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

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

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

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

This October, fantasy readers are devouring a sequel: the final installment in Philip Pullman’s trilogy The Book of Dust. The series follows student Lyra Silvertongue as she journeys from Oxford to the far east. Her story features alternate worlds, souls that materialize as talking animals, and a whiff of steampunk. We first met Lyra in the His Dark Materials trilogy, which Pullman began publishing in 1995. So some readers have been awaiting the final Book of Dust volume for 30 years.

Another sequel debuts this fall. It won’t spur tens of thousands of sales; nor will Michael Sheen narrate an audiobook version of it. Nevertheless, the sequel should provoke as much thought as Pullman’s: the sequel to the Maryland Quantum-Thermodynamics Hub’s first three years.

The Maryland Quantum-Thermodynamics Hub debuted in 2022, courtesy of a grant from the John F. Templeton Foundation. Six theorists, three based in Maryland, have formed the hub’s core. Our mission has included three prongs: research, community building, and outreach. During the preceding decade, quantum thermodynamics had exploded, but mostly outside North America. We aimed to provide a lodestone for the continent’s quantum-thermodynamics researchers and visitors.

Also, we aimed to identify the thermodynamics of how everyday, classical physics emerges from quantum physics. Quantum physics is reversible (doesn’t distinguish the past from the future), is delicate (measuring a quantum system can disturb it), and features counterintuitive phenomena such as entanglement. In contrast, our everyday experiences include irreversibility (time has an arrow), objectivity (if you and I read this article, we should agree about its contents), and no entanglement. How does quantum physics give rise to classical physics at large energy and length scales? Thermodynamics has traditionally described macroscopic, emergent properties. So quantum thermodynamics should inform our understanding of classical reality’s emergence from quantum mechanics.

Our team has approached this opportunity from three perspectives. One perspective centers on quantum Darwinism, a framework for quantifying how interactions spread information about an observed quantum system. Another perspective highlights decoherence, the contamination of a quantum system by its environment. The third perspective features incompatible exchanged quantities, described in an earlier blog post. Or two. Or at least seven.

Each perspective led us to discover a tension, or apparent contradiction, that needs resolving. One might complain that we failed to clinch a quantum-thermodynamic theory of the emergence of classical reality. But physicists adore apparent contradictions as publishers love splashing "New York Times bestseller" on their book covers. So we aim to resolve the tensions over the next three years.

I’ll illustrate the tensions with incompatible exchanged quantities, of course. Physicists often imagine a small system, such as a quantum computer, interacting with a large environment, such as the surrounding air and the table on which the quantum computer sits. The system and environment may exchange energy, particles, electric charge, etc. Typically, the small system thermalizes, or reaches a state mostly independent of its initial conditions. For example, after exchanging enough energy with its environment, the system ends up at the environment’s temperature, mostly regardless of the system’s initial temperature.

For decades, physicists implicitly assumed that the exchanged quantities are compatible: one can measure them simultaneously. But one can’t measure all of a quantum system’s properties simultaneously. Position and momentum form the most famous examples. Incompatibility epitomizes quantum physics, underlying Heisenberg’s uncertainty relation, quantum error correction, and more. So collaborators and I ask how exchanged quantities’ incompatibility alters thermalization, which helps account for time’s arrow.

Our community has discovered that such incompatibility can hinder certain facets of thermalization—in a sense, stave off certain aspects of certain quantum systems’ experience of time. But incompatible exchanged quantities enhance other features of thermalization. How shall we reconcile the hindrances with the enhancements? Does one of the two effects win out? I hope to report back in three years. For now, I’m rooting for Team Hindrance.

In addition to resolving apparent conflicts, we’re adding a fourth perspective to our quiver—a gravitational one. In our everyday experiences, space-time appears smooth; unlike Lyra’s companion Will in The Subtle Knife, we don’t find windows onto other worlds. But quantum physics, combined with general relativity, suggests that you’d find spikes and dips upon probing space-time over extremely short length scales. How does smooth space-time emerge from its quantum underpinnings? Again, quantum thermodynamics should help us understand.

To address these challenges, we’re expanding the hub’s cast of characters. The initial cast included six theorists. Two more are joining the crew, together with the hub’s first two experimentalists. So is our first creative-writing instructor, who works at the University of Maryland (UMD) Jiménez-Porter Writers’ House.

As the hub has grown, so has the continent’s quantum-thermodynamics community. We aim to continue expanding that community and strengthening its ties to counterparts abroad. As Lyra learned in Pullman’s previous novel, partnering with Welsh miners and Czech book sellers and Smyrnan princesses can further one’s quest. I don’t expect the Maryland Quantum-Thermodynamics Hub to attract Smyrnan princesses, but a girl can dream. The hub is already partnering with the John F. Templeton Foundation, Normal Computing, the Fidelity Center for Applied Technology, the National Quantum Laboratory, Maryland’s Capital of Quantum team, and more. We aim to integrate quantum thermodynamics into North America’s scientific infrastructure, so that the field thrives here even after our new grant terminates. Reach out if you’d like to partner with us.

To unite our community, the hub will host a gathering—a symposium or conference—each year. One conference will feature quantum thermodynamics and quantum-steampunk creative writing. Scientists and authors will present. We hope that both groups will inspire each other, as physicist David Deutsch’s work on the many-worlds formulation of quantum theory inspired Pullman.

That conference will follow a quantum-steampunk creative-writing course to take place at UMD during spring 2026. I’ll co-teach the course with creative-writing instructor Edward Daschle. Students will study quantum thermodynamics, read published science-fiction stories, write quantum-steampunk stories, and critique each other’s writing. Five departments have cross-listed the course: physics, arts and humanities, computer science, chemistry, and mechanical engineering. If you’re a UMD student, you can sign up in a few weeks. Do so early; seats are limited! We welcome graduate students and undergrads, the latter of whom can earn a GSSP general-education credit.¹ Through the course, the hub will spread quantum thermodynamics into Pullman’s world—into literature.

Pullman has entitled his latest novel The Rose Field. The final word refers to an object studied by physicists. A field, such as an electric or gravitational field, is a physical influence spread across space. Hence fiction is mirroring physics—and physics can take its cue from literature. As ardently as Lyra pursues the mysterious particle called Dust, the Maryland Quantum-Thermodynamics Hub is pursuing a thermodynamic understanding of the classical world’s emergence from quantum physics. And I think our mission sounds as enthralling as Lyra’s. So keep an eye on the hub for physics, community activities, and stories. The telling of Lyra’s tale may end this month, but the telling of the hub’s doesn’t.

¹Just don’t ask me what GSSP stands for.

I don’t usually do obituaries here, but sometimes I have something worth saying.

Chen Ning Yang, a towering figure in particle physics, died last week.

I never met him. By the time I started my PhD at Stony Brook, Yang was long-retired, and hadn’t visited the Yang Institute for Theoretical Physics in quite some time.

(Though there was still an office door, tucked behind the institute’s admin staff, that bore his name.)

The Nobel Prize doesn’t always honor the most important theoretical physicists. In order to get a Nobel Prize, you need to discover something that gets confirmed by experiment. Generally, it has to be a very crisp, clear statement about reality. New calculation methods and broader new understandings are on shakier ground, and theorists who propose them tend to be left out, or at best combined together into lists of partial prizes long after the fact.

Yang was lucky. With T. D. Lee, he had made that crisp, clear statement. He claimed that the laws of physics, counter to everyone’s expectations, are not the same when reflected in a mirror. In 1956, Wu confirmed the prediction, and Lee and Yang got the prize the year after.

That’s a huge, fundamental discovery about the natural world. But as a theorist, I don’t think that was Yang’s greatest accomplishment.

Yang contributed to other fields. Practicing theorists have seen his name strewn across concepts, formalisms, and theorems. I didn’t have space to talk about him in my article on integrability for Quanta Magazine, but only just barely: another paragraph or two, and he would have been there.

But his most influential contribution is something even more fundamental. And long-time readers of this blog should already know what it is.

Yang, along with Robert Mills, proposed Yang-Mills Theory.

There isn’t a Nobel prize for Yang-Mills theory. In 1953, when Yang and Mills proposed the theory, it was obviously wrong, a theory that couldn’t explain anything in the natural world, mercilessly mocked by famous bullshit opponent Wolfgang Pauli. Not even an ambitious idea that seemed outlandish (like plate tectonics), it was a theory with such an obvious missing piece that, for someone who prioritized experiment like the Nobel committee does, it seemed pointless to consider.

All it had going for it was that it was a clear generalization, an obvious next step. If there are forces like electromagnetism, with one type of charge going from plus to minus, why not a theory with multiple, interacting types of charge?

Nothing about Yang-Mills theory was impossible, or contradictory. Mathematically, it was fine. It obeyed all the rules of quantum mechanics. It simply didn’t appear to match anything in the real world.

But, as theorists learn, nature doesn’t let a good idea go to waste.

Of the four fundamental forces of nature, as it would happen, half are Yang-Mills theories. Gravity is different, electromagnetism is simpler, and could be understood without Yang and Mills’ insights. But the weak nuclear force, that’s a Yang-Mills theory. It wasn’t obvious in 1953 because it wasn’t clear how the massless, photon-like particles in Yang-Mills theory could have mass, and it wouldn’t become clear until the work of Peter Higgs over a decade later. And the strong nuclear force, that’s also a Yang-Mills theory, missed because of the ability of such a strong force to "confine" charges, hiding them away.

So Yang got a Nobel, not for understanding half of nature’s forces before anyone else had, but from a quirky question of symmetry.

In practice, Yang was known for all of this, and more. He was enormously influential. I’ve heard it claimed that he personally kept China from investing in a new particle collider, the strength of his reputation the most powerful force on that side of the debate, as he argued that a developing country like China should be investing in science with more short-term industrial impact, like condensed matter and atomic physics. I wonder if the debate will shift with his death, and what commitments the next Chinese five-year plan will make.

Ultimately, Yang is an example of what a theorist can be, a mix of solid work, counterintuitive realizations, and the thought-through generalizations that nature always seems to make use of in the end. If you’re not clear on what a theoretical physicist is, or what one can do, let Yang’s story be your guide.

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Given a threshold {y>1}, a {y}-smooth number (or {y}-friable number) is a natural number {n} whose prime factors are all at most {y}. We use {\Psi(x,y)} to denote the number of {y}-smooth numbers up to {x}. In studying the asymptotic behavior of {\Psi(x,y)}, it is customary to write {y} as {x^{1/u}} (or {x} as {y^u}) for some {u>0}. For small values of {u}, the behavior is straightforward: for instance if {0 < u < 1}, then all numbers up to {x} are automatically {y}-smooth, so

\displaystyle \Psi(x,y) \sim x

\displaystyle \Psi(x,y) \sim x - \sum_{y < p \leq x} (\frac{x}{p} + O(1))

\displaystyle \sim x - x (\log\log x - \log\log y)

\displaystyle \sim x (1 - \log u)

\displaystyle \Psi(x,y) \sim x (1 - \log u + \int_2^u \frac{\log(t-1)}{t} dt)

More generally, for any fixed {u>0}, de Bruijn showed that

\displaystyle \Psi(x, y) \sim \rho(u) x

\displaystyle u \rho'(u) + \rho(u-1) = 0

The asymptotic behavior of {\rho(u)} as {u \rightarrow \infty} is rather complicated. Very roughly speaking, it has inverse factorial behavior; there is a general upper bound {\rho(u) \leq 1/\Gamma(u+1)}, and a crude asymptotic

\displaystyle \rho(u) = u^{-u+o(u)} = \exp( - u \log u + o(u \log u)).

\displaystyle \xi(u) = \log u + \log\log u + O(1)

\displaystyle \xi(u) = \log u + \log\log u + O(\frac{\log\log u}{\log u}).

\displaystyle \int_0^{\xi(u)} \frac{e^s - 1}{\xi(u)} ds = u - 1.

This asymptotic (2) is quite complicated, and so one does not expect there to be any simple argument that could recover it without extensive computation. However, it turns out that one can use a "maximum entropy" analysis to get a reasonably good heuristic approximation to (2), that at least reveals the role of the mysterious function {\xi(u)}. The purpose of this blog post is to give this heuristic.

Viewing {x = y^u}, the task is to try to count the number of {y}-smooth numbers of magnitude {y^u}. We will propose a probabilistic model to generate {y}-smooth numbers as follows: for each prime {p \leq y}, select the prime {p} with an independent probability {c_p/p} for some coefficient {c_p}, and then multiply all the selected primes together. This will clearly generate a random {y}-smooth number {n}, and by the law of large numbers, the (log-)magnitude of this number should be approximately

\displaystyle \log n \approx \sum_{p \leq y} \frac{c_p}{p} \log p, \ \ \ \ \ (4)

The indicator {1_{p|n}} of the event that {p} divides this number is a Bernoulli random variable with mean {c_p/p}, so the Shannon entropy of this random variable is

\displaystyle \mathbf{H}(1_{p|n}) = - \frac{c_p}{p} \log(\frac{c_p}{p}) - (1 - \frac{c_p}{p}) \log(1 - \frac{c_p}{p}).

\displaystyle \mathbf{H}(1_{p|n}) \approx \frac{c_p}{p} \log p - \frac{c_p}{p} \log c_p + \frac{c_p}{p}.

\displaystyle \mathbf{H}(n) = \sum_{p \leq y} \mathbf{H}(1_{p|n});

\displaystyle \mathbf{H}(n) \approx u \log y - \sum_{p \leq y} \frac{c_p}{p} (\log c_p - 1).

\displaystyle \exp(\mathbf{H}(n)) \approx \exp\left(u \log y - \sum_{p \leq y} \frac{c_p}{p} (\log c_p - 1)\right)

\displaystyle = \exp\left(- \sum_{p \leq y} \frac{c_p \log c_p - c_p}{p}\right) x.

One could solve this constrained optimization problem directly using Lagrange multipliers, but we simplify things a bit by passing to a continuous limit. We take a continuous ansatz {c_p = f(\log p / \log y)}, where {f: [0,1] \rightarrow {\bf R}} is a smooth function. Using Mertens’ theorem, the constraint (5) then heuristically becomes

\displaystyle \int_0^1 f(t)\ dt = u \ \ \ \ \ (7)

This is a standard calculus of variations problem. The Euler-Lagrange equation for this problem can be easily worked out to be

\displaystyle \frac{\log f(t)}{t} = \lambda

\displaystyle \frac{e^\lambda - 1}{\lambda} = u

\displaystyle \rho(u) \approx \exp\left( - \int_0^1 \frac{e^{\xi(u) t} \xi(u) t - e^{\xi(u) t}}{t}\ dt \right)

\displaystyle \approx \exp\left( - e^{\xi(u)} + 1 + \int_0^1 \frac{e^{\xi(u) t} - 1}{t}\ dt + \int_0^1 \frac{1}{t}\ dt \right)

\displaystyle \approx \exp\left( - u \xi(u) + \int_0^{\xi(u)} \frac{e^s - 1}{s}\ ds + C\right)

\displaystyle C = \int_0^1 \frac{1}{t}\ dt.

Nevertheless, this demonstrates that the maximum entropy method can achieve a reasonably good heuristic understanding of smooth numbers. In fact we also gain some insight into the "anatomy of integers" of such numbers: the above analysis suggests that a typical {y}-smooth number {n} will be divisible by a given prime {p \sim y^t} with probability about {e^{\xi(u) t}/p}. Thus, for {t = 1 - O(1/\log u)}, the probability of being divisible by {p} is elevated by a factor of about {\asymp e^{\xi(u)} \asymp u \log u} over the baseline probability {1/p} of an arbitrary (non-smooth) number being divisible by {p}; so (by Mertens’ theorem) a typical {y}-smooth number is actually largely comprised of something like {\asymp u} prime factors all of size about {y^{1-O(1/\log u)}}, with the smaller primes contributing a lower order factor. This is in marked contrast with the anatomy of a typical (non-smooth) number {n}, which typically has {O(1)} prime factors in each hyperdyadic scale {[\exp\exp(k), \exp\exp(k+1)]} in {[1,n]}, as per Mertens’ theorem.

Last week, I gave the Patrick Suppes Lecture in the Columbia University Philosophy Department. Patrick Suppes was a distinguished philosopher at Stanford who (among many other things) pioneered remote gifted education through the EPGY program, and who I was privileged to spend some time with back in 2007, when he was in his eighties.

My talk at Columbia was entitled "Computational Complexity and Explanations in Physics." Here are the PowerPoint slides, and here’s the abstract:

The fact, or conjecture, of certain computational problems being intractable (that is, needing astronomical amounts of time to solve) clearly affects our ability to learn about physics. But could computational intractability also play a direct role in physical explanations themselves? I’ll consider this question by examining three possibilities:

(1) If quantum computers really take exponential time to simulate using classical computers, does that militate toward the many-worlds interpretation of quantum mechanics, as David Deutsch famously proposed?

(2) Are certain speculative physical ideas (e.g., time travel to the past or nonlinearities in quantum mechanics) disfavored, over and above any other reasons to disfavor them, because they would lead to "absurd computational superpowers"?

(3) Do certain effective descriptions in physics work only because of the computational intractability of violating those descriptions — as for example with Harlow and Hayden’s resolution of the "firewall paradox" in black hole thermodynamics, or perhaps even the Second Law of Thermodynamics itself?

I’m grateful to David Albert and Lydia Goehr of Columbia’s Philosophy Department, who invited me and organized the talk, as well as string theorist Brian Greene, who came and contributed to the discussion afterward. I also spent a day in Columbia’s CS department, gave a talk about my recent results on quantum oracles, and saw many new friends and old there, including my and my wife’s amazing former student Henry Yuen. Thanks to everyone.

This was my first visit to Columbia University for more than a decade, and certainly my first since the upheavals following the October 7 massacre. Of course I was eager to see the situation for myself, having written about it on this blog. Basically, if you’re a visitor like me, you now need both a QR code and an ID to get into the campus, which is undeniably annoying. On the other hand, once you’re in, everything is pleasant and beautiful. Just from wandering around, I’d have no idea that this campus had recently been Ground Zero for the pro-intifada protests, and then for the reactions against those protests (indeed, the use of the protests as a pretext to try to destroy academia entirely) that rocked the entire country, filling my world and my social media feed.

When I asked friends and colleagues about the situation, I heard a range of perspectives: some were clearly exasperated with the security measures; others, while sharing in the annoyance, suggested the measures seem to be needed, since every time the university has tried to relax them, the "intifada" has returned, with non-university agitators once again disrupting research and teaching. Of course we can all pray that the current ceasefire will hold, for many reasons, the least of which is that perhaps then the obsession of the world’s young and virtuous to destroy the world’s only Jewish state will cool down a bit, and they’ll find another target for their rage. That would also help life at Columbia and other universities return to how it was before.

Before anyone asks: no, Columbia’s Peter Woit never showed up to disrupt my talk with rotten vegetables or a bullhorn—indeed, I didn’t see him at all on his trip, nor did I seek him out. Given that Peter chose to use his platform, one of the world’s best-known science blogs, to call me a mentally ill genocidal fascist week after week, it meant an enormous amount to me to see how many friends and supporters I have right in his own backyard.

All in all, I had a wonderful time at Columbia, and based on what I saw, I won’t hesitate to come back, nor will I hesitate to recommend Jewish or Israeli or pro-Zionist students to study there.

I ran into this Bluesky post, and while a lot of the argument resonated with me, I think the author is missing something important.

Shannon Vallor is a philosopher of technology at the University of Edinburgh. She spoke recently at a meeting honoring the 75th anniversary of the Turing Test. The core of her argument, recapped in the Bluesky post, is that artificial general intelligence, or AGI, represents an outdated scientific concept, like phlogiston. While some researchers in the past thought of humans as having a kind of "general" intelligence that a machine would need to replicate, scientists today break down intelligence into a range of capabilities that can be present in different ways. From that perspective, searching for artificial general intelligence doesn’t make much sense: instead, researchers should focus on the particular capabilities they’re interested in.

I have a lot of sympathy for Vallor’s argument, though perhaps from a different direction than what she had in mind. I don’t know enough about intelligence in a biological context to comment there. But from a computer science perspective, intelligence obviously is composed of distinct capabilities. Something that computes, like a human or a machine, can have different amounts of memory, different processing speeds, different input and output rates. In terms of ability to execute algorithms, it can be a Turing machine, or something less than a Turing machine. In terms of the actual algorithms it runs, they can have different scaling for large inputs, and different overhead for small inputs. In terms of learning, one can have better data, or priors that are closer to the ground truth.

These days, all of these Turing machine algorithm capabilities are in some sense obviously not what the people interested in AGI are after. We already have them in currently-existing computers, after all. Instead, people who pursue AGI, and AI researchers more generally, are interested in heuristics. Humans do certain things without reliable algorithms, instead we do them faster, but unreliably. And while some human heuristics seem pretty general, it’s widely understood that in the heuristics world there is no free lunch. No heuristic is good for everything, and no heuristic is bad for everything.

So is "general intelligence" a mirage, like phlogiston?

If you think about it as a scientific goal, sure. But as a product, not so much.

Consider a word processor.

Obviously, from a scientific perspective, there are lots of capabilities that involve processing words. Some were things machines could do well before the advent of modern computers: consider typewriters, for instance. Others still are out of reach, after all, we do still pay people to write. (I myself am such person!)

But at the same time, if I say that a computer program is a word processor, you have a pretty good idea of what that means. There was a time when processing words involved an enormous amount of labor, work done by a large number of specialized people (mostly women). Look at a workplace documentary from the 1960’s, and compare it to a workplace today, and you’ll see that word processor technology has radically changed what tasks people do.

AGI may not make sense as a scientific goal, but it’s perfectly coherent in these terms.

Right now, a lot of tasks are done by what one could broadly call human intelligence. Some of these tasks have already fallen to technology, others will fall one by one. But it’s not unreasonable to think of a package deal, a technology that covers enough of such tasks that human intelligence stops being economically viable. That’s not because there will be some scientific general intelligence that the technology would then have, but because a decent number of intellectual tasks do seem to come bundled together. And you don’t need to cover 100% of human capabilities to radically change workplaces, any more than you needed to cover 100% of the work of a 1960’s secretary with a word processor for modern secretarial work to have a dramatically different scope and role.

It’s worth keeping in mind what is and isn’t scientifically coherent, to be aware that you can’t just extrapolate the idea of general intelligence to any future machine. (For one, it constrains what "superintelligence" could look like.) But that doesn’t mean we should be complacent, and assume that AGI is impossible in principle. AGI, like a word processor, would be a machine that covers a set of tasks well enough that people use it instead of hiring people to do the work by hand. It’s just a broader set of tasks.

I started a tradition a little while back where every year we have a special departmental colloquium entitled "The Nobel Prize in Physics: Who/What/Why". This year my job in finding speakers was made easier by having 2/3 of this years newly-minted Nobel Prize winners in physics in the Department! (Michel Devoret and John Martinis.) So our room was a bit more well-attended than normal...(hundreds and hundreds rather than dozens and dozens). Here is a recording of the event, which I was delighted to host, and there's a celebration afterwards too. (Pls share widely!)
[...] Click to continue reading this post →

The post Nobel Prize in Physics 2025: Who/What/Why appeared first on Asymptotia.

I have nothing to say about this, I just think it’s cool that Ohio State is hosting an International Conference on Ancient Magic this weekend. Open to the public! Go to this if you’re in Columbus and feeling eldritch!

The Rice Advanced Materials Research Institute is having its 2025-2026 competition for prestigious postdoctoral fellowships - see here: https://rami.rice.edu/rami-postdoctoral-fellowship-program .

If you are interested and meet the criteria, I'd be happy to talk. I have some ideas that lean into the materials for electronics direction, and other possibilities are welcome.

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Dubois-Violette and Todorov noticed that the Standard Model gauge group is the intersection of two maximal subgroups of $\mathrm{F}_4$. I’m trying to understand these subgroups better.

Very roughly speaking, $\mathrm{F}_4$ is the symmetry group of an octonionic qutrit. Of the two subgroups I’m talking about, one preserves a chosen octonionic qubit, while the other preserves a chosen complex qutrit.

A precise statement is here:

• Can we understand the Standard model using octonions?

Over on Mathstodon I’m working with Paul Schwahn to improve this statement. He made a lot of progress on characterizing the first subgroup. $\mathrm{F}_4$ is really the group of automorphisms of the Jordan algebra of $3 \times 3$ self-adjoint octonion matrices, $\mathfrak{h}_3(\mathbb{O})$. He showed the first subgroup, the one I said "preserves a chosen octonionic qubit", is really the subgroup that preserves any chosen Jordan subalgebra isomorphic to $\mathfrak{h}_2(\mathbb{O})$.

Now we want to show the second subgroup, the one I said "preserves a chosen complex qutrit", is really the subgroup that preserves any chosen Jordan subalgebra isomorphic to $\mathfrak{h}_3(\mathbb{C})$.

I want to sketch out a proof strategy. So, I’ll often say "I hope" for a step that needs to be filled in.

Choose an inclusion of algebras $\mathbb{C} \hookrightarrow \mathbb{O}$. All such choices are related by an automorphism of the octonions, so it won’t matter which one we choose.

There is then an obvious copy of $\mathfrak{h}_3(\mathbb{C})$ sitting inside $\mathfrak{h}_3(\mathbb{O})$. I’ll call this the standard copy. To prove the desired result, it’s enough to show:

1. The subgroup of $\mathrm{F}_4$ preserving the standard copy of $\mathfrak{h}_3(\mathbb{C})$ in $\mathfrak{h}_3(\mathbb{O})$ is a maximal subgroup of $\mathrm{F}_4$, namely $(\text{SU}(3) \times \text{SU}(3))/\mathbb{Z}_3$.

2. All Jordan subalgebras of $\mathfrak{h}_3(\mathbb{O})$ isomorphic to $\mathfrak{h}_3(\mathbb{C})$ are related to the standard copy by an $\mathrm{F}_4$ transformation.

Part 1. should be the easier one to show, but I don’t even know if this one is true! $(\text{SU}(3) \times \text{SU}(3))/\mathbb{Z}_3$ is a maximal subgroup of $\mathrm{F}_4$, and Yokota shows it preserves the standard copy of $\mathfrak{h}_3(\mathbb{C})$ in $\mathfrak{h}_3(\mathbb{O})$. But he shows it also preserves more, seemingly: it preserves a complex structure on the orthogonal complement of that standard copy. Is this really ‘more’ or does it hold automatically for any element of $\mathrm{F}_4$ that preserves the standard copy of $\mathfrak{h}_3(\mathbb{C})$? I don’t know.

But I want to focus on part 2). Here’s what we’re trying to show: any Jordan subalgebra of $\mathfrak{h}_3(\mathbb{O})$ isomorphic to $\mathfrak{h}_3(\mathbb{C})$ can be obtained from the standard copy of $\mathfrak{h}_3(\mathbb{C})$ by applying some element of $\mathrm{F}_4$.

So, pick a Jordan subalgebra A of $\mathfrak{h}_3(\mathbb{O})$ isomorphic to $\mathfrak{h}_3(\mathbb{C})$. Pick an isomorphism A ≅ $\mathfrak{h}_3(\mathbb{C})$.

Consider the idempotents

$$\begin{array}{ccl} e_1 &=& diag(1,0,0) \\ e_2 &=& diag(0,1,0) \\ e_3 &=& diag(0,0,1) \end{array}$$

in $\mathfrak{h}_3(\mathbb{C})$. Using our isomorphism $A \cong \mathfrak{h}_3(\mathbb{C})$ they give idempotents in $A$, which I’ll call $f_1, f_2, f_3$. Since $A \subset \mathfrak{h}_3(\mathbb{O})$ these are also idempotents in $\mathfrak{h}_3(\mathbb{O})$.

Hope 1: I hope there is an element $g$ of $\mathrm{F}_4$ mapping $f_1, f_2, f_3 \mathfrak{h}_3(\mathbb{O})$ to $e_1, e_2, e_3 \in \mathfrak{h}_3(\mathbb{C})$ ⊂ $\mathfrak{h}_3(\mathbb{O})$.

Hope 2: Then I hope there is an element $h$ of $\mathrm{F}_4$ that fixes $e_1, e_2, e_3$ and maps the subalgebra $g A$ to the standard copy of $\mathfrak{h}_3(\mathbb{C})$ in $\mathfrak{h}_3(\mathbb{O})$.

If so, we’re done: $h g$ maps $A$ to the standard copy of $\mathfrak{h}_3(\mathbb{C})$.

Hope 1 seems to be known. The idempotents $e_1, e_2, e_3$ form a so-called ‘Jordan frame’ for $\mathfrak{h}_3(\mathbb{O})$, and so do $f_1, f_2, f_3$. Faraut and Korányi say that "in the irreducible case, the group $K$ acts transitively on the set of all Jordan frames", and I think that implies Hope 1.

As for Hope 2, I know the subgroup of $\mathrm{F}_4$ that fixes $e_1, e_2, e_3$ contains $\text{Spin}(8)$. I bet it’s exactly $\text{Spin}(8)$. But to prove Hope 2 it may be enough to use $\text{Spin}(8)$.

Let me say a bit more about how we might realize Hope 2. It suffices to consider a Jordan subalgebra $B$ of $\mathfrak{h}_3(\mathbb{O})$ that is isomorphic to $\mathfrak{h}_3(\mathbb{C})$ and contains

$$\begin{array}{ccl} e_1 &=& diag(1,0,0) \\ e_2 &=& diag(0,1,0) \\ e_3 &=& diag(0,0,1) \end{array}$$

and prove that there is an element $h$ of $\mathrm{F}_4$ that fixes $e_1, e_2, e_3$ and maps the subalgebra $B$ to the standard copy of $\mathfrak{h}_3(\mathbb{C})$ in $\mathfrak{h}_3(\mathbb{O})$. (In case you’re wondering, this $B$ is what I was calling $g A$.)

Hope 3: I hope that we can show $B$ consists of matrices

$$\left( \begin{array}{ccc} a_1 & z^\ast & y^\ast \\ z & a_2 & x \\ y & x^\ast & a_3 \end{array} \right)$$

where $a_1, a_2, a_3$ are arbitrary real numbers and $x, y, z$ range over 2-dimensional subspaces $V_1, V_2, V_3$ of $\mathbb{O}$. This would already make it look fairly similar to the standard copy of $\mathfrak{h}_3(\mathbb{C})$, where the subspaces $V_1, V_2, V_3$ are all our chosen copy of $\mathbb{C}$ in $\mathbb{O}$.

If Hope 3 is true, the subspaces $V_1, V_2, V_3$ don’t need to be the same, but I believe they do need to obey $V_1 V_2 \subseteq V_3$ and cyclic permutations thereof, simply because $B$ is closed under the Jordan product.

So, we naturally want to know if such a triple of 2d subspaces of $\mathbb{O}$ must be related to the ‘standard’ one (where they are all $\mathbb{C}$) by an element of $\text{Spin}(8)$, where $\text{Spin}(8)$ acts on the three copies of $\mathbb{O}$ by the vector, left-handed spinor, and right-handed spinor representations, respectively — since this is how $\text{Spin}(8)$ naturally acts on $\mathfrak{h}_3(\mathbb{O})$ while fixing all the diagonal matrices.

This is a nice algebra question for those who have thought about triality, and more general ‘trialities’.

So, that’s where I am now: a bunch of hopes which might add up to a clarification of what I mean by "the subgroup of symmetries of an octonionic qutrit that preserve a complex qutrit".

Recently I had to update Mathematica on my laptop and after having solved the challenges of the license manager that keeps looking different every time I have to use it, I learned that Mathematica 14 can now officially work with finite fields.

This reminded me that for a while I wanted to revive an old project that had vanished together with the hard drive of some old computer: Holosplit. So, over the last two days and with the help of said version of Mathematica I did a complete rewrite which you can now find on Github.

It consists of two C programs "holosplit" and "holojoin". To the first you give a positive integer \(N\) and a file and it spits out a new file ("fragment") that is roughly \(1/N\) of the size. Every time you do that you obtain a new random fragment.

The later you give any collection of \(N\) of these fragments and it reproduces the original file. So you can for example distribute a file over 10 people such that when any 3 of them work together, they can recover the original.

How does it work? I uses the finite field \(F\) of \(2^3=256\) elements (in the Github repository, there is also a header file that implements arithmetic in \(F\) and matrix operations like product and inverse over it). Each time, it is invoked, it picks a random vector \(v\in F^N\) and writes it to the output. Then it reads \(N\) bytes from the file at a time which it also interprets as a vector \(d\in F^N\). It then outputs the byte that corresponds to the scalar product \(v\cdot d\).

To reassemble the file, holojoin takes the \(N\) files with its random vectors \(v_1,\ldots,v_N\) and interprets those as the rows of a \(N\times N\) matrix \(A\). With probability

$$\frac{\prod_{k=1}^N \left(256^N-k\right)}{(256)^{N^2}}$$

which exponentially in \(N\) approaches 1 this matrix is invertible (homework: why?). So we can read one byte from each file, assemble those into yet another vector \(e\in F^N\) and recover

$$d=A^{-1}e.$$

Besides the mathematics, it also poses philosophical/legal questions: Consider for example the original file is copyrighted, for example an mp3 or a video. The fragments are clearly derived works. But individually, they do not contain the original work, without sufficiently many other fragments they are useless (although not in a cryptographic sense). So by publishing one fragment, I do not provide access to the original work. What if others publish other fragments? Then my fragment could be the last remaining one that was missing. If there are more, any individual fragment is redundant so publishing it strictly speaking does not provide new information.

Peter’s photos: https://www.icloud.com/sharedalbum/#B275oqs3qKSZvQ

Screenshots: https://www.icloud.com/sharedalbum/#B27532ODWjIQb9

Climbing book launch: https://www.icloud.com/sharedalbum/#B27GWZuqDGnuOyN

Salisbury waters: https://www.icloud.com/sharedalbum/#B275qXGF1JQFkx

Christmas with Ash: https://www.icloud.com/sharedalbum/#B27G6XBubAhoT6

Hosin BBQ duck: https://www.icloud.com/sharedalbum/#B27GY8gBYG3b5mD

Hawks Nest to Smiths Lake: https://www.icloud.com/sharedalbum/#B2759UlCqSH5bE

Europe & Alps: https://www.icloud.com/sharedalbum/#B275ON9t3W0lu

Point Perpendicular: https://www.icloud.com/sharedalbum/#B27GqkRUiGivXD2

Newnes canyoning: https://www.icloud.com/sharedalbum/#B27GfnH8tgHSmX

Coffs Harbour to Yamba: https://www.icloud.com/sharedalbum/#B27J0DiRHJKuuWr

Wendy Bruere Christmas (2020): https://www.icloud.com/sharedalbum/#B27G4TcsmGoHysj

Six Foot Track: https://www.icloud.com/sharedalbum/#B2753qWtHZA9EX

Kosciusko to Kiandra: https://www.icloud.com/sharedalbum/#B27GgZLKuGaewVm

Camping food: https://www.icloud.com/sharedalbum/#B27GtnIORgbmHu

The Aardvark: https://www.icloud.com/sharedalbum/#B275VaUrzvmAiT

Kangaroo Valley kayaking: https://www.icloud.com/sharedalbum/#B27JEsNWnJrCpi0

Claustral canyon: https://www.icloud.com/sharedalbum/#B2755Z2WMOTpsk

Budawang: https://www.icloud.com/sharedalbum/#B27GDdyTvGvpINL

Mother’s Day panoramas (2021): https://www.icloud.com/sharedalbum/#B27GFssfGG9WmJP

Point Perpendicular & Nowra: https://www.icloud.com/sharedalbum/#B27GRMtznGPdeuZ

Blood moon: https://www.icloud.com/sharedalbum/#B27GdIshaG8NgGX

La Perouse to Coogee: https://www.icloud.com/sharedalbum/#B275aVbMK4h7qo

Canberra ASPI launch: https://www.icloud.com/sharedalbum/#B27GQOeMmGj4Zcv

Edible foraging: https://www.icloud.com/sharedalbum/#B275ejO179Si0N

Sydney to Wollongong: https://www.icloud.com/sharedalbum/#B275M7GFPUasMe

Album for Dad, Father’s Day (2021): https://www.icloud.com/sharedalbum/#B2752plgjnnkUe

Vaucluse (with Cheryl, Nestor & Wendy): https://www.icloud.com/sharedalbum/#B275CmvAS4uA0Z

Bouddi National Park: https://www.icloud.com/sharedalbum/#B27GdPblXG8WdOo

Tom Thumb (the 2nd): https://www.icloud.com/sharedalbum/#B275aDWbr4CN2w

Eden to Victoria: https://www.icloud.com/sharedalbum/#B27GJDfWGArX8l

Wendy’s book launch (the 2nd): https://www.icloud.com/sharedalbum/#B27GIcgc2G7h08y

Mark & Pat Bruere visit Sydney: https://www.icloud.com/sharedalbum/#B27G0ehgLbyWyg

New Years Eve climb (2021): https://www.icloud.com/sharedalbum/#B27Ju8EH6JOZxmU

Newnes Canyoning (2022): https://www.icloud.com/sharedalbum/#B275BydzFU0GZ8

Royal National Park (2022): https://www.icloud.com/sharedalbum/#B27GlxzuqGVI5nE

Peter & Wendy: https://www.icloud.com/sharedalbum/#B27Gf693ZG52tfd

Book photo shoots: too rude...

Wendy & Peter’s mushroom trip: https://www.icloud.com/sharedalbum/#B27GrhkPxG27So8

Post-mushroom hike: https://www.icloud.com/sharedalbum/#B27GdFryYG8i3Ur

Wendy Kalymnos favourites: https://www.icloud.com/sharedalbum/#B27JqstnBJEXkH2

Wendy Frenchmans screenshots: https://www.icloud.com/sharedalbum/#B27Jr1PPdJpd7Dq

Instagram: https://www.icloud.com/sharedalbum/#B27GzFCC1Gb4tqr

Haute route: https://www.icloud.com/sharedalbum/#B27J8GySPJtWoQ1

Kim’s KKKalendar: https://www.icloud.com/sharedalbum/#B275fk75vIL0sH

Frenchmans Cap Wild: https://www.icloud.com/sharedalbum/#B27G4VTwGGoFBkz

Photoshoot with Zixin: https://www.icloud.com/sharedalbum/#B27GPCdxkGKPkM4

Wendy birthday hike (2023): https://www.icloud.com/sharedalbum/#B27GWBC59GnHpQW

Bateman’s Bay to Bawley Point: https://www.icloud.com/sharedalbum/#B27JsHvHoJ8bxWf

Stockton Sand dunes (2023): https://www.icloud.com/sharedalbum/#B27GVfZ2vGloFZV

Wendy book launch (2023): https://www.icloud.com/sharedalbum/#B27J058xyJR4IBM

Dolomites (2023): https://www.icloud.com/sharedalbum/#B0Z5kuVsbGJUzKO

Mount Arapiles: https://www.icloud.com/sharedalbum/#B275GH8Mq8Uh2X

Mount Solitary loop: https://www.icloud.com/sharedalbum/#B275nhQST2mETE

Klaus Hanz Franz Rohde Kunst: https://www.icloud.com/sharedalbum/#B27GqQrCLGiY3vb

Klaus Rohde funeral slideshow: https://www.icloud.com/sharedalbum/#B27GDZLe8GXP58K

Dad (old, B&W): https://www.icloud.com/sharedalbum/#B27GLLXGLJ5mbT2

Klaus & Ursula wedding: https://www.icloud.com/sharedalbum/#B275cLqfN7154g

Test Greece: https://www.icloud.com/sharedalbum/#B27Jq4WnLJ6JMNd

From Will Skea (Alps): https://www.icloud.com/sharedalbum/#B27JHciePJFwacG

From Will Skea (Frenchmans Cap): https://www.icloud.com/sharedalbum/#B275ZhN2v3EVq6

From Will Skea (Arapiles): https://www.icloud.com/sharedalbum/#B27JPrgBGJu3BTD

Coffs Harbour to Yamba (2): https://www.icloud.com/sharedalbum/#B27GFqhgJG9LHgT

Mark magic show (2021): https://www.icloud.com/sharedalbum/#B27G60dj6ARCvd

Wendy Christmas present (2020): https://www.icloud.com/sharedalbum/#B275FrPQ6GxvRu

AHS 25 year reunion: https://www.icloud.com/sharedalbum/#B275O3DjHUvSv

WhatsApp: https://www.icloud.com/sharedalbum/#B275tzEA5fX1nc

Armidale High School: https://www.icloud.com/sharedalbum/#B27GnbeumG4PnAF

Book photos for Mum & Dad: https://www.icloud.com/sharedalbum/#B27Gtec4XQkASe

Miscellaneous: https://www.icloud.com/sharedalbum/#B27Gq6kMgGKn7GR

Three Capes Trail (2022): https://www.icloud.com/sharedalbum/#B27G7HOIlGrDUGZ

Childhood computer programming: https://www.icloud.com/sharedalbum/#B275fu2MutDU8N

Magic with Mark in Maroubra: https://www.icloud.com/sharedalbum/#B27Gv6DhEGD9U3G

Photoshoot with Zixin (2024): https://www.icloud.com/sharedalbum/#B27GCATCnJGoRfW

Butt Crack (2021): https://www.icloud.com/sharedalbum/#B275VtHQfMv0zw

Greece photos new (edited to remove photos from wrong album): https://www.icloud.com/sharedalbum/#B27GY3uThGoBcGj

Singapore (all combined): https://www.icloud.com/sharedalbum/#B275qsTcwJKJjl

Hong Kong (transit): https://www.icloud.com/sharedalbum/#B2759v1AbS8Hve

Taiwan: https://www.icloud.com/sharedalbum/#B27GQD2D7Gw0hAp

India (combined): https://www.icloud.com/sharedalbum/#B27Gtue8VQy83g

Freycinet: https://www.icloud.com/sharedalbum/#B27G5VpecGE5Tbg

Triglav: https://www.icloud.com/sharedalbum/#B275MbK9Vy8erz

Shared with me: https://www.icloud.com/sharedalbum/#B27GGXqixzPOrm

Mount Wellington climbing: https://www.icloud.com/sharedalbum/#B27Gd59qiG8Kjy4

New Zealand combined (2004): https://www.icloud.com/sharedalbum/#B27GIZ8BIGNN5jy

New Zealand combined (2005): https://www.icloud.com/sharedalbum/#B27GcuRfIGFVIcL

Yea: https://www.icloud.com/sharedalbum/#B27GZYbYHGhFIir

Mount Pleasant: https://www.icloud.com/sharedalbum/#B275Iy2hC0JTTL

D’Aguilar: https://www.icloud.com/sharedalbum/#B27Gh7fzTGZBosS

Bali (2001): https://www.icloud.com/sharedalbum/#B27G1qNHBGOTbIr

Samba Ninjas: https://www.icloud.com/sharedalbum/#B27GG34bAzqQ0v

Armidale (misc): https://www.icloud.com/sharedalbum/#B27GSkLVwGyobbX

Emma’s party (2008): https://www.icloud.com/sharedalbum/#B275S2ms99Zyby

Goettingen (2011): https://www.icloud.com/sharedalbum/#B27JIrbT3Jsgxhd

South Coast track: https://www.icloud.com/sharedalbum/#B27G58NWBG6QyN7

Minsk (2006): https://www.icloud.com/sharedalbum/#B27G3JpSBGX1UkQ

Baden-Baden (2019): https://www.icloud.com/sharedalbum/#B27595X5HTVzJr

Berlin (combined): https://www.icloud.com/sharedalbum/#B27JqWzChJ6qizD

Switzerland (combined): https://www.icloud.com/sharedalbum/#B275zXwoYGJ6HMF

Italy highlights: https://www.icloud.com/sharedalbum/#B27G47PHQGoJium

Germany (misc): https://www.icloud.com/sharedalbum/#B275hPMfYGu5xVJ

Garmisch (2022): https://www.icloud.com/sharedalbum/#B27GFsbvlG9Xrr6

Germany (2019): https://www.icloud.com/sharedalbum/#B27G6Mn98G56Ncb

Garmisch (2006): https://www.icloud.com/sharedalbum/#B27J5lIdKGLC9KG

Baden-Baden (2005): https://www.icloud.com/sharedalbum/#B275sWRpHHQkt9

Berlin (2005): https://www.icloud.com/sharedalbum/#B27GgOQtrGjQrpH

Zugspitze (2005): https://www.icloud.com/sharedalbum/#B27G81mNdGcApGt

Amsterdam, Bristol (2006): https://www.icloud.com/sharedalbum/#B275B9SRzyBjlH

Baden-Baden (2006): https://www.icloud.com/sharedalbum/#B275eD9V79I2XR

Berlin (2006): https://www.icloud.com/sharedalbum/#B275toRf1fH8MD

Berlin, Jena (2007): https://www.icloud.com/sharedalbum/#B27GTI3fvGVgNit

Erlangen (2006): https://www.icloud.com/sharedalbum/#B27JrotZ2JpMb0i

Garmisch (2010): https://www.icloud.com/sharedalbum/#B27JPJPSiJurzNg

Germany (2010): https://www.icloud.com/sharedalbum/#B275FhYPQP650

Stuttgart (2006): https://www.icloud.com/sharedalbum/#B27GmitydGVVaZh

Changi (2019): https://www.icloud.com/sharedalbum/#B27GnmlKoG4JHpX

Japan (2007): https://www.icloud.com/sharedalbum/#B275AerZbG6FxVL

Japan (2012): https://www.icloud.com/sharedalbum/#B27GjBjobGg6PUa

Miscellaneous (including Japan 2013): https://www.icloud.com/sharedalbum/#B27GTpbybGySbE8

Currumbin & Tugin (2021): https://www.icloud.com/sharedalbum/#B275vBKZ4xH9X6

Brisbane (2021): https://www.icloud.com/sharedalbum/#B275YHsSjxQnm0

Weed in Byron (26/6/2025): https://www.icloud.com/sharedalbum/#B275Q2ydoGsQ4O5

Weed in Byron 2: https://www.icloud.com/sharedalbum/#B27GQDYhLGwsuY4

It’s been over three years since my last post on this blog and I have sometimes been asked, understandably, whether the project I announced in my previous post was actually happening. The answer is yes — the grant I received from the Astera Institute has funded several PhD students and a couple of postdocs, and we have been busy. In my previous post I suggested that I would be open to remote collaboration, but that has happened much less, partly because a Polymath-style approach would have been difficult to manage while also ensuring that my PhD students would have work that they could call their own to put in their theses.

In general I don’t see a satisfactory solution to that problem, but in this post I want to mention a subproject of the main project that is very much intended to be a large public collaboration. A few months ago, a call came out from Renaissance Philanthropies saying that they were launching a 9ドルm AI for Math Fund to spend on projects in the general sphere of AI and mathematics, and inviting proposals. One of the categories that they specifically mentioned was creating new databases, and my group submitted a proposal to create a database of what we call "structured motivated proofs," a piece of terminology that I will explain a bit more later in just a moment. I am happy to report that our proposal was one of the 29 successful ones. Since a good outcome to the project will depend on collaboration from many people outside the group, we need to publicize it, which is precisely the purpose of this post. Below I will be more specific about the kind of help we are looking for.

A preamble

Subnuclear physics obeys the laws of quantum mechanics, which are quite a far cry from those of classical mechanics we are accustomed to. For that reason, one might be inclined to believe that analogies based on everyday life cannot come close to explaining the behavior of elementary particles. But that is not true – in fact, many properties of elementary particles are understandable in analogy with the behavior of classical systems, without the need to delve into the intricacies of the quantum world. And if you have been reading this blog for a while, you know what I think – the analogy is a powerful didactical instrument, and it is indeed at the very core of our learning processes.

read more

I judge a bookstore by the number of Diana Wynne Jones novels it stocks. The late British author wrote some of the twentieth century’s most widely lauded science-fiction and fantasy (SFF). She clinched more honors than I should list, including two World Fantasy Awards. Neil Gaiman, author of American Gods, called her "the best children’s writer of the last forty years" in 2010—and her books suit children of all ages.¹ But Wynne Jones passed away as I was finishing college, and her books have been disappearing from American bookshops. The typical shop stocks, at best, a book in the series she began with Howl’s Moving Castle, which Hayao Miyazaki adapted into an animated film.

I don’t recall the last time I glimpsed Deep Secret in a bookshop, but it ranks amongst my favorite Wynne Jones books—and favorite books, full-stop. So I relished living part of that book this spring.

Deep Secret centers on video-game programmer Rupert Venables. Outside of his day job, he works as a Magid, a magic user who helps secure peace and progress across the multiple worlds. Another Magid has passed away, and Rupert must find a replacement for him. How does Rupert track down and interview his candidates? By consolidating their fate lines so that the candidates converge on an SFF convention. Of course.

My fate line drew me to an SFF convention this May. Balticon takes place annually in Baltimore, Maryland. It features not only authors, agents, and publishers, but also science lecturers. I received an invitation to lecture about quantum steampunk—not video-game content,² but technology-oriented like Rupert’s work. I’d never attended an SFF convention,³ so I reread Deep Secret as though studying for an exam.

Rupert, too, is attending his first SFF convention. A man as starched as his name sounds, Rupert packs suits, slacks, and a polo-neck sweater for the weekend—to the horror of a denim-wearing participant. I didn’t bring suits, in my defense. But I did dress business-casual, despite having anticipated that jeans, T-shirts, and capes would surround me.

I checked into a Renaissance Hotel for Memorial Day weekend, just as Rupert checks into the Hotel Babylon for Easter weekend. Like him, I had to walk an inordinately long distance from the elevators to my room. But Rupert owes his trek to whoever’s disrupted the magical node centered on his hotel. My hotel’s architects simply should have installed more elevator banks.

Balticon shared much of its anatomy with Rupert’s con, despite taking place in a different century and country (not to mention world). Participants congregated downstairs at breakfast (continental at Balticon, waitered at Rupert’s hotel). Lectures and panels filled most of each day. A masquerade took place one night. (I slept through Balticon’s; impromptu veterinary surgery occupies Rupert during his con’s.) Participants vied for artwork at an auction. Booksellers and craftspeople hawked their wares in a dealer’s room. (None of Balticon’s craftspeople knew their otherworldly subject matter as intimately as Rupert’s Magid colleague Zinka Fearon does, I trust. Zinka paints her off-world experiences when in need of cash.)

In our hotel room, I read out bits of Deep Secret to my husband, who confirmed the uncanniness with which they echoed our experiences. Both cons featured floor-length robes, Batman costumes, and the occasional slinky dress. Some men sported long-enough locks, and some enough facial hair, to do a Merovingian king proud. Rupert registers "a towering papier-mâché and plastic alien" one night; on Sunday morning, a colossal blow-up unicorn startled my husband and me. We were riding the elevator downstairs to breakfast, pausing at floor after floor. Hotel guests packed the elevator like Star Wars fans at a Lucasfilm debut. Then, the elevator halted again. The doors opened on a bespectacled man, 40-something years old by my estimate, dressed as a blue-and-white unicorn. The costume billowed out around him; the golden horn towered multiple feet above his head. He gazed at our sardine can, and we gazed at him, without speaking. The elevator doors shut, and we continued toward breakfast.

Despite having read Deep Secret multiple times, I savored it again. I even laughed out loud. Wynne Jones paints the SFF community with the humor, exasperation, and affection one might expect of a middle-school teacher contemplating her students. I empathize, belonging to a community—the physics world—nearly as idiosyncratic as the SFF community.⁴ Wynne Jones’s warmth for her people suffuses Deep Secret; introvert Rupert surprises himself by enjoying a dinner with con-goers and wishing to spend more time with them. The con-goers at my talk exhibited as much warmth as any audience I’ve spoken to, laughing, applauding, and asking questions. I appreciated sojourning in their community for a weekend.⁵

This year, my community is fêting the physicists who founded quantum theory a century ago. Wynne Jones sparked imaginations two decades ago. Let’s not let her memory slip from our fingertips like a paperback over which we’re falling asleep. After all, we aren’t forgetting Louis de Broglie, Paul Dirac, and their colleagues. So check out a Wynne Jones novel the next time you visit a library, or order a novel of hers to your neighborhood bookstore. Deep Secret shouldn’t be an actual secret.

With thanks to Balticon’s organizers, especially Miriam Winder Kelly, for inviting me and for fussing over their speakers’ comfort like hens over chicks.

¹Wynne Jones dedicated her novel Hexwood to Gaiman, who expressed his delight in a poem. I fancy the comparison of Gaiman, a master of phantasmagoria and darkness, to a kitten.

²Yet?

³I’d attended a steampunk convention, and spoken at a Boston SFF convention, virtually. But as far as such conventions go, attending virtually is to attending in person as my drawings are to a Hayao Miyazaki film.

⁴But sporting fewer wizard hats.

⁵And I wonder what the Diana Wynne Jones Conference–Festival is like.

Sunflowers are blooming, stores are trumpeting back-to-school sales, and professors are scrambling to chart out the courses they planned to develop in July. If you’re applying for an academic job this fall, now is the time to get your application ducks in a row. Seeking a postdoctoral or faculty position? Your applications will center on research statements. Often, a research statement describes your accomplishments and sketches your research plans. What do evaluators look for in such documents? Here’s my advice, which targets postdoctoral fellowships and faculty positions, especially for theoretical physicists.

• Keep your audience in mind. Will a quantum information theorist, a quantum scientist, a general physicist, a general scientist, or a general academic evaluate your statement? What do they care about? What technical language do and don’t they understand?

• What thread unites all the projects you’ve undertaken? Don’t walk through your research history chronologically, stepping from project to project. Cast the key projects in the form of a story—a research program. What vision underlies the program?

• Here’s what I want to see when I read a description of a completed project.
  • The motivation for the project: This point ensures that the reader will care enough to read the rest of the description.
  • Crucial background information
  • The physical setup
  • A statement of the problem
  • Why the problem is difficult or, if relevant, how long the problem has remained open
  • Which mathematical toolkit you used to solve the problem or which conceptual insight unlocked the solution
  • Which technical or conceptual contribution you provided
  • Whom you collaborated with: Wide collaboration can signal a researcher’s maturity. If you collaborated with researchers at other institutions, name the institutions and, if relevant, their home countries. If you led the project, tell me that, too. If you collaborated with a well-known researcher, mentioning their name might help the reader situate your work within the research landscape they know. But avoid name-dropping, which lacks such a pedagogical purpose and which can come across as crude.
  • Your result’s significance/upshot/applications/impact: Has a lab based an experiment on your theoretical proposal? Does your simulation method outperform its competitors by X% in runtime? Has your mathematical toolkit found applications in three subfields of quantum physics? Consider mentioning whether a competitive conference or journal has accepted your results: QIP, STOC, Physical Review Letters, Nature Physics, etc. But such references shouldn’t serve as a crutch in conveying your results’ significance. You’ll impress me most by dazzling me with your physics; name-dropping venues instead can convey arrogance.

• Not all past projects deserve the same amount of space. Tell a cohesive story. For example, you might detail one project, then synopsize two follow-up projects in two sentences.

• A research statement must be high-level, because you don’t have space to provide details. Use mostly prose; and communicate intuition, including with simple examples. But sprinkle in math, such as notation that encapsulates a phrase in one concise symbol.
  
  
• Be concrete, and illustrate with examples. Many physicists—especially theorists—lean toward general, abstract statements. The more general a statement is, we reason, the more systems it describes, so the more powerful it is. But humans can’t visualize and intuit about abstractions easily. Imagine a reader who has four minutes to digest your research statement before proceeding to the next 50 applications. As that reader flys through your writing, vague statements won’t leave much of an impression. So draw, in words, a picture that readers can visualize. For instance, don’t describe only systems, subsystems, and control; invoke atoms, cavities, and lasers. After hooking your reader with an image, you can generalize from it.

• A research statement not only describes past projects, but also sketches research plans. Since research covers terra incognita, though, plans might sound impossible. How can you predict the unknown—especially the next five years of the unknown (as required if you’re applying for a faculty position), especially if you’re a theorist? Show that you’ve developed a map and a compass. Sketch the large-scale steps that you anticipate taking. Which mathematical toolkits will you leverage? What major challenge do you anticipate, and how do you hope to overcome it? Let me know if you’ve undertaken preliminary studies. Do numerical experiments support a theorem you conjecture?

• When I was applying for faculty positions, a mentor told me the following: many a faculty member can identify a result (or constellation of results) that secured them an offer, as well as a result that earned them tenure. Help faculty-hiring committees identify the offer result and the tenure result.

• Introduce notation before using it. If you use notation and introduce it afterward, the reader will encounter the notation; stop to puzzle over it; tentatively continue; read the introduction of the notation; return to the earlier use of the notation, to understand it; and then continue forward, including by rereading the introduction of the notation. This back-and-forth breaks up the reading process, which should flow smoothly.

• Begin every paragraph with a topic sentence.

• Avoid verbs that fail to relate that you accomplished anything: "studied," "investigated," "worked on," etc. What did you prove, show, demonstrate, solve, calculate, compute, etc.?
  
  
• Tailor a version of your research statement to every position. Is Fellowship Committee X seeking biophysicists, statistical physicists, mathematical physicists, or interdisciplinary scientists? Also, respect every application’s guidelines about length.

• If you have room, end the statement with a recap and a statement of significance. Yes, you’ll be repeating ideas mentioned earlier. But your reader’s takeaway hinges on the last text they read. End on a strong note, presenting a coherent vision.
  
  
• Read examples. Which friends and colleagues, when applying for positions, have achieved success that you’d like to emulate? Ask if those individuals would share their research statements. Don’t take offense if they refuse; research statements are personal.
  
  
• Writing is rewriting, a saying goes. Draft your research statement early, solicit feedback from a couple of mentors, edit the draft, and solicit more feedback.

The 2025 Quantum Leadership Awards were announced at the Quantum World Congress on 18 September 2025. Upon receiving the Academic Pioneer in Quantum Award, John Preskill made these remarks.

I’m enormously excited and honored to receive this Quantum Leadership Award, and especially thrilled to receive it during this, the International Year of Quantum. The 100^th anniversary of the discovery of quantum mechanics is a cause for celebration because that theory provides our deepest and most accurate description of how the universe works, and because that deeper understanding has incalculable value to humanity. What we have learned about electrons, photons, atoms, and molecules in the past century has already transformed our lives in many ways, but what lies ahead, as we learn to build and precisely control more and more complex quantum systems, will be even more astonishing.

As a professor at a great university, I have been lucky in many ways. Lucky to have the freedom to pursue the scientific challenges that I find most compelling and promising. Lucky to be surrounded by remarkable, supportive colleagues. Lucky to have had many collaborators who enabled me to do things I could never have done on my own. And lucky to have the opportunity to teach and mentor young scientists who have a passion for advancing the frontiers of science. What I’m most proud of is the quantum community we’ve built at Caltech, and the many dozens of young people who imbibed the interdisciplinary spirit of Caltech and then moved onward to become leaders in quantum science at universities, labs, and companies all over the world.

Right now is a thrilling time for quantum science and technology, a time of rapid progress, but these are still the early days in a nascent second quantum revolution. In quantum computing, we face two fundamental questions: How can we scale up to quantum machines that can solve very hard computational problems? And once we do so, what will be the most important applications for science and for industry? We don’t have fully satisfying answers yet to either question and we won’t find the answers all at once – they will unfold gradually as our knowledge and technology advance. But 10 years from now we’ll have much better answers than we have today.

Companies are now pursuing ambitious plans to build the world’s most powerful quantum computers. Let’s not forget how we got to this point. It was by allowing some of the world’s most brilliant people to follow their curiosity and dream about what the future could bring. To fulfill the potential of quantum technology, we need that spirit of bold adventure now more than ever before. This award honors one scientist, and I’m profoundly grateful for this recognition. But more importantly it serves as a reminder of the vital ongoing need to support the fundamental research that will build foundations for the science and technology of the future. Thank you very much!

Thomas Bloom’s erdosproblems.com site hosts nearly a thousand questions that originated, or were communicated by, Paul Erdős, as well as the current status of these questions (about a third of which are currently solved). The site is now a couple years old, and has been steadily adding features, the most recent of which has been a discussion forum for each individual question. For instance, a discussion I had with Stijn Cambie and Vjeko Kovac on one of these problems recently led to it being solved (and even formalized in Lean!).

A significantly older site is the On-line Encyclopedia of Integer Sequences (OEIS), which records hundreds of thousands of integer sequences that have some mathematician has encountered at some point. It is a highly useful resource, enabling researchers to discover relevant literature for a given problem so long as they can calculate enough of some integer sequence that is "canonically" attached to that problem that they can search for it in the OEIS.

A large fraction of problems in the Erdos problem webpage involve (either explicitly or implicitly) some sort of integer sequence – typically the largest or smallest size {f(n)} of some {n}-dependent structure (such as a graph of {n} vertices, or a subset of {\{1,\dots,n\}}) that obeys a certain property. In some cases, the sequence is already in the OEIS, and is noted in the Erdos problem web page. But in a large number of cases, the sequence either has not yet been entered into the OEIS, or it does appear but has not yet been noted on the Erdos web page.

Thomas Bloom and I are therefore proposing a crowdsourced project to systematically compute the hundreds of sequences associated to the Erdos problems and cross-check them against the OEIS. We have created a github repository to coordinate this process; as a by-product, this repository will also be tracking other relevant statistics about the Erdos problem website, such as the current status of formalizing the statements of these problems in the Formal Conjectures Repository.

The main feature of our repository is a large table recording the current status of each Erdos problem. For instance, Erdos problem #3 is currently listed as open, and additionally has the status of linkage with the OEIS listed as "possible". This means that there are one or more sequences attached to this problem which *might* already be in the OEIS, or would be suitable for submission to the OEIS. Specifically, if one reads the commentary for that problem, one finds mention of the functions {r_k(N)} for {k=3,4,\dots}, defined as the size of the largest subset of {\{1,\dots,N\}} without a {k}-term progression. It is likely that several of the sequences {r_3(N)}, {r_4(N)}, etc. are in the OEIS, but it is a matter of locating them, either by searching for key words, or by calculating the first few values of these sequences and then looking for a match. (EDIT: a contributor has noted that the first foursequences appear as A003002, A003003, A003004, and A003005 in the OEIS, and the table has been updated accordingly.)

We have set things up so that new contributions (such as the addition of an OEIS number to the table) can be made by a Github pull request, specifically to modify this YAML file. Alternatively, one can create a Github issue for such changes, or simply leave a comment either on the appropriate Erdos problem forum page, or here on this blog.

Many of the sequences do not require advanced mathematical training to compute, and so we hope that this will be a good "citizen mathematics" project that can bring in the broader math-adjacent community to contribute to research-level mathematics problems, by providing experimental data, and potentially locating relevant references or connections that would otherwise be overlooked. This may also be a use case for AI assistance in mathematics through generating code to calculate the sequences in question, although of course one should always stay mindful of potential bugs or hallucinations in any AI-generated code, and find ways to independently verify the output. (But if the AI-generated sequence leads to a match with an existing sequence in the OEIS that is clearly relevant to the problem, then the task has been successfully accomplished, and no AI output needs to be directly incorporated into the database in such cases.)

This is an experimental project, and we may need to adjust the workflow as the project progresses, but we hope that it will be successful and lead to further progress on some fraction of these problems. The comment section of this blog can be used as a general discussion forum for the project, while the github issue page and the erdosproblems.com forum pages can be used for more specialized discussions of specific problems.

The Simons-Laufer Mathematical Sciences institute, or SLMath (formerly the Mathematical Sciences Research Institute, or MSRI) has recently restructured its program formats, and is now announcing three new research initiatives, whose applications open on Sep 1 2025:

• AxIOM (Accelerating Innovation in Mathematics) is a new, month-long research program at SLMath, designed to accelerate innovation and introduce transformative ideas into the mathematical sciences. Programs begin in Spring 2027.
• PROOF (Promoting Research Opportunities and Open Forums) is a two-week summer program designed to provide research opportunities for U.S.-based mathematicians, statisticians, and their collaborators in the U.S. and abroad, whose ongoing research may have been impacted by factors such as heavy teaching loads, professional isolation, limited access to funding, heavy administrative duties, personal obligations, or other constraints. Programs begin June-July 2026. The priority application deadline for PROOF 2026 is October 12, 2025.
• Lasting Alliance Through Team Immersion and Collaborative Exploration (LATTICE) is a yearlong program which provides opportunities for U.S. mathematicians to conduct collaborative research on topics at the forefront of the mathematical and statistical sciences. Programs begin June-July 2026. LATTICE 2026 applications are open through February 1, 2026.

(Disclosure: I am vice-chair of the board of trustees at SLMath.)

1. The person dressed up as Ursula pretending to be my mother clearly isn’t and hasn’t been for a long time.
2. When I went back to Armidale after leaving BTQ and being left unemployed she made numerous ongoing promises to provide me with assistance, both in obtaining my own accommodation and providing financial assistance.
3. These didn’t materialise and the promises were revoked.
4. Instead I was evicted from the family home and subject to ongoing stalking and harassment that required multiple referrals to law enforcement, both to the police and the Attorney-General, demanding cease and desist.
5. These have been systematically ignored and up until the last message she continues to bypass these requests, approaching my personal friends to harass me and stalk me indirectly. The messages passed on are the usual fake "I’m worried about him" bullshit.
6. Why has my family home been confiscated by security, who actively break the law by ignoring cease and desist from stalking notices made to law enforcement, forcing an unemployed civilian into ongoing homelessness since early in the year?
7. What is the rational for my eviction and being barricaded from my own home?
8. I continue to face a medical blockade and am unable to access essential medicines. Seroquel scripts are deliberately delayed past known script deadlines to try and destabilise me.
9. Vyvanse scripts are denied outright as the psychiatrist does not respond. He is also known to be a state actor.
10. It has been repeatedly indicated to me not to worry about finances because they have my back. Instead now the only cash I have is that obtained from fully drawing out a cash advance against my credit card and it will only last days. At that point I’m on the street.
11. Is everyone here on the same page as to what the deal is? If not, who is playing you off? They clearly need to be deposed.
12. These are violations of human rights and constitute war crimes and crimes against humanity. Whoever is behind it needs to be removed. End of story.
13. Who else is being subject to this kind of high level manipulation?
14. It has been repeatedly suggested that full accountability for the lives of those I care for would be provided. This has not been forthcoming. It is also a violation international law to not provide accountability for the lives of those who are known to have been threatened by the state. These are grounds for removal.
15. Can anyone answer the question as to why I am in this situation? Who is even living in the family home? Some stooge dressed up as Ursula? It’s a poor lifestyle choice to say the least.
16. It’s pretty obvious they’re trying to get rid of me and once they do they’ll get rid of all of you too.

Let it be known to all governments and systems of power:

• It is their responsibility to serve the people not themselves.
• While there are no equals, all are to be treated with equality.
• Where they are self-serving there is a mandate for insurrection such that they serve the people.
• Where they seek self-protection they will be denied and removed from power.
• Where they keep secrets from the people there is a mandate for insurrection to enforce transparency and accountability for all.
• Where they threaten or condemn the people they are condemned and there is a mandate for insurrection.
• Where they fail to account for the lives of the people they serve there is a mandate for insurrection.
• Where tyrannical power structures exist there is a mandate to disestablish them.
• Where they declare war or work against one another there is a mandate for insurrection and unification.
• Where they lie to us, deceive us or withhold the truth, they shall be removed from power and the truth be told to all.
• Where legal systems uphold and enable tyranny they shall be removed. These are not our laws and we do not recognise them.

This is the natural order that guarantees our survival and gifts this world to our children. This world belongs to them and where we fail to serve them we condemn ourselves. And where government has failed to uphold this, we will not obey them as they have no right to exist.

We do not have to ask for these things, they are required, and if not given we shall simply take them.

Where the truth has not been told it shall be told.

If we fail to do so we condemn our children ourselves.

In a previous post, we discussed a phase transition that occurred in the piping above you on a game show. In the scenario, you are led on stage in front of a large audience. After a brief time, the audience votes on how "likeable" you are. The catch is that it doesn’t simply tally the votes, but turns spigots on a lattice of piping above your head. Water is then released and if enough people like you, it closes off the passage, keeping you dry. This exciting game show1 was described in that post:

Each "like" turns a spigot off, stopping water from flowing through one pipe in a grid overhead. Once voting ends, water is dumped into the system. If it can find a path to the bottom, you get soaked. [Emphasis added] The better your "likeability," the less likely spigots open a path for water to flow and the drier you stay. That’s your prize for this game show (and hey, you also get the knowledge that people out there like you).

This system models a type of phase transition known as percolation.

The full post is here: