the coins into two piles, the piles will either be increasing or decreasing, so in either case Bob can take both. The first interesting case is . Bob can again always take the two largest piles, guaranteeing himself of the coins. On the other hand, if Alice almost divides the coins evenly, for instance into piles for some small , then Bob cannot take all three piles as they are non-monotone, and so can only take two of them, allowing Alice to limit the payout fraction to be arbitrarily close to . So we conclude that .

An hour after Desmond’s comment, Stijn Cambie noted (though not in the language I used above) that a similar construction to the one above, in which Alice divides the coins into pairs that are almost even, in such a way that the longest monotone sequence is of length , gives the upper bound . It is also easy to see that is a non-increasing function of , so this gives a general bound . Less than an hour after that, Wouter van Doorn noted that the Hanani result mentioned above gives the lower bound , and posed the problem of determining the asymptotic limit of as , given that this was now known to range between and . This version was accepted by Thomas Bloom, the moderator of the Erdős problem site, as a valid interpretation of the original problem.

The next day, Stijn computed the first few values of exactly:

The problem then lay dormant for almost two months, until December 7, 2025, in which Boris Alexeev, as part of a systematic sweep of the Erdős problems using the AI tool Aristotle, was able to get this tool to autonomously solve this conjecture in the proof assistant language Lean. The proof converted the problem to a rectangle-packing problem.

This was one further addition to a recent sequence of examples where an Erdős problem had been automatically solved in one fashion or another by an AI tool. Like the previous cases, the proof turned out to not be particularly novel. Within an hour, Koishi Chan gave an alternate proof deriving the required bound from the original Erdős-Szekeres theorem by a standard “blow-up” argument which we can give here in the Alice-Bob formulation. Take a large , and replace each pile of coins with new piles, each of size , chosen so that the longest monotone subsequence in this collection is . Among all the new piles, the longest monotone subsequence has length . Applying Erdős-Szekeres, one concludes the bound

Once this proof was found, it was natural to try to see if it had already appeared in the literature. AI deep research tools have successfully located such prior literature in the past, but in this case they did not succeed, and a more “old-fashioned” Google Scholar job turned up some relevant references: a 2016 paper by Tidor, Wang and Yang contained this precise result, citing an earlier paper of Wagner as inspiration for applying “blowup” to the Erdős-Szekeres theorem.

But the story does not end there! Upon reading the above story the next day, I realized that the problem of estimating was a suitable task for AlphaEvolve, which I have used recently as mentioned in this previous post. Specifically, one could task to obtain upper bounds on by directing it to produce real numbers (or integers) summing up to a fixed sum (I chose ) with a small a value of as possible. After an hour of run time, AlphaEvolve produced the following upper bounds on for , with some intriguingly structured potential extremizing solutions:

Proposition 1 If and , then .

Proof: Consider a sequence of numbers clustered around the “red number” and “blue number” , consisting of blocks of “blue” numbers, followed by blocks of “red” numbers, and then further blocks of “blue” numbers. When , one should take all blocks to be slightly decreasing within each block, but the blue blocks should be are increasing between each other, and the red blocks should also be increasing between each other. When , all of these orderings should be reversed. The total number of elements is indeed

Here is a figure illustrating the above construction in the case (obtained after starting with a ChatGPT-provided file and then manually fixing a number of placement issues):

Here is a plot of (produced by ChatGPT Pro), showing that it is basically a piecewise linear approximation to the square root function:

Shortly afterwards, Lawrence Wu clarified the connection between this problem and a square packing problem, which was also due to Erdős (Problem 106). Let be the least number such that, whenever one packs squares of sidelength into a square of sidelength , with all sides parallel to the coordinate axes, one has

Proposition 2 For any , one has

Proof: Given and , let be the maximal sum over all increasing subsequences ending in , and be the maximal sum over all decreasing subsequences ending in . For , we have either (if ) or (if ). In particular, the squares and are disjoint. These squares pack into the square , so by definition of , we have

This idea of using packing to prove Erdős-Szekeres type results goes back to a 1959 paper of Seidenberg, although it was a discrete rectangle-packing argument that was not phrased in such an elegantly geometric form. It is possible that Aristotle was “aware” of the Seidenberg argument via its training data, as it had incorporated a version of this argument in its proof.

Here is an illustration of the above argument using the AlphaEvolve-provided example

for to convert it to a square packing (image produced by ChatGPT Pro):

At this point, Lawrence performed another AI deep research search, this time successfully locating a paper from just last year by Baek, Koizumi, and Ueoro, where they show that

Theorem 3 For any , one has

which, when combined with a previous argument of Praton, implies

Theorem 4 For any and with , one has

There just remained the issue of putting everything together. I did feed all of the above information into a large language model, which was able to produce a coherent proof of (1) assuming the results of Baek-Koizumi-Ueoro and Praton. Of course, LLM outputs are prone to hallucination, so it would be preferable to formalize that argument in Lean, but this looks quite doable with current tools, and I expect this to be accomplished shortly. But I was also able to reproduce the arguments of Baek-Koizumi-Ueoro and Praton, which I include below for completeness.

Proof: (Proof of Theorem 3, adapted from Baek-Koizumi-Ueoro) We can normalize . It then suffices to show that if we pack the length torus by axis-parallel squares of sidelength , then

Pick . Then we have a grid

UPDATE: Actually, the above argument also proves Theorem 4 with only minor modifications. Nevertheless, we give the original derivation of Theorem 4 using the embedding argument of Praton below for sake of completeness.

Proof: (Proof of Theorem 4, adapted from Praton) We write with . We can rescale so that the square one is packing into is . Thus, we pack squares of sidelength into , and our task is to show that

One striking feature of this story for me is how important it was to have a diverse set of people, literature, and tools to attack this problem. To be able to state and prove the precise formula for required multiple observations, including some version of the following:

• The sequence can be numerically computed as a sequence of rational numbers.
• When appropriately normalized and arranged, visible patterns in this sequence appear that allow one to conjecture the form of the sequence.
• This problem is a weighted version of the Erdős-Szekeres theorem.
• Among the many proofs of the Erdős-Szekeres theorem is the proof of Seidenberg in 1959, which can be interpreted as a discrete rectangle packing argument.
• This problem can be reinterpreted as a continuous square packing problem, and in fact is closely related to (a generalized axis-parallel form of) Erdős problem 106, which concerns such packings.
• The axis-parallel form of Erdős problem 106 was recently solved by Baek-Koizumi-Ueoro.
• The paper of Praton shows that Erdős Problem 106 implies the generalized version needed for this problem. This implication specializes to the axis-parallel case.

Another key ingredient was the balanced AI policy on the Erdős problem website, which encourages disclosed AI usage while strongly discouraging undisclosed use. To quote from that policy: “Comments prepared with the assistance of AI are permitted, provided (a) this is disclosed, (b) the contents (including mathematics, code, numerical data, and the existence of relevant sources) have been carefully checked and verified by the user themselves without the assistance of AI, and (c) the comment is not unreasonably long.”

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

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

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

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

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

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

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

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

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

• Applied category theory for modeling.

These days, the most common question I get goes something like this:

A decade ago, you told people that scalable quantum computing wasn’t imminent. Now, though, you claim it plausibly is imminent. Why have you reversed yourself??

I appreciated the friend of mine who paraphrased this as follows: “A decade ago you said you were 35. Now you say you’re 45. Explain yourself!”

A couple weeks ago, I was delighted to attend Q2B in Santa Clara, where I gave a keynote talk entitled “Why I Think Quantum Computing Works” (link goes to the PowerPoint slides). This is one of the most optimistic talks I’ve ever given. But mostly that’s just because, uncharacteristically for me, here I gave short shrift to the challenge of broadening the class of problems that achieve huge quantum speedups, and just focused on the experimental milestones achieved over the past year. With every experimental milestone, the little voice in my head that asks “but what if Gil Kalai turned out to be right after all? what if scalable QC wasn’t possible?” grows quieter, until now it can barely be heard.

Going to Q2B was extremely helpful in giving me a sense of the current state of the field. Ryan Babbush gave a superb overview (I couldn’t have improved a word) of the current status of quantum algorithms, while John Preskill’s annual where-we-stand talk was “magisterial” as usual (that’s the word I’ve long used for his talks), making mine look like just a warmup act for his. Meanwhile, Quantinuum took a victory lap, boasting of their recent successes in a way that I considered basically justified.

After returning from Q2B, I then did an hour-long podcast with “The Quantum Bull” on the topic “How Close Are We to Fault-Tolerant Quantum Computing?” You can watch it here:

As far as I remember, this is the first YouTube interview I’ve ever done that concentrates entirely on the current state of the QC race, skipping any attempt to explain amplitudes, interference, and other basic concepts. Despite (or conceivably because?) of that, I’m happy with how this interview turned out. Watch if you want to know my detailed current views on hardware—as always, I recommend 2x speed.

Or for those who don’t have the half hour, a quick summary:

• In quantum computing, there are the large companies and startups that might succeed or might fail, but are at least trying to solve the real technical problems, and some of them are making amazing progress. And then there are the companies that have optimized for doing IPOs, getting astronomical valuations, and selling a narrative to retail investors and governments about how quantum computing is poised to revolutionize optimization and machine learning and finance. Right now, I see these two sets of companies as almost entirely disjoint from each other.
  
• The interview also contains my most direct condemnation yet of some of the wild misrepresentations that IonQ, in particular, has made to governments about what QC will be good for (“unlike AI, quantum computers won’t hallucinate because they’re deterministic!”)
  
• The two approaches that had the most impressive demonstrations in the past year are trapped ions (especially Quantinuum but also Oxford Ionics) and superconducting qubits (especially Google but also IBM), and perhaps also neutral atoms (especially QuEra but also Infleqtion and Atom Computing).
  
• Contrary to a misconception that refuses to die, I haven’t dramatically changed my views on any of these matters. As I have for a quarter century, I continue to profess a lot of confidence in the basic principles of quantum computing theory worked out in the mid-1990s, and I also continue to profess ignorance of exactly how many years it will take to realize those principles in the lab, and of which hardware approach will get there first.
  
• But yeah, of course I update in response to developments on the ground, because it would be insane not to! And 2025 was clearly a year that met or exceeded my expectations on hardware, with multiple platforms now boasting >99.9% fidelity two-qubit gates, at or above the theoretical threshold for fault-tolerance. This year updated me in favor of taking more seriously the aggressive pronouncements—the “roadmaps”—of Google, Quantinuum, QuEra, PsiQuantum, and other companies about where they could be in 2028 or 2029.
  
• One more time for those in the back: the main known applications of quantum computers remain (1) the simulation of quantum physics and chemistry themselves, (2) breaking a lot of currently deployed cryptography, and (3) eventually, achieving some modest benefits for optimization, machine learning, and other areas (but it will probably be a while before those modest benefits win out in practice). To be sure, the detailed list of quantum speedups expands over time (as new quantum algorithms get discovered) and also contracts over time (as some of the quantum algorithms get dequantized). But the list of known applications “from 30,000 feet” remains fairly close to what it was a quarter century ago, after you hack away the dense thickets of obfuscation and hype.

I’m going to close this post with a warning. When Frisch and Peierls wrote their now-famous memo in March 1940, estimating the mass of Uranium-235 that would be needed for a fission bomb, they didn’t publish it in a journal, but communicated the result through military channels only. As recently as February 1939, Frisch and Meitner had published in Nature their theoretical explanation of recent experiments, showing that the uranium nucleus could fission when bombarded by neutrons. But by 1940, Frisch and Peierls realized that the time for open publication of these matters had passed.

Similarly, at some point, the people doing detailed estimates of how many physical qubits and gates it’ll take to break actually deployed cryptosystems using Shor’s algorithm are going to stop publishing those estimates, if for no other reason than the risk of giving too much information to adversaries. Indeed, for all we know, that point may have been passed already. This is the clearest warning that I can offer in public right now about the urgency of migrating to post-quantum cryptosystems, a process that I’m grateful is already underway.

Update: Someone on Twitter who’s “long $IONQ” says he’ll be posting about and investigating me every day, never resting until UT Austin fires me, in order to punish me for slandering IonQ and other “pure play” SPAC IPO quantum companies. And also, because I’ve been anti-Trump and pro-Biden. He confabulates that I must be trying to profit from my stance (eg by shorting the companies I criticize), it being inconceivable to him that anyone would say anything purely because they care about what’s true.

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

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

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

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

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

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

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

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

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

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

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

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

In particular we have

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

so composing with the exceptional isomorphisms:

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

we get a 2-1 homomorphism

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

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

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

but there are three obvious inclusions

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

namely the left one:

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

the right one:

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

and the diagonal one:

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

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

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

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

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

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

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

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

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

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

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

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

The left subgroup

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The diagonal subgroup

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• John Baez, Division algebras and quantum theory.

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

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

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

However, we know this representation is

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

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

So, it must be that $G_\ell$ is not a subgroup of $\text{Spin}(9)$.

Proof of Theorem 10

To complete the proof of Theorem 10 we just need to see why there are just two conjugacy classes of subgroups of $\text{Spin}(10)$ isomorphic to $\text{SU}(2) \times \text{SU}(3)$. But in fact Will Sawin proved a stronger result! He was answering this question of mine:

Define the Standard Model gauge group to be $\text{S}(\text{U}(2) \times \text{U}(3))$, the subgroup of $\text{SU}(5)$ consisting of block diagonal matrices with a $2 \times 2$ block and then a $3 \times 3$ block. (This is isomorphic to the quotient of $\text{U}(1) \times \text{SU}(2) \times \text{SU}(3)$ by the subgroup of elements $(\alpha, \alpha^{-3}, \alpha^2$) where $\alpha$ is a 6th root of unity.)

Up to conjugacy, how many subgroups isomorphic to the Standard Model gauge group does $\text{Spin}(10)$ have?

This question is relevant to grand unified theories of particle physics, as explained here:

• John C. Baez and John Huerta, The algebra of grand unified theories, Bull. Amer. Math. Soc. 47 (2010), 483-552.

This paper focuses on one particular copy of $\text{S}(\text{U}(2) \times \text{U}(3))$ in $\text{Spin}(10)$, given as follows. By definition we have an inclusion $\text{S}(\text{U}(2) \times \text{U}(3)) \hookrightarrow \text{SU}(5)$, and we also have an inclusion $\text{SU}(5) \hookrightarrow \text{Spin}(10)$ because for any $n$ we have an inclusion $\text{SU}(n) \hookrightarrow \text{SO}(2n)$, and $\text{SU}(n)$ is simply connected so this gives a homomorphism $\text{SU}(n) \hookrightarrow \text{Spin}(2n)$.

However I think there is also an inclusion $\text{S}(\text{U}(2) \times \text{U}(3)) \hookrightarrow \text{Spin}(9)$, studied by Krasnov:

• Kirill Krasnov, SO(9) characterisation of the Standard Model gauge group.

Composing this with $\text{Spin}(9) \hookrightarrow \text{Spin}(10)$, this should give another inclusion $\text{S}(\text{U}(2) \times \text{U}(3)) \hookrightarrow \text{Spin}(10)$, and I believe this one is ‘truly different from’ — i.e., not conjugate to — the first one I mentioned.

So I believe my current answer to my question is “at least two”. But that’s not good enough.

Sawin’s answer relies heavily on the 3-fold way — that’s why I told you that stuff about orthogonal and symplectic representatinos. When we embed the group $\text{SU}(2) \times \text{SU}(3)$ in $\text{Spin}(10)$, we are automatically giving this group an orthogonal 10-dimensional representation, thanks to the map $\text{Spin}(10) \to \text{SO}(10)$. We can classify the possibilities.

He writes:

There are infinitely many embeddings. However, all but one of them is “essentially the same as” the one you studied as they become equal to the one you studied on restriction to $\text{SU}(2)\times \text{SU}(3)$. The remaining one is the one studied by Krasnov.

I follow the strategy suggested by Kenta Suzuki.

$\text{SU}(3)$ has irreducible representations of dimensions $1,3,3,6,8,6, 10, 10$, and higher dimensions. The $10$-dimensional ones are dual to each other, as are the $6$-dimensional ones, so they can’t appear. The $3$-dimensional ones are dual to each other and can only appear together. So the only $10$-dimensional self-dual representations of $\text{SU}(3)$ decompose as irreducibles as $8+1+1$, $3+3+1+1+1+1$, or ten $1$s. All of these are orthogonal because the 8-dimensional representation is orthogonal. However, the ten $1$s cannot appear because then $\text{SU}(3)$ would act trivially.

A representation of $\text{SU}(3) \times \text{SU}(2)$ is a sum of tensor products of irreducible representations of $\text{SU}(3)$ and irreducible representations of $\text{SU}(2)$. Restricted to $\text{SU}(3)$, each tensor product splits into a sum of copies of the same irreducible representation. So $\text{SU}(2)$ can only act nontrivially when the same representation appears multiple times. Since the $3+3$ is two different $3$-dimensional representation, only the $1$-dimensional representation can occur twice. Thus, our 10-dimensional orthogonal representation of $\text{SU}(3) \times \text{SU}(2)$ necessarily splits as either the $8$-dimensional adjoint repsentation of $\text{SU}(3)$ plus a $2$-dimensional orthogonal representation of $\text{SU}(2)$ or the $6$-dimensional sum of standard and conjugate [i.e., dual] representations of $\text{SU}(3)$ plus a $4$-dimensional orthogonal representation of $\text{SU}(2)$. However, $\text{SU}(2)$ has a unique nontrivial representation of dimension $2$ and it isn’t orthgonal, so only the second case can appear. $\text{SU}(2)$ has representations of dimension $1,2,3,4$ of which the $2$ and $4$-dimensional ones are symplectic and so must appear with even multiplicity in any orthogonal representation, so the only nontrivial $4$-dimensional orthogonal ones are $2+2$ or $3+1$.

So there are two ten-dimensional orthogonal representations of $\text{SU}(2) \times \text{SU}(3)$ that are nontrivial on both factors, those being the sum of two different $3$-dimensional irreducible representations of $\text{SU}(3)$ with either two copies of the two-dimensional irreducible representation of $\text{SU}(2)$ or the three-dimensional and the one-dimensional irreducible representation of $\text{SU}(2)$. The orthogonal structure is unique up to isomorphisms, so these give two conjugacy classes of homomorphisms $\text{SU}(2) \times \text{SU}(3) \to SO(10)$ and thus two conjugacy classes of homomorphisms $\text{SU}(2) \times \text{SU}(3) \to \text{Spin}(10)$. The first one corrresponds to the embedding you studied while only the second one restricts to $\text{Spin}(9)$ so indeed these are different.

To understand how to extend these to $\text{S}(\text{U}(2) \times \text{U}(3))$, I consider the centralizer of the representation within $\text{Spin}(10)$. Since the group is connected, this is the same as the centralizer of its Lie algebra, which is therefore the inverse image of the centralizer in $\text{SO}(10)$. Now there is a distinction between the two examples because the example with irrep dimensions $3+3+2+2$ has centralizer with identity component $\text{U}(1) \times \text{SU}(2)$ while the example with irrep dimensions $3+3+3+1$ has centralizer with identity component $\text{U}(1)$. In the second case, the image of $\text{U}(2) \times \text{U}(3)$ must be the image of $\text{SU}(2) \times \text{SU}(3)$ times the centralizer of the image of $\text{SU}(2) \times \text{SU}(3)$, so this gives a unique example, which must be the one considered by Krasnov.

In the first case, we can restrict attention to a torus $\text{U}(1) \times \text{U}(1)$ in $\text{SU}(2) \times \text{SU}(2)$. The center of $\text{S}(\text{U}(2) \times \text{U}(3))$ maps to a one-dimensional subgroup of this torus, which can be described by a pair of integers. Explicitly, given a two-by-two-unitary matrix $A$ and a three-by-three unitary matrix $B$ with $\det(A) \det(B) =1$, we can map to $\text{U}(5)$ by sending $(A,B)$ to $A \gamma^a \oplus B \gamma^b$ where $\gamma = \det (A) = \det(B)^{-1}$, and then map from $\text{U}(5)$ to $SO(10)$. This lifts to the spin group if and only if the determinant in $\text{U}(5)$ is a perfect square. The determinant is $\gamma^{ 1 + 2a - 1 + 3b} = \gamma^{2a+3b}$ so a lift exists if and only if $b$ is even.

The only possible kernel of this embedding is the scalars. The scalar $A = \lambda^3 I_2, B = \lambda^{-2} I_3$ maps to $\lambda^{3+ 6a} I_2 \oplus \lambda^{-2 + 6b} I_3$ and so the kernel is trivial if and only if $\gcd(3+6a,-2 + 6b)=1$.

However, there are infinitely many integer solutions $a,b$ to $\gcd(3+6a,-2a+6b)=1$ with $b$ even (in fact, a random $a$ and even $b$ works with probability $9/\pi^2$), so this gives infinitely many examples.

• Part 1. How to define octonion multiplication using complex scalars and vectors, much as quaternion multiplication can be defined using real scalars and vectors. This description requires singling out a specific unit imaginary octonion, and it shows that octonion multiplication is invariant under $\mathrm{SU}(3)$.
• Part 2. A more polished way to think about octonion multiplication in terms of complex scalars and vectors, and a similar-looking way to describe it using the cross product in 7 dimensions.
• Part 3. How a lepton and a quark fit together into an octonion — at least if we only consider them as representations of $\mathrm{SU}(3)$, the gauge group of the strong force. Proof that the symmetries of the octonions fixing an imaginary octonion form precisely the group $\mathrm{SU}(3)$.
• Part 4. Introducing the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$: the $3 \times 3$ self-adjoint octonionic matrices. A result of Dubois-Violette and Todorov: the symmetries of the exceptional Jordan algebra preserving their splitting into complex scalar and vector parts and preserving a copy of the $2 \times 2$ adjoint octonionic matrices form precisely the Standard Model gauge group.
• Part 5. How to think of $2 \times 2$ self-adjoint octonionic matrices as vectors in 10d Minkowski spacetime, and pairs of octonions as left- or right-handed spinors.
• Part 6. The linear transformations of the exceptional Jordan algebra that preserve the determinant form the exceptional Lie group $\mathrm{E}_6$. How to compute this determinant in terms of 10-dimensional spacetime geometry: that is, scalars, vectors and left-handed spinors in 10d Minkowski spacetime.
• Part 7. How to describe the Lie group $\mathrm{E}_6$ using 10-dimensional spacetime geometry. This group is built from the double cover of the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
• Part 8. A geometrical way to see how $\mathrm{E}_6$ is connected to 10d spacetime, based on the octonionic projective plane.
• Part 9. Duality in projective plane geometry, and how it lets us break the Lie group $\mathrm{E}_6$ into the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
• Part 10. Jordan algebras, their symmetry groups, their invariant structures — and how they connect quantum mechanics, special relativity and projective geometry.
• Part 11. Particle physics on the spacetime given by the exceptional Jordan algebra: a summary of work with Greg Egan and John Huerta.
• Part 12. The bioctonionic projective plane and its connections to algebra, geometry and physics.
• Part 13. Two ways to embed $\text{SU}(2) \times \text{SU}(3)$ in $\text{Spin}(10)$, and their consequences for particle physics.

A $5 million grant for my old friend Richard Rusczyk and the MATHCOUNTS foundation. Great news! Richard founded Art of Problem Solving, which is the only online math course platform I’ve ever found to be worth a damn. I love the way they emphasize depth over acceleration. When you do algebra with them, you don’t do the high school algebra curriculum in six weeks, you spend the whole year but you do a hell of a lot more algebra. That’s the way it should be! The point of learning math, if you love math, isn’t to get to the highest course number at the earliest age possible; it’s to learn deeply, richly, broadly, and well. I have taught too many grad students who can say “a representation is a functor from a groupoid with one object to the category k-Vect” but who have never sat down and worked out the character tables of a dozen different groups.

I also like the way AOPS is resolutely monomedia. No voice, no video. Just text! That makes sense. Math is made of words.

I met Richard when we were in high school, both in training for the Math Olympiad. He was from Alabama and he taught me how to say “my bad” for “sorry about that.” I’d never heard it! In 1988 nobody outside the South said “my bad.” I was impressed. I started saying “my bad” when I got home, just one of my many 1988 affectations.

A recent Correspondence piece in Nature Machine Intelligence points at an issue with using LLMs to write journal articles. LLMs are trained on enormous amounts of scholarly output, but the result is quite opaque: it is usually impossible to tell which sources influence a specific LLM-written text. That means that when a scholar uses an LLM, they may get a result that depends on another scholar’s work, without realizing it or documenting it. The ideas’ provenance gets lost, and the piece argues this is damaging, depriving scholars of credit and setting back progress.

It’s a good point. Provenance matters. If we want to prioritize funding for scholars whose ideas have the most impact, we need a way to track where ideas arise.

However, current publishing norms make essentially no effort to do this. Academic citations are not used to track provenance, and they are not typically thought of as tracking provenance. Academic citations track priority.

Priority is a central value in scholarship, with a long history. We give special respect to the first person to come up with an idea, make an observation, or do a calculation, and more specifically, the first person to formally publish it. We do this even if the person’s influence was limited, and even if the idea was rediscovered independently later on. In an academic context, being first matters.

In a paper, one is thus expected to cite the sources that have priority, that came up with an idea first. Someone who fails to do so will get citation request emails, and reviewers may request revisions to the paper to add in those missing citations.

One may also cite papers that were helpful, even if they didn’t come first. Tracking provenance in this way can be nice, a way to give direct credit to those who helped and point people to useful resources. But it isn’t mandatory in the same way. If you leave out a secondary source and your paper doesn’t use anything original to that source (like new notation), you’re much less likely to get citation request emails, or revision requests from reviewers. Provenance is just much lower priority.

In practice, academics track provenance in much less formal ways. Before citations, a paper will typically have an Acknowledgements section, where the authors thank those who made the paper possible. This includes formal thanks to funding agencies, but also informal thanks for “helpful discussions” that don’t meet the threshold of authorship.

If we cared about tracking provenance, those acknowledgements would be crucial information, an account of whose ideas directly influenced the ideas in the paper. But they’re not treated that way. No-one lists the number of times they’ve been thanked for helpful discussions on their CV, or in a grant application, no-one considers these discussions for hiring or promotion. You can’t look them up on an academic profile or easily graph them in a metascience paper. Unlike citations, unlike priority, there is essentially no attempt to measure these tracks of provenance in any organized way.

Instead, provenance is often the realm of historians or history-minded scholars, writing long after the fact. For academics, the fact that Yang and Mills published their theory first is enough, we call it Yang-Mills theory. For those studying the history, the story is murkier: it looks like Pauli came up with the idea first, and did most of the key calculations, but didn’t publish when it looked to him like the theory couldn’t describe the real world. What’s more, there is evidence suggesting that Yang knew about Pauli’s result, that he had read a letter from him on the topic, that the idea’s provenance goes back to Pauli. But Yang published, Pauli didn’t. And in the way academia has worked over the last 75 years, that claim of priority is what actually mattered.

Should we try to track provenance? Maybe. Maybe the emerging ubiquitousness of LLMs should be a wakeup call, a demand to improve our tracking of ideas, both in artificial and human neural networks. Maybe we need to demand interpretability from our research tools, to insist that we can track every conclusion back to its evidence for every method we employ, to set a civilizational technological priority on the accurate valuation of information.

What we shouldn’t do, though, is pretend that we just need to go back to what we were doing before.

I’ve just uploaded to the arXiv my preprint The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree. This paper resolves (in the asymptotic regime of sufficiently high degree) an old question about the polynomial lemniscates

(The images here were generated using AlphaEvolve and Gemini.) A reasonably well-known conjecture of Erdős, Herzog, and Piranian (Erdős problem 114) asserts that this is indeed the maximizer, thus for all monic polynomials of degree .

There have been several partial results towards this conjecture. For instance, Eremenko and Hayman verified the conjecture when . Asympotically, bounds of the form had been known for various such as , , or ; a significant advance was made by Fryntov and Nazarov, who obtained the asymptotically sharp upper bound

I recently explored this problem with the optimization tool AlphaEvolve, where I found that when I assigned this tool the task of optimizing for a given degree , that the tool rapidly converged to choosing to be equal to (up to the rotation and translation symmetries of the problem). This suggested to me that the conjecture was true for all , though of course this was far from a rigorous proof. AlphaEvolve also provided some useful visualization code for these lemniscates which I have incorporated into the paper (and this blog post), and which helped build my intuition for this problem; I view this sort of “vibe-coded visualization” as another practical use-case of present-day AI tools.

In this paper, we iteratively improve upon the Fryntov-Nazarov method to obtain the following bounds, in increasing order of strength:

• (i) .
• (ii) .
• (iii) .
• (iv) for sufficiently large .

The proof of these bounds is somewhat circuitious and technical, with the analysis from each part of this result used as a starting point for the next one. For this blog post, I would like to focus on the main ideas of the arguments.

A key difficulty is that there are relatively few tools for upper bounding the arclength of a curve; indeed, the coastline paradox already shows that curves can have infinite length even when bounded. Thus, one needs to utilize some smooth or algebraic structure on the curve to hope for good upper bounds. One possible approach is via the Crofton formula, using Bezout’s theorem to control the intersection of the curve with various lines. This is already good enough to get bounds of the form (for instance by combining it with other known tools to control the diameter of the lemniscate), but it seems challenging to use this approach to get bounds close to the optimal .

Instead, we follow Fryntov–Nazarov and utilize Stokes’ theorem to convert the arclength into an area integral. A typical identity used in that paper is

But this argument does not fully capture the oscillating nature of the phase on one hand, and the oscillating nature of on the other. Fryntov–Nazarov exploited these oscillations with some additional decompositions and integration by parts arguments. By optimizing these arguments, I was able to establish an inequality of the form

One can heuristically justify (1) as follows. Suppose we work in a region where the functions , are roughly constant: , . For simplicity let us normalize to be real, and to be negative real. In order to have a non-trivial lemniscate in this region, should be close to . Because the unit circle is tangent to the line at , the lemniscate condition is then heuristically approximated by the condition that . On the other hand, the hypothesis suggests that for some amplitude , which heuristically integrates to . Writing in polar coordinates as and in Cartesian coordinates as , the condition can then be rearranged after some algebra as

A graphic illustration of (1) (provided by Gemini) is shown below, where the dark spots correspond to small values of that act to “repel” (and shorten) the lemniscate. (The bright spots correspond to the critical points of , which in this case consist of six critical points at the origin and one at both of and .)

By choosing parameters appropriately, one can show that and , yielding the first bound . However, by a more careful inspection of the arguments, and in particular measuring the defect in the triangle inequality

At this point, the only remaining cases that need to be handled are the ones with bounded dispersion: . In this case, one can do some elementary manipulations of the factorization

The Rice Advanced Materials Institute has an open search in the area of computational materials.   The full position posting is here.  Please spread the word!

The brief description:  

The Rice Advanced Materials Institute (RAMI) at Rice University, located in Houston TX, seeks applications for tenure-track Assistant and/or Associate Professor position in the area of advanced computational and/or applied artificial intelligence (AI)/machine learning (ML)-informed materials research with an anticipated start date of July 1, 2026 to January 1, 2027. RAMI, working in conjunction with the Schools of Engineering and Computing and Natural Sciences and the Departments therein, seeks applicants from diverse backgrounds for potential tenure-track faculty positions in Departments to be determined by the best natural fit and input of the candidate (e.g., potential departments could include, but not be limited to, Chemical and Biomolecular Engineering, Chemistry, Electrical and Computer Engineering, Materials Science and NanoEngineering, Mechanical Engineering, or Physics and Astronomy). More experienced candidates nearing the transition to or having already recently transitioned to the Associate Professor level and conducting transformative research projects will be considered. 

We seek outstanding candidates with research interests in theoretical, computational, modeling, and/or simulation of materials science (and related fields) and the application of advanced AI and ML approaches to accelerate, improve, or revolutionize the same. Research spanning all of aspects of materials including soft/hard matter, inorganic/organic materials, etc. is welcomed, as long as that research maps significantly to one or more of the RAMI core research areas, including: 

• Next-generation Electronics/Photonics – Developing materials to enable a new paradigm in microelectronics ranging from memory/logic to communications to sensors and beyond, with a focus on low-power/voltage function.

• Energy Materials (Systems) – Developing materials innovations to transform energy storage and conversion/harvesting.

• Materials for the Environment – Developing materials innovations to assure responsible use of natural resources and long-term stewardship of our air, soil, and water resources.

This effort is being initiated through RAMI, a campus-wide institute with the goal of expanding Rice’s already strong efforts in materials research across many departments in science and engineering. RAMI includes >70 current faculty members working on a wide array of research that overlap with RAMI’s core research focus areas. Rice considers efforts to bolster these areas as a focal point of materials research in the coming years.

The selected candidates will be expected to teach and develop undergraduate and graduate courses within their expertise and home department; perform high-quality research in their specialized area and present findings from their research in peer-reviewed publications and conferences; establish a strong research program supported by extramural funding; be involved in service to the university and broader scientific community; and collaborate with faculty in diverse disciplines. Successful candidates will have a strong commitment to teaching, advising, and mentoring undergraduate and graduate students from diverse backgrounds. 

Applications will be reviewed in a rolling fashion but should be received by 11:59 PM (eastern time) on January 4, 2026. Applications received after this deadline will be considered as they arrive and until the position is filled.

This (taken in Kiel, Germany in 1931 and then colorized) is one of the most famous photographs in Jewish history, but it acquired special resonance this weekend. It communicates pretty much everything I’d want to say about the Bondi Beach massacre in Australia, more succinctly than I could in words.

But I can’t resist sharing one more photo, after GPT5-Pro helpfully blurred the faces for me. This is my 8-year-old son Daniel, singing a Chanukah song at our synagogue, at an event the morning after the massacre, which was packed despite the extra security needed to get in.

Alright, one more photo. This is Ahmed Al Ahmed, the hero who tackled and disarmed one of the terrorists, recovering in the hospital from his gunshot wounds. Facebook and Twitter and (alas) sometimes the comment section of this blog show me the worst of humanity, day after day after day, so it’s important to remember the best of humanity as well.

Chanukah, of course, is the most explicitly Zionist of all Jewish holidays, commemorating as it does the Maccabees’ military victory against the Seleucid Greeks, in their (historically well-attested) wars of 168-134BCE to restore an independent Jewish state with its capital in Jerusalem. In a sense, then, the terrorists were precisely correct, when they understood the cry “globalize the intifada” to mean “murder random Jews anywhere on earth, even halfway around the world, who might be celebrating Chanukah.” By the lights of the intifada worldview, Chabadniks in Sydney were indeed perfectly legitimate targets. By my worldview, though, the response is equally clear: to abandon all pretense, and say openly that now, as in countless generations past, Jews everywhere are caught up in a war, not of our choosing, which we “win” merely by surviving with culture and memory intact.

Happy Chanukah.

During the spring of 2022, I felt as though I kept dashing backward and forward in time. 

At the beginning of the season, hay fever plagued me in Maryland. Then, I left to present talks in southern California. There—closer to the equator—rose season had peaked, and wisteria petals covered the ground near Caltech’s physics building. From California, I flew to Canada to present a colloquium. Time rewound as I traveled northward; allergies struck again. After I returned to Maryland, the spring ripened almost into summer. But the calendar backtracked when I flew to Sweden: tulips and lilacs surrounded me again.

The zigzagging through horticultural time disoriented my nose, but I couldn’t complain: it echoed the quantum information processing that collaborators and I would propose that summer. We showed how to improve quantum metrology—our ability to measure things, using quantum detectors—by simulating closed timelike curves.

A closed timelike curve is a trajectory that loops back on itself in spacetime. If on such a trajectory, you’ll advance forward in time, reverse chronological direction to advance backward, and then reverse again. Author Jasper Fforde illustrates closed timelike curves in his novel The Eyre Affair. A character named Colonel Next buys an edition of Shakespeare’s works, travels to the Elizabethan era, bestows them on a Brit called Will, and then returns to his family. Will copies out the plays and stages them. His colleagues publish the plays after his death, and other editions ensue. Centuries later, Colonel Next purchases one of those editions to take to the Elizabethan era.¹ 

Closed timelike curves can exist according to Einstein’s general theory of relativity. But do they exist? Nobody knows. Many physicists expect not. But a quantum system can simulate a closed timelike curve, undergoing a process modeled by the same mathematics.

How can one formulate closed timelike curves in quantum theory? Oxford physicist David Deutsch proposed one formulation; a team led by MIT’s Seth Lloyd proposed another. Correlations distinguish the proposals. 

Two entities share correlations if a change in one entity tracks a change in the other. Two classical systems can correlate; for example, your brain is correlated with mine, now that you’ve read writing I’ve produced. Quantum systems can correlate more strongly than classical systems can, as by entangling. 

Suppose Colonel Next correlates two nuclei and gives one to his daughter before embarking on his closed timelike curve. Once he completes the loop, what relationship does Colonel Next’s nucleus share with his daughter’s? The nuclei retain the correlations they shared before Colonel Next entered the loop, according to Seth and collaborators. When referring to closed timelike curves from now on, I’ll mean ones of Seth’s sort.

We can simulate closed timelike curves by subjecting a quantum system to a circuit of the type illustrated below. We read the diagram from bottom to top. Along this direction, time—as measured by a clock at rest with respect to the laboratory—progresses. Each vertical wire represents a qubit—a basic unit of quantum information, encoded in an atom or a photon or the like. Each horizontal slice of the diagram represents one instant. 

At the bottom of the diagram, the two vertical wires sprout from one curved wire. This feature signifies that the experimentalist prepares the qubits in an entangled state, represented by the symbol . Farther up, the left-hand wire runs through a box. The box signifies that the corresponding qubit undergoes a transformation (for experts: a unitary evolution). 

At the top of the diagram, the vertical wires fuse again: the experimentalist measures whether the qubits are in the state they began in. The measurement is probabilistic; we (typically) can’t predict the outcome in advance, due to the uncertainty inherent in quantum physics. If the measurement yields the yes outcome, the experimentalist has simulated a closed timelike curve. If the no outcome results, the experimentalist should scrap the trial and try again.

So much for interpreting the diagram above as a quantum circuit. We can reinterpret the illustration as a closed timelike curve. You’ve probably guessed as much, comparing the circuit diagram to the depiction, farther above, of Colonel Next’s journey. According to the second interpretation, the loop represents one particle’s trajectory through spacetime. The bottom and top show the particle reversing chronological direction—resembling me as I flew to or from southern California.

How can we apply closed timelike curves in quantum metrology? In Fforde’s books, Colonel Next has a brother, named Mycroft, who’s an inventor.² Suppose that Mycroft is studying how two particles interact (e.g., by an electric force). He wants to measure the interaction’s strength. Mycroft should prepare one particle—a sensor—and expose it to the second particle. He should wait for some time, then measure how much the interaction has altered the sensor’s configuration. The degree of alteration implies the interaction’s strength. The particles can be quantum, if Mycroft lives not merely in Sherlock Holmes’s world, but in a quantum-steampunk one.

But how should Mycroft prepare the sensor—in which quantum state? Certain initial states will enable the sensor to acquire ample information about the interaction; and others, no information. Mycroft can’t know which preparation will work best: the optimal preparation depends on the interaction, which he hasn’t measured yet. 

Mycroft can overcome this dilemma via a strategy published by my collaborator David Arvidsson-Shukur, his recent student Aidan McConnell, and me. According to our protocol, Mycroft entangles the sensor with a third particle. He subjects the sensor to the interaction (coupling the sensor to particle #2) and measures the sensor. 

Then, Mycroft learns about the interaction—learns which state he should have prepared the sensor in earlier. He effectively teleports this state backward in time to the beginning-of-protocol sensor, using particle #3 (which began entangled with the sensor).³ Quantum teleportation is a decades-old information-processing task that relies on entanglement manipulation. The protocol can transmit quantum states over arbitrary distances—or, effectively, across time.

We can view Mycroft’s experiment in two ways. Using several particles, he manipulates entanglement to measure the interaction strength optimally (with the best possible precision). This process is mathematically equivalent to another. In the latter process, Mycroft uses only one sensor. It comes forward in time, reverses chronological direction (after Mycroft learns the optimal initial state’s form), backtracks to an earlier time (to when the sensing protocol began), and returns to progressing forward in time (informing Mycroft about the interaction).

In Sweden, I regarded my work with David and Aidan as a lark. But it’s led to an experiment, another experiment, and two papers set to debut this winter. I even pass as a quantum metrologist nowadays. Perhaps I should have anticipated the metamorphosis, as I should have anticipated the extra springtimes that erupted as I traveled between north and south. As the bard says, there’s a time for all things.

¹In the sequel, Fforde adds a twist to Next’s closed timelike curve. I can’t speak for the twist’s plausibility or logic, but it makes for delightful reading, so I commend the novel to you.

²You might recall that Sherlock Holmes has a brother, named Mycroft, who’s an inventor. Why? In Fforde’s novel, an evil corporation pursues Mycroft, who’s built a device that can transport him into the world of a book. Mycroft uses the device to hide from the corporation in Sherlock Holmes’s backstory.

³Experts, Mycroft implements the effective teleportation as follows: He prepares a fourth particle in the ideal initial sensor state. Then, he performs a two-outcome entangling measurement on particles 3 and 4: he asks “Are particles 3 and 4 in the state in which particles 1 and 3 began?” If the measurement yields the yes outcome, Mycroft has effectively teleported the ideal sensor state backward in time. He’s also simulated a closed timelike curve. If the measurement yields the no outcome, Mycroft fails to measure the interaction optimally. Figure 1 in our paper synopsizes the protocol.

In school, kids learn about different types of energy. They learn about solar energy and wind energy, nuclear energy and chemical energy, electrical energy and mechanical energy, and potential energy and kinetic energy. They learn that energy is conserved, that it can never be created or destroyed, but only change form. They learn that energy makes things happen, that you can use energy to do work, that energy is different from matter.

Some, between good teaching and good students, manage to impose order on the jumble of concepts and terms. Others end up envisioning the whole story a bit like Pokemon, with different types of some shared “stuff”.

Energy isn’t “stuff”, though. So what is it? What relates all these different types of things?

Energy is something which is conserved.

The mathematician Emmy Noether showed that, when the laws of physics are symmetrical, they come with a conserved quantity. For example, because the laws of the physics are the same from place to place, momentum is conserved. Similarly, because the laws of physics are the same from one time to another, Noether’s theorem states that there must be some quantity related to time, some number we can calculate, that is conserved, even as other things change. We call that number energy.

If energy is that simple, why are there all those types?

Energy is a number we can calculate. It’s a number we can calculate for different things. If you have a detailed description of how something in physics works, you can use that description to calculate that thing’s energy. In school, you memorize formulas like and . These are all formulas that, with a bit more knowledge, you could calculate. They are the things that, for a something that meets the conditions, are conserved. They are things that, according to Noether’s theorem, stay the same.

Because of this, you shouldn’t think of energy as a substance, or a fuel. Energy is something we can do: we physicists, and we students of physics. We can take a physical system, and see what about it ought to be conserved. Energy is an action, a calculation, a conceptual tool that can be used to make predictions.

Most things are, in the end.

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Not long ago William MacAskill, the founder of the Effective Altruist movement, visited Austin, where I got to talk with him in person for the first time. I was a fan of his book What We Owe the Future, and found him as thoughtful and eloquent face-to-face as I did on the page. Talking to Will inspired me to write the following short reflection on how I should spend my time, which I’m now sharing in case it’s of interest to anyone else.

By inclination and temperament, I simply seek the clearest possible understanding of reality.  This has led me to spend time on (for example) the Busy Beaver function and the P versus NP problem and quantum computation and the foundations of quantum mechanics and the black hole information puzzle, and on explaining whatever I’ve understood to others.  It’s why I became a professor.

But the understanding I’ve gained also tells me that I should try to do things that will have huge positive impact, in what looks like a pivotal and even terrifying time for civilization.  It tells me that seeking understanding of the universe, like I’ve been doing, is probably nowhere close to optimizing any values that I could defend.  It’s self-indulgent, a few steps above spending my life learning to solve Rubik’s Cube as quickly as possible, but only a few.  Basically, it’s the most fun way I could make a good living and have a prestigious career, so it’s what I ended up doing.  I should be skeptical that such a course would coincidentally also maximize the good I can do for humanity.

Instead I should plausibly be figuring out how to make billions of dollars, in cryptocurrency or startups or whatever, and then spending it in a way that saves human civilization, for example by making AGI go well.  Or I should be convincing whatever billionaires I know to do the same.  Or executing some other galaxy-brained plan.  Even if I were purely selfish, as I hope I’m not, still there are things other than theoretical computer science research that would bring more hedonistic pleasure.  I’ve basically just followed a path of least resistance.

On the other hand, I don’t know how to make billions of dollars.  I don’t know how to make AGI go well.  I don’t know how to influence Elon Musk or Sam Altman or Peter Thiel or Sergey Brin or Mark Zuckerberg or Marc Andreessen to do good things rather than bad things, even when I have gotten to talk to some of them.  Past attempts in this direction by extremely smart and motivated people—for example, those of Eliezer Yudkowsky and Sam Bankman-Fried—have had, err, uneven results, to put it mildly.  I don’t know why I would succeed where they failed.

Of course, if I had a better understanding of reality, I might know how better to achieve prosocial goals for humanity.  Or I might learn why they were actually the wrong goals, and replace them with better goals.  But then I’m back to the original goal of understanding reality as clearly as possible, with the corresponding danger that I spend my time learning to solve Rubik’s Cube faster.

AI promises to revolutionize the way we do science, which raises a central technological question of our time: Can classical AI understand all natural phenomena, or are some fundamentally beyond its reach? Many proponents of artificial intelligence argue that any pattern that can be generated or found in nature can be efficiently discovered and modeled by a classical learning algorithm, implying that AI is a universal and sufficient tool for science.

The word “classical” is important here to contrast with quantum computation. Nature is quantum mechanical, and the insights of Shor’s algorithm [1] along with quantum error correction [2,3,4] teach us that there are quantum systems, at least ones that have been heavily engineered, that can have trajectories that are fundamentally unpredictable¹ by any classical algorithm, including AI. This opens the possibility that there are complex quantum phenomena occurring naturally in our universe where classical AI is insufficient, and we need a quantum computer in order to model them.

This essay uses the perspective of computational complexity to unpack this nuanced question. We begin with quantum sampling, arguing that despite clear quantum supremacy, it does not represent a real hurdle for AI to predict quantum phenomena. We then shift to the major unsolved problems in quantum physics and quantum chemistry, examining how quantum computers could empower AI in these domains. Finally, we’ll consider the possibility of truly complex quantum signals in nature, where quantum computers might prove essential for prediction itself.

Quantum Sampling

In 2019 Google demonstrated quantum supremacy on a digital quantum device [5], and in 2024 their latest chip performed a task in minutes where our best classical computers would take 10^25 years [6]. The task they performed is to prepare a highly entangled many-body quantum state and to sample from the corresponding distribution over classical configurations. Quantum supremacy on such sampling problems is on firm ground, with results in complexity theory backing up the experimental claims [7].² Moreover, the classical hardness of quantum sampling appears to be generic in quantum physics. A wide range of quantum systems will generate highly entangled many-body states where sampling becomes classically hard.

However, quantum sampling alone does not refute the universality of classical AI. The output of quantum sampling often appears completely featureless, which cannot be verified by any classical or quantum algorithm, or by any process in our universe for that matter. For example, running the exact same sampling task a second time will produce a list of configurations that will appear unrelated to the original. In order for a phenomenon to be subject to scientific prediction, there must be an experiment that can confirm or deny the prediction. So if quantum sampling has no features that can be experimentally verified, there is nothing to predict, and no pattern for the AI to discover and model.

Quantum Chemistry and Condensed Matter Physics

There are many unsolved problems in quantum chemistry and condensed matter physics that are inaccessible using our best classical simulation algorithms and supercomputers. For example, these occur in the strongly correlated regime of electronic structure in quantum chemistry, and around low-temperature phase transitions of condensed matter systems. We do not understand the electronic structure of FeMoco, the molecule responsible for nitrogen fixation in the nitrogenase enzyme, nor do we understand the phase diagram of the 2D Fermi-Hubbard lattice and whether or not it exhibits superconductivity.

It is possible there are no fundamental barriers for a sufficiently advanced AI to solve these problems. Researchers in the field have achieved major breakthroughs using neural networks to predict complex biological structures like protein folding. One could imagine similar specialized AI models that predict the electronic structure of molecules, or that predict quantum phases of matter. Perhaps the main reason it is currently out of reach is a lack of sufficient training data. Here lies a compelling opportunity for quantum computing: The only feasible way to generate an abundance of accurate training data may be to use a quantum computer, since physical experiments are too difficult, too unreliable, and too slow.

How should we view these problems in physics and chemistry from the perspective of computational complexity? Physicists and chemists often consider systems with a fixed number of parameters, or even single instances. Although computing physical quantities may be extremely challenging, single instances cannot form computationally hard problems, since ultimately only a constant amount of resources is required to solve it. Systems with a fixed number of parameters often behave similarly, since physical quantities tend to depend smoothly on the parameters which allows for extrapolation and learning [8]. Here we can recognize a familiar motif from machine learning: Ab initio prediction is challenging, but prediction becomes efficient after learning from data. Quantum computers are useful for generating training data, but then AI is able to learn from this data and perform efficient inference.

Truly Complex Quantum Signals

While AI might be able to learn much of the patterns of physics and chemistry from quantum-generated data, there remains a deeper possibility: The quantum universe may produce patterns that AI cannot compress and understand. If there are quantum systems that display signals that are truly classically complex, then predicting the pattern will require a quantum computer at inference time, not only in training.

We’ll now envision how such a signal could arise. Imagine a family of quantum systems of arbitrary size N, and at each size N there is a number of independent parameters that is polynomial in N, for example the coefficients of a Hamiltonian or the rotation angles of a quantum circuit. Suppose the system has some physical feature whose signal we would like to compute as a function of the parameters, and this signal has the following properties:

• (Signal) There is a quantum algorithm that efficiently computes the signal. For example, the signal cannot be exponentially small in N.
• (Verifiable) The signal is verifiable, at least by an ideal quantum computer. For example, the task could be to compute an expectation value.
• (Typically complex) When the parameters are chosen randomly, the signal is computationally hard to classically compute in the average case.

If these properties hold, then it’s possible that no machine learning model using a polynomial amount of classical compute can perform the task, even with the help of training data.

The requirement of verifiability by a quantum process ensures that the signal being computed is a robust phenomenon where there is some “fact of the matter”, and a prediction can be confirmed or denied by nature. For example, this holds for any task where the output is the expectation value of some observable. The average-case hardness ensures that hard instances really exist and can be easily generated, rather than only existing in some abstract worst-case that cannot be instantiated.

There is a connection between the verifiability of a computation and its utility to us. Suppose we use a computer to help us design a high-temperature superconductor. If our designed material indeed works as a high-temperature superconductor when fabricated, this forms a verification of the predictions made by our computer. Utility implies verifiability, and likewise, unverifiable computations cannot be useful. However, since nature is quantum, a computation need not be classically verifiable in order to be useful, but only quantumly verifiable. In our high-temperature superconductor example, nature has verified our computer by performing a quantum process.

Making progress

John Preskill’s “entanglement frontier” seeks to understand the collective behavior of many interacting quantum particles [10]. In order to shed light on the fundamental limits of classical AI and the utility of quantum computers in this regime, we must understand if the exponential Hilbert space of quantum theory remains mostly hidden, or if it reveals itself in observable phenomena. The search for classically complex signals forms an exciting research program for making progress. Google recently performed the first demonstration of a classically complex signal on a quantum device: The out-of-time-order correlators³ of random quantum circuits [9]. We can seek to find more such examples, first in abstract models, and then in the real world, to understand how abundant they are in nature.

1. Under widely accepted cryptographic assumptions. ︎
2. Classical computers cannot perform quantum sampling unless the polynomial hierarchy collapses. ︎
3. More precisely, the higher moments of the out-of-time-order correlator. ︎

References

[1] Shor, Peter W. “Algorithms for quantum computation: discrete logarithms and factoring.” Proceedings 35th annual symposium on foundations of computer science. Ieee, 1994.

[2] Shor, Peter W. “Scheme for reducing decoherence in quantum computer memory.” Physical review A 52.4 (1995): R2493.

[3] Shor, Peter W. “Fault-tolerant quantum computation.” Proceedings of 37th conference on foundations of computer science. IEEE, 1996.

[4] Kitaev, A. Yu. “Fault-tolerant quantum computation by anyons.” Annals of physics 303.1 (2003): 2-30.

[5] Arute, Frank, et al. “Quantum supremacy using a programmable superconducting processor.” Nature 574.7779 (2019): 505-510.

[6] Morvan, Alexis, et al. “Phase transitions in random circuit sampling.” Nature 634.8033 (2024): 328-333.

[7] Aaronson, Scott, and Alex Arkhipov. “The computational complexity of linear optics.” Proceedings of the forty-third annual ACM symposium on Theory of computing. 2011.

[8] Huang, Hsin-Yuan, et al. “Provably efficient machine learning for quantum many-body problems.” Science 377.6613 (2022): eabk3333.

[9] Abanin, Dmitry A., et al. “Constructive interference at the edge of quantum ergodic dynamics.” arXiv preprint arXiv:2506.10191 (2025).

[10] Preskill, John. “Quantum computing and the entanglement frontier.” arXiv preprint arXiv:1203.5813 (2012).

Matthew Bolan, Joachim Breitner, Jose Brox, Nicholas Carlini, Mario Carneiro, Floris van Doorn, Martin Dvorak, Andrés Goens, Aaron Hill, Harald Husum, Hernán Ibarra Mejia, Zoltan Kocsis, Bruno Le Floch, Amir Livne Bar-on, Lorenzo Luccioli, Douglas McNeil, Alex Meiburg, Pietro Monticone, Pace P. Nielsen, Emmanuel Osalotioman Osazuwa, Giovanni Paolini, Marco Petracci, Bernhard Reinke, David Renshaw, Marcus Rossel, Cody Roux, Jérémy Scanvic, Shreyas Srinivas, Anand Rao Tadipatri, Vlad Tsyrklevich, Fernando Vaquerizo-Villar, Daniel Weber, Fan Zheng, and I have just uploaded to the arXiv our preprint The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale. This is the final report for the Equational Theories Project, which was proposed in this blog post and also showcased in this subsequent blog post. The aim of this project was to see whether one could collaboratively achieve a large-scale systematic exploration of a mathematical space, which in this case was the implication graph between 4694 equational laws of magmas. A magma is a set equipped with a binary operation (or, equivalently, a multiplication table). An equational law is an equation involving this operation and a number of indeterminate variables. Some examples of equational laws, together with the number that we assigned to that law, include

• (Trivial law):
• (Singleton law):
• (Commutative law):
• (Central groupoid law):
• (Associative law):

The aim of the project was to work out which of these laws imply which others. For instance, all laws imply the trivial law , and conversely the singleton law implies all the others. On the other hand, the commutative law does not imply the associative law (because there exist magmas that are commutative but not associative), nor is the converse true. All in all, there are implications of this type to settle; most of these are relatively easy and could be resolved in a matter of minutes by an expert in college-level algebra, but prior to this project, it was impractical to actually do so in a manner that could be feasibly verified. Also, this problem is known to become undecidable for sufficiently long equational laws. Nevertheless, we were able to resolve all the implications informally after two months, and have them completely formalized in Lean after a further five months.

After a rather hectic setup process (documented in this personal log), progress came in various waves. Initially, huge swathes of implications could be resolved first by very simple-minded techniques, such as brute-force searching all small finite magmas to refute implications; then, automated theorem provers such as Vampire or Mace4 / Prover9 were deployed to handle a large fraction of the remainder. A few equations had existing literature that allowed for many implications involving them to be determined. This left a core of just under a thousand implications that did not fall to any of the “cheap” methods, and which occupied the bulk of the efforts of the project. As it turns out, all of the remaining implications were negative; the difficulty was to construct explicit magmas that obeyed one law but not another. To do this, we discovered a number of general constructions of magmas that were effective at this task. For instance:

• Linear models, in which the carrier was a (commutative or noncommutative) ring and the magma operation took the form for some coefficients , turned out to resolve many cases.
• We discovered a new invariant of an equational law, which we call the “twisting semigroup” of that law, which also allowed us to construct further examples of magmas that obeyed one law but not another , by starting with a base magma that obeyed both laws, taking a Cartesian power of that magma, and then “twisting” the magma operation by certain permutations of designed to preserve but not .
• We developed a theory of “abelian magma extensions”, similar to the notion of an abelian extension of a group, which allowed us to flexibly build new magmas out of old ones in a manner controlled by a certain “magma cohomology group ” which were tractable to compute, and again gave ways to construct magmas that obeyed one law but not another .
• Greedy methods, in which one fills out an infinite multiplication table in a greedy manner (somewhat akin to naively solving a Sudoku puzzle), subject to some rules designed to avoid collisions and maintain a law , as well as some seed entries designed to enforce a counterexample to a separate law . Despite the apparent complexity of this method, it can be automated in a manner that allowed for many outstanding implications to be refuted.
• Smarter ways to utilize automated theorem provers, such as strategically adding in additional axioms to the magma to help narrow the search space, were developed over the course of the project.

Even after applying all these general techniques, though, there were about a dozen particularly difficult implications that resisted even these more powerful methods. Several ad hoc constructions were needed in order to understand the behavior of magmas obeying such equations as E854, E906, E1323, E1516, and E1729, with the latter taking months of effort to finally solve and then formalize.

A variety of GUI interfaces were also developed to facilitate the collaboration (most notably the Equation Explorer tool mentioned above), and several side projects were also created within the project, such as the exploration of the implication graph when the magma was also restricted to be finite. In this case, we resolved all of the implications except for one (and its dual):

Problem 1 Does the law (E677) imply the law (E255) for finite magmas?

See this blueprint page for some partial results on this problem, which we were unable to resolve even after months of effort.

Interestingly, modern AI tools did not play a major role in this project (but it was largely completed in 2024, before the most recent advanced models became available); while they could resolve many implications, the older “good old-fashioned AI” of automated theorem provers were far cheaper to run and already handled the overwhelming majority of the implications that the advanced AI tools could. But I could imagine that such tools would play a more prominent role in future similar projects.

Wouter van Doorn and I have uploaded to the arXiv our paper “Growth rates of sequences governed by the squarefree properties of its translates“. In this paper we answer a number of questions of Erdős} (Problem 1102 and Problem 1103 on the Erdős problem web site) regarding how quickly a sequence of increasing natural numbers can grow if one constrains its translates to interact with the set of squarefree numbers in various ways. For instance, Erdős defined a sequence to have “Property ” if each of its translates only intersected in finitely many points. Erdős believed this to be quite a restrictive condition on , writing “Probably a sequence having property P must increase fairly fast, but I have no results in this direction.”. Perhaps surprisingly, we show that while these sequences must be of density zero, they can in fact grow arbitrary slowly in the sense that one can have for all sufficiently large and any specified function that tends to infinity as . For instance, one can find a sequence that grows like . The density zero claim can be proven by a version of the Maier matrix method, and also follows from known moment estimates on the gaps between squarefree numbers; the latter claim is proven by a greedy construction in which one slowly imposes more and more congruence conditions on the sequence to ensure that various translates of the sequence stop being squarefree after a certain point.

Erdős also defined a somewhat complementary property , which asserts that for infinitely many , all the elements of for are square-free. Since the squarefree numbers themselves have density , it is easy to see that a sequence with property must have (upper) density at most (because it must be “admissible” in the sense of avoiding one residue class modulo for each ). Erdős observed that any sufficiently rapidly growing (admissible) sequence would obey property but beyond that, Erdős writes “I have no precise information about the rate of increase a sequence having property Q must have.”. Our results in this direction may also be surprising: we show that there exist sequences with property with density exactly (or equivalently, ). This requires a recursive sieve construction, in which one starts with an initial scale and finds a much larger number such that is squarefree for most of the squarefree numbers (and all of the squarefree numbers ). We quantify Erdős’s remark by showing that an (admissible) sequence will necessarily obey property once it grows significantly faster than , but need not obey this property if it only grows like . This is achieved through further application of sieve methods.

A third property studied by Erdős is the property of having squarefree sums, so that is squarefree for all . Erdős writes, “In fact one can find a sequence which grows exponentially. Must such a sequence really increase so fast? I do not expect that there is such a sequence of polynomial growth.” Here our results are relatively weak: we can construct such a sequence that grows like , but do not know if this is optimal; the best lower bound we can produce on the growth, coming from the large sieve, is . (Somewhat annoyingly, the precise form of the large sieve inequality we needed was not in the literature, so we have an appendix supplying it.) We suspect that further progress on this problem requires advances in inverse sieve theory.

A weaker property than squarefree sums (but stronger than property ), referred to by Erdős as property , asserts that there are infinitely many such that all elements of (not just the small ones) are square-free. Here, the situation is close to, but not quite the same, as that for property ; we show that sequences with property must have upper density strictly less than , but can have density arbitrarily close to this value.

Finally, we looked at a further question of Erdős on the size of an admissible set . Because the squarefree numbers are admissible, the maximum number of elements of an admissible set up to (OEIS A083544) is at least the number of squarefree elements up to (A013928). It was observed by Ruzsa that the former sequence is greater than the latter for infinitely many . Erdős asked, “Probably this holds for all large x. It would be of some interest to estimate A(x) as accurately as possible.”

We are able to show

However, we do not currently have enough numerical data for the sequence to completely confirm the conjecture in all cases. This could potentially be a crowdsourced project (similar to the Erdős-Guy-Selfridge project reported on in this previous blog post).

I should preface this by saying I don’t actually know that much about education. I taught a bit in my previous life as a professor, yes, but I probably spent more time being taught how to teach than actually teaching.

Recently, the Atlantic had a piece about testing accommodations for university students, like extra time on exams, or getting to do an exam in a special distraction-free environment. The piece quotes university employees who are having more and more trouble satisfying these accommodations, and includes the statistic that 20 percent of undergraduate students at Brown and Harvard are registered as disabled.

The piece has kicked off a firestorm on social media, mostly focused on that statistic (which conveniently appears just before the piece’s paywall). People are shocked, and cynical. They feel like more and more students are cheating the system, getting accommodations that they don’t actually deserve.

I feel like there is a missing mood in these discussions, that the social media furor is approaching this from the wrong perspective. People are forgetting what exams actually ought to be for.

Exams are for the students.

Exams are measurement tools. An exam for a class says whether a student has learned the material, or whether they haven’t, and need to retake the class or do more work to get there. An entrance exam, or a standardized exam like the SAT, predicts a student’s future success: whether they will be able to benefit from the material at a university, or whether they don’t yet have the background for that particular program of study.

These are all pieces of information that are most important to the students themselves, that help them structure their decisions. If you want to learn the material, should you take the course again? Which universities are you prepared for, and which not?

We have accommodations, and concepts like disability, because we believe that there are kinds of students for whom the exams don’t give this information accurately. We think that a student with more time, or who can take the exam in a distraction-free environment, would have a more accurate idea of whether they need to retake the material, or whether they’re ready for a course of study, than a student who has to take the exam under ordinary conditions. And we think we can identify the students who this matters for, and the students for whom this doesn’t matter nearly as much.

These aren’t claims about our values, or about what students deserve. They’re empirical claims, about how test results correlate with outcomes the students want. The conversation, then, needs to be built on top of those empirical claims. Are we better at predicting the success of students that receive accommodations, or worse? Can we measure that at all, or are we just guessing? And are we communicating the consequences accurately to students, that exam results tell them something useful and statistically robust that should help them plan their lives?

Values come in later, of course. We don’t have infinite resources, as the Atlantic piece emphasizes. We can’t measure everyone with as much precision as we would like. At some level, generalization takes over and accuracy is lost. There is absolutely a debate to be had about which measurements we can afford to make, and which we can’t.

But in order to have that argument at all, we first need to agree on what we’re measuring. And I feel like most of the people talking about this piece haven’t gotten there yet.

Having spent a lot of time pondering the octonionic projective plane and its possible role in the Standard Model of particle physics, I’m now getting interested in the ‘bioctonionic plane’, which is based on the bioctonions $\mathbb{C} \otimes \mathbb{O}$ rather than the octonions $\mathbb{O}$.

The bioctonionic plane also has intriguing mathematically connections to the Standard Model. But it’s not a projective plane in the axiomatic sense — and it can’t be constructed by straightforwardly copying the way you build a projective plane over a division algebra, since unlike the octonions, the bioctonions are not a division algebra. Nonetheless we can define points and lines in the bioctonionic plane. The twist is that now some pairs of distinct lines intersect in more than one point — and dually, some pairs of distinct points lie on more than one line. It obeys some subtler axioms, so people call it a Hjelmslev plane.

I am not ready to give a really good explanation of the bioctonionic plane! Instead, I just want to lay out some basic facts about how it fits into mathematics — and possibly physics.

Latham Boyle works at the University of Edinburgh, which is where I am now. Being able to talk to someone who deeply understands octonions and particle physics is very energizing. I’m especially fascinated by this paper of his:

• Latham Boyle, The Standard Model, the exceptional Jordan algebra, and triality.

It gives a convincing argument that the bioctonionic plane may be better than the octonionic projective plane for particle physics. The reason is that the tangent space of any point of the bioctonionic plane is a copy of $(\mathbb{C} \otimes \mathbb{O})^2$, a 16-dimensional complex vector space. The symmetry group of the bioctonionic plane is the exceptional Lie group $\text{E}_6$. Sitting inside the stabilizer group of any given point is a copy of the Standard Model gauge group. And — here’s the cool part — this group acts on $(\mathbb{C} \otimes \mathbb{O})^2$ just as it does on one generation of fermions (not their antiparticles). If we try the same trick using the octonionic projective plane, we can fit the Standard Model gauge group in the stabilizer group of a point in a very natural way, but its action on the tangent space is its action only on left-handed fermions.

I want to explain this in detail, but not today. Instead, I want to skim through some basic facts about the bioctonionic plane.

First, this plane is one of the four Rosenfeld planes:

• the octonionic projective plane $\mathbb{O}\text{P}^2$, a 16-dimensional compact Riemannian manifold on which the compact Lie group $\text{F}_4$ acts transitively as isometries, with the stabilizer of any point being $\text{Spin}(9)$. This is a symmetric space, and as such it’s called FII in Cartan’s classification.

• the bioctonionic plane $(\mathbb{C} \otimes \mathbb{O})\mathbb{P}^2$, a 32-dimensional compact Riemannian manifold on which the compact Lie group $\text{E}_6$ acts transitively as isometries, with the stabilizer of any point being $(\text{Spin}(10) \times \text{U}(1))/\mathbb{Z}_4$. This is the symmetric space EIII.

• the quateroctonionic plane $(\mathbb{H} \otimes \mathbb{O}) \mathbb{P}^2$, a 64-dimensional compact Riemannian manifold on which the compact Lie group $\text{E}_7$ acts transitively as isometries, with the stabilizer of any point being $(\text{Spin}(12) \times \text{Sp}(1))/\mathbb{Z}_2$. This is the symmetric space EVI.

• the octooctonionic plane $(\mathbb{O} \otimes \mathbb{O}) \mathbb{P}^2$, a 128-dimensional compact Riemannian manifold on which the compact Lie group $\text{E}_8$ acts transitively as isometries, with the stabilizer of any point being $\text{Spin}(16)/\mathbb{Z}_2$. This is the symmetric space EVIII.

There’s a nice network of systematic approaches to these spaces: they form one row of the so-called magic square, so one way to learn about the bioctonionic plane is to study the magic square, for example here:

• Chris H. Barton and Anthony Sudbery, Magic squares of Lie algebras. Available as arXiv:math/0001083; see also arXiv:0203010 for a “streamlined and extended” version, which has more yet also less.

Here you can also find lots of references to earlier work, e.g. to Freudenthal and Tits. The basic idea of the magic square is that you start with two normed division algebras $\mathbb{K}, \mathbb{K}'$ and from them you build a Lie algebra, which gives a Lie group $G(\mathbb{K},\mathbb{K}')$. There’s also a way to get a subgroup $H(\mathbb{K}, \mathbb{K}')$, and the quotient space

$$(\mathbb{K} \otimes \mathbb{K}')\text{P}^2 = G(\mathbb{K},\mathbb{K}')/H(\mathbb{K}, \mathbb{K}')$$

is a kind of ‘plane’ on which the group $G(\mathbb{K},\mathbb{K}')$ acts. If you take $\mathbb{K}' = \mathbb{O}$, this construction gives you the four Rosenfeld planes listed above.

Each one of these planes is a compact Riemannian symmetric space: this means it’s a connected compact Riemannian manifold $M$ such that for each point $p \in M$ there’s an isometry

$$\sigma_p \colon M \to M$$

that fixes $p$, acts as $-1$ on its tangent space, and squares to the identity. This map is called ‘reflection across $p$’ for the obvious reason. For example, a round 2-sphere is a symmetric space, with $\sigma_p$ switching the red and blue tangent vectors if $p$ is the black dot:

Cartan classified compact Riemannian symmetric spaces, and there’s a nice theory of them. Any compact simple Lie group is one, and most of the rest come in infinite series connected to real and complex Clifford algebras, as I explained here. But there are 9 extra ones, all related to the octonions and exceptional Lie groups. The Rosenfeld planes are four of these.

You can learn this material starting on Wikipedia and then going to a textbook, ideally this:

• Sigurdur Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978.

Helgason taught me Lie theory when I was a grad student at MIT, so I have a fondness for his book—but it’s also widely accepted as the most solid text on symmetric spaces!

The bioctonionic plane is even better: it’s a compact hermitian symmetric space: a compact Riemannian symmetric space $M$ where each tangent space $T_p M$ has a complex structure

$$J \colon T_p M \to T_p M , \qquad J^2 = -1$$

compatible with the metric, and reflection about each point preserves this complex structure. I mentioned that the bioctonionic plane is

$$(\mathbb{C} \otimes \mathbb{O})\text{P}^2 \cong G(\mathbb{C},\mathbb{O})/H(\mathbb{C},\mathbb{O})$$

where

$$G(\mathbb{C},\mathbb{O}) = \text{E}_6$$

acts transitively, and the stabilizer of a point is

$$H(\mathbb{C}, \mathbb{O}) = (\text{Spin}(10) \times \text{U}(1))/\mathbb{Z}_4$$

The $\text{U}(1)$ here comes from the complex structure!

Wikipedia is especially thorough on hermitian symmetric spaces, so if you want to delve into those, start here:

• Wikipedia, Hermitian symmetric space.

Another tack is to focus on the exceptional Lie groups $\text{F}_4, \text{E}_6, \text{E}_7$ and $\text{E}_8$ and their connection to the nonassociative algebras $\mathfrak{h}_3(\mathbb{O})$, $\mathfrak{h}_3(\mathbb{C} \otimes \mathbb{O})$, $\mathfrak{h}_3(\mathbb{H} \otimes \mathbb{O})$ and $\mathfrak{h}_3(\mathbb{O} \otimes \mathbb{O})$, respectively. Here I recommend this:

• Ichiro Yokota, Exceptional Lie Groups. Available as arXiv:0902.0431. (See especially Chapter 3, for $\text{E}_6$ and the complexified exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{C} \otimes \mathbb{O})$.)

If you have a fondness for algebra you may also want to learn how symmetric spaces arise from Jordan triple systems or Jordan pairs. This is important if we wish to see the bioctonionic plane as the space of pure states of some exotic quantum system!

Now, this is much easier to do for the octonionic plane, because that’s the space of pure states for the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$, which is a Euclidean Jordan algebra, meaning one in which a sum of squares can only be zero if all those squares are zero. You can think of a Euclidean algebra $A$ as consisting of observables, with the sums of squares being ‘nonnegative’ observables. These nonnegative observables form a convex cone $K \subseteq A$. The dual vector space $A^\ast$ contains a cone of linear functionals $f$ that send these nonnegative observables to nonnegative real numbers — I’ll call this the dual cone $K^\ast$. The functionals $f \in K^\ast$ with $f(1) = 1$ are called states. The states form a convex set, and the extreme points are called pure states. All of this fits nicely into a modern framework for understanding quantum theory and potential generalizations, called ‘generalized probabilistic theories’:

• Howard Barnum, Alexander Wilce, Post-classical probability theory, in Quantum Theory: Informational Foundations and Foils, eds. Giulio Chiribella, Robert W. Spekkens, Springer, 2016. (See Section 5 for Jordan algebras, and ignore the fact that they say the exceptional Jordan algebra consists of $2 \times 2$ matrices: they know perfectly well that they’re $3 \times 3$.)

The underlying math, with a lot more about symmetric spaces, cones and Euclidean Jordan algebras but with none of the physics interpretation, is wonderfully explained here:

• Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford U. Press, 1994.

A crucial fact throughout this book is that when you start with a Euclidean Jordan algebra $A$, its cone $K$ of nonnegative observables is self-dual: there’s an isomorphism of vector spaces $A \cong A^\ast$ that maps $K$ to $K^\ast$ in a one-to-one and onto way. The cone $K$ is also homogeneous, meaning that the group of invertible linear transformations of $A$ preserving $K$ acts transitively on the interior of $K$. Faraut and Korányi call a self-dual homogeneous cone a symmetric cone — and they show that any symmetric cone comes from a Euclidean Jordan algebra! This result plays an important role in modern work on the foundations of quantum theory.

Unfortunately, I’m telling you all this nice stuff about Euclidean Jordan algebras and symmetric cones just to say that while all this applies to the octonionic projective plane, sadly, it does not apply to the bioctonionic plane! The bioctonionic plane does not come from a Euclidean Jordan algebra or a symmetric cone. Thus, to understand it as a space of pure states, we’d have to resort to a more general formalism.

There are a few papers that attempt exactly this:

• Lawrence C. Biedenharn and Piero Truini, An $\mathcal{E}_6 \otimes \mathcal{U}(1)$ invariant quantum mechanics for a Jordan pair, Journal of Mathematical Physics 23 (1982), 1327-1345.

• Lawrence C. Biedenharn and Piero Truini, Exceptional groups and elementary particle structures, Physica A: Statistical Mechanics and its Applications 14 (1982), 257–270.

• Lawrence C. Biedenharn, G. Olivieri and Piero Truini, Three graded exceptional algebras and symmetric spaces, Zeitschrift Physik C — Particles and Fields 33 (1986), 47–65.

Here’s the basic idea. We can define $\mathfrak{h}_3(\mathbb{C} \otimes \mathbb{O})$ to consist of $3 \times 3$ hermitian matrices with entries in $\mathbb{C} \otimes \mathbb{O}$, where ‘hermitian’ is defined using the star-algebra structure on $\mathbb{C} \otimes \mathbb{O}$ where we conjugate the octonion part but not the complex part! Then $\mathfrak{h}_3(\mathbb{C} \otimes \mathbb{O})$ is just the complexification of $\mathfrak{h}_3(\mathbb{O})$:

$$\mathfrak{h}_3(\mathbb{C} \otimes \mathbb{O}) \cong \mathbb{C} \otimes \mathfrak{h}_3(\mathbb{O})$$

Then because $\mathfrak{h}_3(\mathbb{O})$ is a Jordan algebra over $\mathbb{R}$, $\mathfrak{h}_3(\mathbb{C} \otimes \mathbb{O})$ is a Jordan algebra over $\mathbb{C}$. So we can do a lot with it. But it’s not a Euclidean Jordan algebra.

Puzzle. Show that it’s not.

So, Biedenharn and Truini need a different approach to relate $\mathfrak{h}_3(\mathbb{C} \otimes \mathbb{O})$ to some sort of exotic quantum system. And they use an approach already known to mathematicians: namely, the theory of Jordan pairs! Here you work, not with a single element of $\mathfrak{h}_3(\mathbb{C} \otimes \mathbb{O})$, but with a pair.

Jordan triple systems and Jordan pairs are two closely related generalizations of Jordan algebras. I’ve been working on the nLab articles about these concepts, so click the links if you want to learn more about them. I explain how either of these things gives you a 3-graded Lie algebra — that is, a $\mathbb{Z}$-graded Lie algebra that is nonvanishing only in the middle 3 grades:

$$\mathfrak{g} = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$$

And from a 3-graded Lie algebra you can get a symmetric space $G/H$ where the Lie algebra of $G$ is $\mathfrak{g}$ and the Lie algebra of $H$ is $\mathfrak{g}_0$. Each tangent space of this symmetric space is thus isomorphic to $\mathfrak{g}_{-1} \oplus \mathfrak{g}_1$.

In the case relevant to the bioctonionic plane, the 3-graded Lie algebra is

$$\mathfrak{e}_6 = (\mathbb{C} \otimes \mathbb{O}) \; \oplus \; (\mathfrak{so}(10) \oplus \mathbb{R}) \; \oplus \; (\mathbb{C} \otimes \mathbb{O})$$

So, the bioctonionic plane is a symmetric space on which $\text{E}_6$ acts, with stabilizer group $\text{Spin}(10) \times \text{U}(1)$ (up to covering spaces), and with tangent space isomorphic to $(\mathbb{C} \otimes \mathbb{O})^2$.

So all this is potentially very nice. For much more on this theory, try the work of Ottmar Loos:

• Ottmar Loos, Symmetric Spaces, Volume 1: General Theory, W. A. Benjamin, 1969.

• Ottmar Loos, Symmetric Spaces, Volume 2: Compact Spaces and Classification, W. A. Benjamin, 1969.

• Ottmar Loos, Jordan Pairs, Lecture Notes in Mathematics 460, Springer, Berlin, 1975.

• Ottmar Loos, Jordan triple systems, $R$-spaces, and bounded symmetric domains, Bulletin of the American Mathematical Society 77 (1971), 558–561.

• Ottmar Loos, Bounded Symmetric Domains and Jordan Pairs, Mathematical Lectures, University of California, Irvine, 1977.

That’s a lot of stuff! For a quick overview of Loos’ work, I find this helpful:

• Kevin McCrimmon, Jordan Pairs, by Ottmar Loos, review in Bulletin of the American Mathematical Society 84 4 (1978), 685–690.

Unfortunately Loos does not delve into examples, particularly the bioctonionic plane. For that, try Biedenharn and Truini, and also these:

• Daniele Corradetti, Alessio Marrani, David Chester and R. Aschheim, Conjugation matters: bioctonionic Veronese vectors and Cayley-Rosenfeld planes, Int. J. Geom. Meth. Mod. Phys. 19 (2022), 2250142.

• Jian Qiu, Plücker coordinates and the Rosenfeld planes, Journal of Geometry and Physics 206 (2024), 105331.

To wrap things up, I should say a bit about ‘Hjelmslev planes’, since the bioctonionic plane is supposed to be one of these. Axiomatically, a Hjelmslev plane is a set $P$ of points, a set $L$ of lines, and an incidence relation between points and lines. We require that for any two distinct points there is at least one line incident to both, and for any two distinct line there is at least one point incident to both. If two points are incident to more than one line we say they are neighbors. If two lines are incident to more than one point we say they are neighbors. We demand that both these ‘neighbor’ relations are equivalence relations, and that if we quotient $P$ and $L$ by these equivalence relations, we get an axiomatic projective plane.

Challenge. What projective plane do we get if we apply this quotient construction to the bioctonionic plane?

My only guess is that we get the octonionic projective plane — but I don’t know why.

The literature on Hjelmslev planes seems a bit difficult, but I’m finding this to be a good introduction:

• Joanne L. Hall and Asha Rao, An algorithm for constructing Hjelmslev planes, in Algebraic Design Theory and Hadamard Matrices, Springer, 2014.

The answer to my puzzle should be here, because they’re talking about Hjelmslev planes built using split octonion algebras (like $\mathbb{C} \otimes \mathbb{O}$):

• Tonny A. Springer and Ferdinand D. Veldkamp, On Hjelmslev–Moufang planes, Mathematicsche Zeitschrift 107 (1968), 249–263.

But I don’t see the answer here yet!

• Part 1. How to define octonion multiplication using complex scalars and vectors, much as quaternion multiplication can be defined using real scalars and vectors. This description requires singling out a specific unit imaginary octonion, and it shows that octonion multiplication is invariant under $\mathrm{SU}(3)$.
• Part 2. A more polished way to think about octonion multiplication in terms of complex scalars and vectors, and a similar-looking way to describe it using the cross product in 7 dimensions.
• Part 3. How a lepton and a quark fit together into an octonion — at least if we only consider them as representations of $\mathrm{SU}(3)$, the gauge group of the strong force. Proof that the symmetries of the octonions fixing an imaginary octonion form precisely the group $\mathrm{SU}(3)$.
• Part 4. Introducing the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$: the $3 \times 3$ self-adjoint octonionic matrices. A result of Dubois-Violette and Todorov: the symmetries of the exceptional Jordan algebra preserving their splitting into complex scalar and vector parts and preserving a copy of the $2 \times 2$ adjoint octonionic matrices form precisely the Standard Model gauge group.
• Part 5. How to think of $2 \times 2$ self-adjoint octonionic matrices as vectors in 10d Minkowski spacetime, and pairs of octonions as left- or right-handed spinors.
• Part 6. The linear transformations of the exceptional Jordan algebra that preserve the determinant form the exceptional Lie group $\mathrm{E}_6$. How to compute this determinant in terms of 10-dimensional spacetime geometry: that is, scalars, vectors and left-handed spinors in 10d Minkowski spacetime.
• Part 7. How to describe the Lie group $\mathrm{E}_6$ using 10-dimensional spacetime geometry. This group is built from the double cover of the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
• Part 8. A geometrical way to see how $\mathrm{E}_6$ is connected to 10d spacetime, based on the octonionic projective plane.
• Part 9. Duality in projective plane geometry, and how it lets us break the Lie group $\mathrm{E}_6$ into the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
• Part 10. Jordan algebras, their symmetry groups, their invariant structures — and how they connect quantum mechanics, special relativity and projective geometry.
• Part 11. Particle physics on the spacetime given by the exceptional Jordan algebra: a summary of work with Greg Egan and John Huerta.
• Part 12. The bioctonionic projective plane and its connections to algebra, geometry and physics.

A while ago on this blog, Tom posted a question about teaching calculus: what do you tell students the value of $\displaystyle\int \frac{1}{x}\,dx$ is? The standard answer is $\ln{|x|}+C$, with $C$ an “arbitrary constant”. But that’s wrong if $\displaystyle\int$ means (as we also usually tell students it does) the “most general antiderivative”, since

$$F(x) = \begin{cases} \ln{|x|} + C^- &\text{if}\;x\lt 0\\ \ln{|x|} + C^+ &\text{if}\;x\gt 0 \end{cases}$$

is a more general antiderivative, for two arbitrary constants $C^-$ and $C^+$. (I’m writing $\ln$ for the natural logarithm function that Tom wrote as $\log$, for reasons that will become clear later.)

In the ensuing discussion it was mentioned that other standard indefinite integrals like $\displaystyle\int \frac{1}{x^2}\,dx = -\frac{1}{x} + C$ are just as wrong. This happens whenever the domain of the integrand is disconnected: the “arbitrary constant” $C$ is really only locally constant. Moreover, Mark Meckes pointed out that believing in such formulas can lead to mistaken calculations such as

$$\int_{-1}^1 \frac{1}{x^2}\,dx = \left.-\frac{1}{x}\right]_{-1}^1 = -2$$

which is “clearly nonsense” since the integrand is everywhere positive.

In this post I want to argue that there’s actually a very natural perspective from which $\displaystyle\int \frac{1}{x^2}\,dx = -\frac{1}{x} + C$ is correct, while $\displaystyle\int \frac{1}{x}\,dx = \ln{|x|}+C$ is wrong for a different reason.

The perspective in question is complex analysis. Most of the functions encountered in elementary calculus are actually complex-analytic — the only real counterexamples are explicit “piecewise” functions and things like ${|x|}$, which are mainly introduced as counterexamples to illustrate the meaning of continuity and differentiability. Therefore, it’s not unreasonable to interpret the indefinite integral $\displaystyle\int f(x)\,dx$ as asking for the most general complex-analytic antiderivative of $f$. And the complex domain of $\frac{1}{z}$ and $\frac{1}{z^2}$ is $\mathbb{C}\setminus \{0\}$, which is connected!

Thus, for instance, since $\frac{d}{d z}\left[-\frac{1}{z}\right] = \frac1{z^2}$, it really is true that the most general (complex-analytic) antiderivative of $\frac{1}{z^2}$ is $-\frac{1}{z}+C$ for a single arbitrary constant $C$, so we can write $\displaystyle\int \frac{1}{z^2}\,dz = -\frac{1}{z} + C$. Note that any such antiderivative has the same domain $\mathbb{C}\setminus \{0\}$ as the original function.

In addition, the dodgy calculation

$$\int_{-1}^1 \frac{1}{z^2}\,dz = \left.-\frac{1}{z}\right]_{-1}^1 = -2$$

is actually correct if we interpret $\int_{-1}^1$ to mean the integral along some (in fact, any) curve in $\mathbb{C}$ from $-1$ to $1$ that doesn’t pass through the singularity $z=0$. Of course, this doesn’t offend against signs because any such path must pass through non-real numbers, whose squares can contribute negative real numbers to the integral.

The case of $\displaystyle\int \frac{1}{z}\,dz$ is a bit trickier, because the complex logarithm is multi-valued. However, if we’re willing to work with multi-valued functions (which precisely means functions whose domain is a Riemann surface covering some domain in $\mathbb{C}$), we have such a multi-valued function that I’ll denote $\log$ (in contrast to the usual real-number function $\ln$) defined on a connected domain, and there we have $\frac{d}{d z}\left[ \log(z) \right] = \frac{1}{z}$. Thus, the most general (complex-analytic) antiderivative of $\frac{1}{z}$ is $\log(z)+C$ where $C$ is a single arbitrary constant, so we can write $\displaystyle\int \frac{1}{z}\,dz = \log(z) + C$.

What happened to $\ln{|x|}$? Well, as it happens, if $x$ is a negative real number and $Log$ denotes the principal branch of the complex logarithm, then $Log(x) = \ln{|x|} + i\pi$, hence $\ln{|x|} = Log(x) - i\pi$. Therefore, the antiderivative $\ln{|x|}$ for negative real $x$ is of the form $\Log(x)+C$, where $\Log$ is a branch of the complex logarithm and $C$ is a constant (namely, $-i\pi$).