Planet Musings

The delicious chaos that (almost always) eventually tames into a tasty flaky pastry crust… it’s always a worrying mess to start out, but you trust to your experience, and you carry on, with hope. #thanksgiving

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In Kenneth Grahame’s 1908 novel The Wind in the Willows, a Mole meets a Water Rat who lives on a River. The Rat explains how the River permeates his life: “It’s brother and sister to me, and aunts, and company, and food and drink, and (naturally) washing.” As the River plays many roles in the Rat’s life, so does Carnot’s theorem play many roles in a thermodynamicist’s.

Nicolas Léonard Sadi Carnot lived in France during the turn of the 19th century. His father named him Sadi after the 13th-century Persian poet Saadi Shirazi. Said father led a colorful life himself,¹ working as a mathematician, engineer, and military commander for and before the Napoleonic Empire. Sadi Carnot studied in Paris at the École Polytechnique, whose members populate a “Who’s Who” list of science and engineering. 

As Carnot grew up, the Industrial Revolution was humming. Steam engines were producing reliable energy on vast scales; factories were booming; and economies were transforming. France’s old enemy Britain enjoyed two advantages. One consisted of inventors: Englishmen Thomas Savery and Thomas Newcomen invented the steam engine. Scotsman James Watt then improved upon Newcomen’s design until rendering it practical. Second, northern Britain contained loads of coal that industrialists could mine to power her engines. France had less coal. So if you were a French engineer during Carnot’s lifetime, you should have cared about engines’ efficiencies—how effectively engines used fuel.²

Carnot proved a fundamental limitation on engines’ efficiencies. His theorem governs engines that draw energy from heat—rather than from, say, the motional energy of water cascading down a waterfall. In Carnot’s argument, a heat engine interacts with a cold environment and a hot environment. (Many car engines fall into this category: the hot environment is burning gasoline. The cold environment is the surrounding air into which the car dumps exhaust.) Heat flows from the hot environment to the cold. The engine siphons off some heat and converts it into work. Work is coordinated, well-organized energy that one can directly harness to perform a useful task, such as turning a turbine. In contrast, heat is the disordered energy of particles shuffling about randomly. Heat engines transform random heat into coordinated work.

An engine’s efficiency is the bang we get for our buck—the upshot we gain, compared to the cost we spend. Running an engine costs the heat that flows between the environments: the more heat flows, the more the hot environment cools, so the less effectively it can serve as a hot environment in the future. An analogous statement concerns the cold environment. So a heat engine’s efficiency is the work produced, divided by the heat spent.

Carnot upper-bounded the efficiency achievable by every heat engine of the sort described above. Let denote the cold environment’s temperature; and , the hot environment’s. The efficiency can’t exceed . What a simple formula for such an extensive class of objects! Carnot’s theorem governs not only many car engines (Otto engines), but also the Stirling engine that competed with the steam engine, its cousin the Ericsson engine, and more.

In addition to generality and simplicity, Carnot’s bound boasts practical and fundamental significances. Capping engine efficiencies caps the output one can expect of a machine, factory, or economy. The cap also prevents engineers from wasting their time on daydreaming about more-efficient engines. 

More fundamentally than these applications, Carnot’s theorem encapsulates the second law of thermodynamics. The second law helps us understand why time flows in only one direction. And what’s deeper or more foundational than time’s arrow? People often cast the second law in terms of entropy, but many equivalent formulations express the law’s contents. The formulations share a flavor often synopsized with “You can’t win.” Just as we can’t grow younger, we can’t beat Carnot’s bound on engines. 

One might expect no engine to achieve the greatest efficiency imaginable: , called the Carnot efficiency. This expectation is incorrect in one way and correct in another. Carnot did design an engine that could operate at his eponymous efficiency: an eponymous engine. A Carnot engine can manifest as the thermodynamicist’s favorite physical system: a gas in a box topped by a movable piston. The gas undergoes four strokes, or steps, to perform work. The strokes form a closed cycle, returning the gas to its initial conditions.³ 

Steampunk artist Todd Cahill beautifully illustrated the Carnot cycle for my book. The gas performs useful work because a teapot sits atop the piston. Pushing the piston upward, the gas lifts the teapot. You can find a more detailed description of Carnot’s engine in Chapter 4 of the book, but I’ll recap the cycle here.

The gas expands during stroke 1, pushing the piston and so outputting work. Maintaining contact with the hot environment, the gas remains at the temperature . The gas then disconnects from the hot environment. Yet the gas continues to expand throughout stroke 2, lifting the teapot further. Forfeiting energy, the gas cools. It ends stroke 2 at the temperature .

The gas contacts the cold environment throughout stroke 3. The piston pushes on the gas, compressing it. At the end of the stroke, the gas disconnects from the cold environment. The piston continues compressing the gas throughout stroke 4, performing more work on the gas. This work warms the gas back up to .

In summary, Carnot’s engine begins hot, performs work, cools down, has work performed on it, and warms back up. The gas performs more work on the piston than the piston performs on it. Therefore, the teapot rises (during strokes 1 and 2) more than it descends (during strokes 3 and 4). 

At what cost, if the engine operates at the Carnot efficiency? The engine mustn’t waste heat. One wastes heat by roiling up the gas unnecessarily—by expanding or compressing it too quickly. The gas must stay in equilibrium, a calm, quiescent state. One can keep the gas quiescent only by running the cycle infinitely slowly. The cycle will take an infinitely long time, outputting zero power (work per unit time). So one can achieve the perfect efficiency only in principle, not in practice, and only by sacrificing power. Again, you can’t win.

Carnot’s theorem may sound like the Eeyore of physics, all negativity and depression. But I view it as a companion and backdrop as rich, for thermodynamicists, as the River is for the Water Rat. Carnot’s theorem curbs diverse technologies in practical settings. It captures the second law, a foundational principle. The Carnot cycle provides intuition, serving as a simple example on which thermodynamicists try out new ideas, such as quantum engines. Carnot’s theorem also provides what physicists call a sanity check: whenever a researcher devises a new (for example, quantum) heat engine, they can confirm that the engine obeys Carnot’s theorem, to help confirm their proposal’s accuracy. Carnot’s theorem also serves as a school exercise and a historical tipping point: the theorem initiated the development of thermodynamics, which continues to this day. 

So Carnot’s theorem is practical and fundamental, pedagogical and cutting-edge—brother and sister, and aunts, and company, and food and drink. I just wouldn’t recommend trying to wash your socks in Carnot’s theorem.

¹To a theoretical physicist, working as a mathematician and an engineer amounts to leading a colorful life.

²People other than Industrial Revolution–era French engineers should care, too.

³A cycle doesn’t return the hot and cold environments to their initial conditions, as explained above.

Trump’s cronies are already going after US government employees involved in the response to climate change. You can read about it here:

• Hadas Gold and Rene Marsh, Elon Musk publicized the names of government employees he wants to cut. It’s terrifying federal workers, CNN, 27 November 2024.

When President-elect Donald Trump said Elon Musk and Vivek Ramaswamy would recommend major cuts to the federal government in his administration, many public employees knew that their jobs could be on the line.

Now they have a new fear: becoming the personal targets of the world’s richest man—and his legions of followers.

Last week, in the midst of the flurry of his daily missives, Musk reposted two X posts that revealed the names and titles of people holding four relatively obscure climate-related government positions. Each post has been viewed tens of millions of times, and the individuals named have been subjected to a barrage of negative attention. At least one of the four women named has deleted her social media accounts.

Although the information he posted on those government positions is available through public online databases, these posts target otherwise unknown government employees in roles that do not deal directly with the public.

[…]

It appears [one] woman Musk targeted has since gone dark on social media, shutting down her accounts. The agency, the US International Development Finance Corporation, says it supports investment in climate mitigation, resilience and adaptation in low-income countries experiencing the most devastating effects of climate change. A DFC official said the agency does not comment on individual personnel positions or matters.

Musk also called out the Department of Energy’s chief climate officer in its loan programs office. The office funds fledgling energy technologies in need of early investment and awarded $465 million to Tesla Motors in 2010, helping to position Musk’s electric vehicle company as an EV industry leader. The chief climate officer works across agencies to “reduce barriers and enable clean energy deployment” according to her online bio.

Another woman, who serves as senior advisor on environmental justice and climate change at the Department of Health and Human Services, was another Musk target. HHS focuses on protecting the public health from pollution and other environmental hazards, especially in low-income communities and communities of color that are experiencing a higher share of exposures and impacts. The office first launched at Health and Human Services under the Biden administration in 2022.

A senior adviser to climate at the Department of Housing and Urban Development was also singled out. The original X post said the woman “should not be paid $181,648.00 by the US taxpayer to be the ‘Climate advisor’ at HUD.” Musk reposted with the comment: “But maybe her advice is amazing.” Followed by two laughing emojis.

This revives fears that US climate change policies will be rolled back. Reporters are interviewing me again about the Azimuth Climate Data Backup Project—because we’re again facing the possibility that a Trump administration could get rid of the US government’s climate data.

From 2016 to 2018, our team backed up up 30 terabytes of US government databases on climate change and the environment, saving it from the threat of a government run by climate change deniers. 627 people contributed a total of $20,427 to our project on Kickstarter to pay for storage space and a server.

That project is done now, with the data stored in a secret permanent location. But that data is old, and there’s plenty more by now.

I don’t have the energy to repeat the process now. As before, I’m hoping that the people at NOAA, NASA, etc. have quietly taken their own precautions. They’re in a much better position to do it! But I don’t know what they’ve done.

First I got interviewed for this New York Times article about the current situation:

• Austyn Gaffney, How Trump’s return could affect climate and weather data, New York Times, 14 November 2024.

Then I got interviewed for a second article, which says a bit more about what the Azimuth Project actually did:

• Chelsea Harvey, Scientists scramble to save climate data from Trump—again, Scientific American, 22 November 2024.

Eight years ago, as the Trump administration was getting ready to take office for the first time, mathematician John Baez was making his own preparations.

Together with a small group of friends and colleagues, he was arranging to download large quantities of public climate data from federal websites in order to safely store them away. Then-President-elect Donald Trump had repeatedly denied the basic science of climate change and had begun nominating climate skeptics for cabinet posts. Baez, a professor at the University of California, Riverside, was worried the information — everything from satellite data on global temperatures to ocean measurements of sea-level rise — might soon be destroyed.

His effort, known as the Azimuth Climate Data Backup Project, archived at least 30 terabytes of federal climate data by the end of 2017.

In the end, it was an overprecaution.

The first Trump administration altered or deleted numerous federal web pages containing public-facing climate information, according to monitoring efforts by the nonprofit Environmental Data and Governance Initiative (EDGI), which tracks changes on federal websites. But federal databases, containing vast stores of globally valuable climate information, remained largely intact through the end of Trump’s first term.

Yet as Trump prepares to take office again, scientists are growing more worried.

Federal datasets may be in bigger trouble this time than they were under the first Trump administration, they say. And they’re preparing to begin their archiving efforts anew.

“This time around we expect them to be much more strategic,” said Gretchen Gehrke, EDGI’s website monitoring program lead. “My guess is that they’ve learned their lessons.”

[….]

Much of the renewed concern about federal data stems from Project 2025, a 900-page conservative policy blueprint spearheaded by the Heritage Foundation that outlines recommendations for the next administration.

Project 2025 calls for major overhauls of some federal science agencies. It suggests that Trump should dismantle NOAA and calls for the next administration to “reshape” the U.S. Global Change Research Program, which coordinates federal research on climate and the environment.

The plan also suggests that the “Biden Administration’s climate fanaticism will need a whole-of-government unwinding.”

A leaked video from the Project 2025 presidential transition project suggested that political appointees “will have to eradicate climate change references from absolutely everywhere.”

Trump has previously distanced himself from Project 2025. In July, he wrote on the social media platform Truth Social that he knew “nothing about Project 2025,” did not know who was behind it and did not have anything to do with the plan.

But since winning the 2024 presidential election, Trump has picked several nominees for his new administration that are credited by name in the conservative policy plan, reviving fears that Project 2025 could influence his priorities.

Trump has also recently named Elon Musk and Vivek Ramaswamy to lead his new so-called Department of Government Efficiency, an external commission tasked with shrinking the federal government, restructuring federal agencies and cutting costs. The announcement has also ignited concerns about job security for federal scientists, including the researchers tasked with maintaining government datasets.

“There are lots and lots of signs that the Trump team is attempting to decapitate the government in the sense of firing lots of people,” said Baez, who co-founded the Azimuth Climate Data Backup Project in 2016 and is currently a professor of the graduate division in the math department at University of California Riverside. “If they manage to do something like that, then these databases could be in more jeopardy.”

Though federal datasets remained largely untouched under the first Trump administration, other climate-related information on federal websites did change or disappear, Gehrke pointed out. EDGI documented about a 40 percent decline in the use of the term “climate change” across 13 federal agencies it monitored during the first term.

A better organized effort could result in more censoring under a second administration, she said.

While groups like EDGI are gearing up for their next efforts, Baez says he has no immediate plans to revamp the Azimuth Climate Data Backup Project — although he hopes other groups will step up instead. One lesson he learned the first time is just how much data exists in the federal ecosystem and how much effort it takes to archive it, even with a dedicated group of volunteers.

“We got sort of a little bit burnt out by that process,” Baez said. “I’m hoping some younger generation of people picks up where we left off.”

If you’re interested in doing this, and want to see what data we backed up, you can see a list here.

read more

I realised just now that I entirely forgot (it seems) to post about an episode of PBS' show Nova called "Decoding the Universe: Cosmos" which aired back in the Spring. I thought they did a good job of talking about some of the advances in our understanding that have happened over the last 50 years (the idea is that it is the 50th anniversary of the show) in areas of astrophysics and cosmology. I was a contributor, filmed at the top of Mount Wilson at the Observatory where Hubble made his famous discoveries about the size of the universe, and its expansion. I talk about some of those discoveries and other ideas in the show. Here's a link to the "Decoding the Universe" site. (You can also find it on YouTube.)

If you follow the link you'll notice another episode up there: "Decoding the Universe: Quantum". That's a companion they made, and it focuses on understanding in quantum physics, connecting it to things in the everyday world. and also back to black holes and things astrophysical and cosmological. It also does a good job of shining a light on many concepts.

I was also a contributor to this episode, and it was a real delight to work with them in a special role: I got to unpack many of the foundational quantum mechanical concepts (transitions in atoms, stimulated emission, tunnelling, etc) to camera by doing line drawings while I explained - and kudos [...] Click to continue reading this post →

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For those of you who keep up with my social media posts, you’ve probably been expecting that I’d eventually announce that I’m transitioning from Twitter to something else… and it is Bluesky. I’ll stay on Twitter for a bit longer while I settle in (and while I wait for people … Click to continue reading this post →

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 A few tidbits that I encountered recently:

• The saga of Ranga Dias at Rochester draws to a close, as described by the Wall Street Journal.  It took quite some time for this to propagate through their system.  This is after multiple internal investigations that somehow were ineffective, an external investigation, and a lengthy path through university procedures (presumably because universities have to be careful not to shortcut any of their processes, or they open themselves up to lawsuits).
• At around the same time, Mikhail Eremets passed away.  He was a pioneer in high pressure measurements of material properties and in superconductivity in hydrides.
• Also coincident, this preprint appeared on the arXiv, a brief statement summarizing some of the evidence for relatively high temperature superconductivity in hydrides at high pressure.
• Last week Carl Bender gave a very nice colloquium at Rice, where he spoke about a surprising result.  When we teach undergrad quantum mechanics, we tell students that the Hamiltonian (the expression with operators that gives the total energy of a quantum system) has to be Hermitian, because this guarantees that the energy eigenvalues have to be real numbers.  Generically, non-hermitian Hamiltonians would imply complex energies, which would imply non-conservation of total probability. That is one way of treating open quantum systems, when particles can come and go, but for closed quantum systems, we like real energies.  Anyway, it turns out that one can write an explicitly complex Hamiltonian that nonetheless has a completely real energy spectrum, and this has deep connections to PT symmetry conservation.  Here is a nice treatment of this.
• Just tossing this out:  The entire annual budget for the state of Arkansas is $6.5B.  The annual budget for Stanford University is $9.5B.  

Space and time seem as fundamental as anything can get. Philosophers like Immanuel Kant thought that they were inescapable, that we could not conceive of the world without space and time. But increasingly, physicists suspect that space and time are not as fundamental as they appear. When they try to construct a theory of quantum gravity, physicists find puzzles, paradoxes that suggest that space and time may just be approximations to a more fundamental underlying reality.

One piece of evidence that quantum gravity researchers point to are dualities. These are pairs of theories that seem to describe different situations, including with different numbers of dimensions, but that are secretly indistinguishable, connected by a “dictionary” that lets you interpret any observation in one world in terms of an equivalent observation in the other world. By itself, duality doesn’t mean that space and time aren’t fundamental: as I explained in a blog post a few years ago, it could still be that one “side” of the duality is a true description of space and time, and the other is just a mathematical illusion. To show definitively that space and time are not fundamental, you would want to find a situation where they “break down”, where you can go from a theory that has space and time to a theory that doesn’t. Ideally, you’d want a physical means of going between them: some kind of quantum field that, as it shifts, changes the world between space-time and not space-time.

What I didn’t know when I wrote that post was that physicists already knew about such a situation in 1993.

Back when I was in pre-school, famous string theorist Edward Witten was trying to understand something that others had described as a duality, and realized there was something more going on.

In string theory, particles are described by lengths of vibrating string. In practice, string theorists like to think about what it’s like to live on the string itself, seeing it vibrate. In that world, there are two dimensions, one space dimension back and forth along the string and one time dimension going into the future. To describe the vibrations of the string in that world, string theorists use the same kind of theory that people use to describe physics in our world: a quantum field theory. In string theory, you have a two-dimensional quantum field theory stuck “inside” a theory with more dimensions describing our world. You see that this world exists by seeing the kinds of vibrations your two-dimensional world can have, through a type of quantum field called a scalar field. With ten scalar fields, ten different ways you can push energy into your stringy world, you can infer that the world around you is a space-time with ten dimensions.

String theory has “extra” dimensions beyond the three of space and one of time we’re used to, and these extra dimensions can be curled up in various ways to hide them from view, often using a type of shape called a Calabi-Yau manifold. In the late 80’s and early 90’s, string theorists had found a similarity between the two-dimensional quantum field theories you get folding string theory around some of these Calabi-Yau manifolds and another type of two-dimensional quantum field theory related to theories used to describe superconductors. People called the two types of theories dual, but Witten figured out there was something more going on.

Witten described the two types of theories in the same framework, and showed that they weren’t two equivalent descriptions of the same world. Rather, they were two different ways one theory could behave.

The two behaviors were connected by something physical: the value of a quantum field called a modulus field. This field can be described by a number, and that number can be positive or negative.

When the modulus field is a large positive number, then the theory behaves like string theory twisted around a Calabi-Yau manifold. In particular, the scalar fields have many different values they can take, values that are smoothly related to each other. These values are nothing more or less than the position of the string in space and time. Because the scalars can take many values, the string can sit in many different places, and because the values are smoothly related to each other, the string can smoothly move from one place to another.

When the modulus field is a large negative number, then the theory is very different. What people thought of as the other side of the duality, a theory like the theories used to describe superconductors, is the theory that describes what happens when the modulus field is large and negative. In this theory, the scalars can no longer take many values. Instead, they have one option, one stable solution. That means that instead of there being many different places the string could sit, describing space, there are no different places, and thus no space. The string lives nowhere.

These are two very different situations, one with space and one without. And they’re connected by something physical. You could imagine manipulating the modulus field, using other fields to funnel energy into it, pushing it back and forth from a world with space to a world of nowhere. Much more than the examples I was aware of, this is a super-clear example of a model where space is not fundamental, but where it can be manipulated, existing or not existing based on physical changes.

We don’t know whether a model like this describes the real world. But it’s gratifying to know that it can be written down, that there is a picture, in full mathematical detail, of how this kind of thing works. Hopefully, it makes the idea that space and time are not fundamental sound a bit more reasonable.

Previously: Part 8. Next: Part 10.

It’s the penultimate week of the course, and up until now we’ve abstained from using the axiom of choice. But this week we gorged on it.

We proved that all the usual things are equivalent to the axiom of choice: Zorn’s lemma, the well ordering principle, cardinal comparability (given two sets, one must inject into the other), and the souped-up version of cardinal comparability that compares not just two sets but an arbitrary collection of them: for any nonempty family of sets $(X_i)_{i \in I}$, there is some $X_i$ that injects into all the others.

The section I most enjoyed writing and teaching was the last one, on unnecessary uses of the axiom of choice. I’m grateful to Todd Trimble for explaining to me years ago how to systematically remove dependence on choice from arguments in basic general topology. (For some reason, it’s very tempting in that subject to use choice unnecessarily.) I talk about this at the very end of the chapter.

Previously: Part 9.

The course is over! The grand finale was the theorem that

$$X \times Y \cong X + Y \cong max(X, Y)$$

for all infinite sets $X$ and $Y$. Proving this required most of the concepts and results from the second half of the course: well ordered sets, the Cantor–Bernstein theorem, the Hartogs theorem, Zorn’s lemma, and so on.

I gave the merest hints of the world of cardinal arithmetic that lies beyond. If I’d had more time, I would have got into large sets (a.k.a. large cardinals), but the course was plenty long enough already.

Thanks very much to everyone who’s commented here so far, but thank you most of all to my students, who really taught me an enormous amount.

Are you interested in using category-theoretic methods to tackle problems in topics like quantum computation, machine learning, numerical analysis or graph theory? Then you might like the Adjoint School! A lot of applied category theorists I know have gotten their start there. It can be a transformative experience, in part thanks to all the people you’ll meet.

You’ll work online on a research project with a mentor and a team of other students for several months. Then you’ll get together for several days at the end of May at the University of Florida, in Gainesville. Then comes the big annual conference on applied category theory, ACT2025.

You can apply here starting November 1st, 2024. The deadline to apply is December 1st.

For more details, including the list of mentors and their research projects, read on.

Important dates

• Application opens: November 1, 2024
• Application deadline: December 1, 2024
• School runs: January-May, 2025
• Research week dates: May 26-30, 2025

Who should apply?

Anyone, from anywhere in the world, who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language—the definition of monoidal category for example—is encouraged.

The school will consider advanced undergraduates, PhD students, post-docs, as well as people working outside of academia. Members of groups which are underrepresented in the mathematics and computer science communities are especially encouraged to apply.

Research projects

Each project team consists of 4-5 students, led by a mentor and a teaching assistant. The school takes place in two phases: an online learning seminar that meets regularly between January and May, and an in-person research week held on the university campus, the week prior to the Applied Category Theory Conference.

During the learning seminar, participants read, discuss, and respond to papers chosen by the project mentors. Every other week a pair of participants will present a paper which will be followed by a group discussion. After the learning seminar, each pair of participants will also write a blog post, based on the paper they presented, for The n-Category Café.

Here are the research project and mentors. For the reading materials, visit the Adjoint School website.

Structuring Quantum Effects with Monads and Arrows

Mentor: Juliana Kaizer Vizzotto

Category theory provides a mathematical framework for understanding structures and their relationships abstractly. We can use the tools from category theory for reasoning about abstraction, modularity, and compositionality, offering a powerful framework for modeling complex systems in computer science. In the context of quantum computing, we need to deal with the properties inherent of quantum information. Traditional categorical frameworks often model computations as sequential transformations, but quantum processes demand a representation that captures: i) the quantum parallelism caused by the non-local wave character of quantum information which is qualitatively different from the classical notion of parallelism; and also ii) the notion of observation, or measurement, in which the observed part of the quantum state and every other part that is entangled with it immediately lose their wave character.

In this project we will investigate the use of monads to model quantum parallelism, inspired by the work of Moggi on modeling computational effects. Moreover, quantum computation introduces additional complexity, particularly with respect to measurement and the collapse of quantum states. Then we will study how to construct a categorical representation for the traditional general model of quantum computing based on density matrices and superoperators using a generalization of monads, called arrows. Finally, we will investigate the use of relative monads in the context of quantum measurement.

Homotopy of Graphs

Mentor: Laura Scull

Graphs are discrete structures made of vertices connected by edges, making examples easily accessible. We take a categorical approach to these, and work in the category of graphs and graph homomorphisms between them. Even though many standard graph theory ideas can be phrased in these terms, this area remains relatively undeveloped.

This project will consider discrete homotopy theory, where we define the notion of homotopy between graph morphisms by adapting definitions from topological spaces. In particular, we will look at the theory of ×-homotopy as developed by Dochtermann and Chih-Scull. The resulting theory has some but not all of the formal properties of classical homotopy of spaces, and diverges in some interesting ways.

Our project will start with learning about the basic category of graphs and graph homomorphisms, and understanding categorical concepts such as limits, colimits and expnentials in this world. This offers an opportunity to play with concrete examples of abstract universal properties. We will then consider the following question: do homotopy limits and colimits exist for graphs? If so, what do they look like? This specific question will be our entry into the larger inquiries around what sort of structure is present in homotopy of graphs, and how it compares to the classical homotopy theory of topological spaces. We will develop this theme further in directions that most interest our group.

Compositional Generalization in Reinforcement Learning

Mentor: Georgios Bakirtzis

Reinforcement learning is a form of semi-supervised learning. In reinforcement learning we have an environment, an agent that acts on this environment through actions, and a reward signal. It is the reward signal that makes reinforcement learning a powerful technique in the control of autonomous systems, but it is also the sparcity of this reward structure that engenders issues. Compositional methods decompose reinforcement learning to parts that are tractable. Categories provide a nice framework to think about compositional reinforcement learning.

An important open problem in reinforcement learning is /compositional generalization. This project will tackle the problem of compositional generalization in reinforcement learning in a category-theoretic computational framework in Julia. Expected outcomes are of this project are category theory derived algorithms and concrete experiments. Participants will be expected to be strong computationally, but not necessarily have experience in reinforcement learning.

Categorical Metric Structures for Numerical Analysis

Mentor: Justin Hsu

Numerical analysis studies computations that use finite approximations to continuous data, e.g., finite precision floating point numbers instead of the reals. A core challenge is to bound the amount of error incurred. Recent work develops several type systems to reason about roundoff error, supported by semantics in categories of metric spaces. This project will focus on categorical structures uncovered by these works, seeking to understand and generalize them.

More specifically, the first strand of work will investigate the neighborhood monad, a novel graded monad in the category of (pseudo)metric spaces. This monad supports the forward rounding error analysis in the NumFuzz type system. There are several known extensions incorporating particular computational effects (e.g., failure, non-determinism, randomization) but a more general picture is currently lacking.

The second strand of work will investigate backwards error lenses, a lens-like structure on metric spaces that supports the backward error analysis in the Bean type system. The construction resembles concepts from the lens literature, but a precise connection is not known. Connecting these lenses to known constructions could enable backwards error analysis for more complex programs.

Organizers

The organizers of Adjoint School 2025 are Elena Dimitriadis Bermejo, Ari Rosenfield, Innocent Obi, and Drew McNeely. The steering committee consists of David Jaz Myers, Daniel Cicala, Elena di Lavore, and Brendan Fong.

Are you interested in using category-theoretic methods to tackle problems in topics like quantum computation, machine learning, numerical analysis or graph theory? Then you might like the Adjoint School! A lot of applied category theorists I know have gotten their start there. It can be a transformative experience, in part thanks to all the people you’ll meet.

You’ll work online on a research project with a mentor and a team of other students for several months. Then you’ll get together for several days at the end of May at the University of Florida, in Gainesville. Then comes the big annual conference on applied category theory, ACT2025.

You can apply here starting November 1st, 2024. The deadline to apply is December 1st.

For more details, including the list of mentors and their research projects, read on.

Important dates

• Application opens: November 1, 2024
• Application deadline: December 1, 2024
• School runs: January-May, 2025
• Research week dates: May 26-30, 2025

Who should apply?

Anyone, from anywhere in the world, who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language — the definition of monoidal category for example — is encouraged.

The school will consider advanced undergraduates, PhD students, post-docs, as well as people working outside of academia. Members of groups which are underrepresented in the mathematics and computer science communities are especially encouraged to apply.

Research projects

Each project team consists of 4-5 students, led by a mentor and a teaching assistant. The school takes place in two phases: an online learning seminar that meets regularly between January and May, and an in-person research week held on the university campus, the week prior to the Applied Category Theory Conference.

During the learning seminar, participants read, discuss, and respond to papers chosen by the project mentors. Every other week a pair of participants will present a paper which will be followed by a group discussion. After the learning seminar, each pair of participants will also write a blog post, based on the paper they presented, for The n-Category Café.

Here are the research project and mentors. For the reading materials, visit the Adjoint School website.

Structuring Quantum Effects with Monads and Arrows

Mentor: Juliana Kaizer Vizzotto

Category theory provides a mathematical framework for understanding structures and their relationships abstractly. We can use the tools from category theory for reasoning about abstraction, modularity, and compositionality, offering a powerful framework for modeling complex systems in computer science. In the context of quantum computing, we need to deal with the properties inherent of quantum information. Traditional categorical frameworks often model computations as sequential transformations, but quantum processes demand a representation that captures: i) the quantum parallelism caused by the non-local wave character of quantum information which is qualitatively different from the classical notion of parallelism; and also ii) the notion of observation, or measurement, in which the observed part of the quantum state and every other part that is entangled with it immediately lose their wave character.

In this project we will investigate the use of monads to model quantum parallelism, inspired by the work of Moggi on modeling computational effects. Moreover, quantum computation introduces additional complexity, particularly with respect to measurement and the collapse of quantum states. Then we will study how to construct a categorical representation for the traditional general model of quantum computing based on density matrices and superoperators using a generalization of monads, called arrows. Finally, we will investigate the use of relative monads in the context of quantum measurement.

Homotopy of Graphs

Mentor: Laura Scull

Graphs are discrete structures made of vertices connected by edges, making examples easily accessible. We take a categorical approach to these, and work in the category of graphs and graph homomorphisms between them. Even though many standard graph theory ideas can be phrased in these terms, this area remains relatively undeveloped.

This project will consider discrete homotopy theory, where we define the notion of homotopy between graph morphisms by adapting definitions from topological spaces. In particular, we will look at the theory of ×-homotopy as developed by Dochtermann and Chih-Scull. The resulting theory has some but not all of the formal properties of classical homotopy of spaces, and diverges in some interesting ways.

Our project will start with learning about the basic category of graphs and graph homomorphisms, and understanding categorical concepts such as limits, colimits and expnentials in this world. This offers an opportunity to play with concrete examples of abstract universal properties. We will then consider the following question: do homotopy limits and colimits exist for graphs? If so, what do they look like? This specific question will be our entry into the larger inquiries around what sort of structure is present in homotopy of graphs, and how it compares to the classical homotopy theory of topological spaces. We will develop this theme further in directions that most interest our group.

Compositional Generalization in Reinforcement Learning

Mentor: Georgios Bakirtzis

Reinforcement learning is a form of semi-supervised learning. In reinforcement learning we have an environment, an agent that acts on this environment through actions, and a reward signal. It is the reward signal that makes reinforcement learning a powerful technique in the control of autonomous systems, but it is also the sparcity of this reward structure that engenders issues. Compositional methods decompose reinforcement learning to parts that are tractable. Categories provide a nice framework to think about compositional reinforcement learning.

An important open problem in reinforcement learning is /compositional generalization. This project will tackle the problem of compositional generalization in reinforcement learning in a category-theoretic computational framework in Julia. Expected outcomes are of this project are category theory derived algorithms and concrete experiments. Participants will be expected to be strong computationally, but not necessarily have experience in reinforcement learning.

Categorical Metric Structures for Numerical Analysis

Mentor: Justin Hsu

Numerical analysis studies computations that use finite approximations to continuous data, e.g., finite precision floating point numbers instead of the reals. A core challenge is to bound the amount of error incurred. Recent work develops several type systems to reason about roundoff error, supported by semantics in categories of metric spaces. This project will focus on categorical structures uncovered by these works, seeking to understand and generalize them.

More specifically, the first strand of work will investigate the neighborhood monad, a novel graded monad in the category of (pseudo)metric spaces. This monad supports the forward rounding error analysis in the NumFuzz type system. There are several known extensions incorporating particular computational effects (e.g., failure, non-determinism, randomization) but a more general picture is currently lacking.

The second strand of work will investigate backwards error lenses, a lens-like structure on metric spaces that supports the backward error analysis in the Bean type system. The construction resembles concepts from the lens literature, but a precise connection is not known. Connecting these lenses to known constructions could enable backwards error analysis for more complex programs.

Organizers

The organizers of Adjoint School 2025 are Elena Dimitriadis Bermejo, Ari Rosenfield, Innocent Obi, and Drew McNeely. The steering committee consists of David Jaz Myers, Daniel Cicala, Elena di Lavore, and Brendan Fong.

Particle physicists describe how elementary particles behave using a set of equations called their “Standard Model.” How did they become so confident that a set of math formulas, ones that can be compactly summarized on a coffee cup, can describe so much of nature?

My previous “Celebrations of the Standard Model” (you can find the full set here) have included the stories of how we know the strengths of the forces, the number of types (“flavors” and “colors”) and the electric charges of the quarks, and the structures of protons and neutrons, among others. Along the way I explained how W bosons, the electrically charged particles involved in the weak nuclear force, quickly decay (i.e. disintegrate into other particles). But I haven’t yet explained how their cousin, the electrically-neutral Z boson, decays. That story brings us to a central feature of the Standard Model.

Here’s the big picture. There’s a super-important number that plays a central role in the Standard Model. It’s a sort of angle (in a sense that will become clearer in Figs. 2 and 3 below), and is called θ_w or θ_weak. Through the action of the Higgs field on the particles, this one number determines many things, including

• the relative masses of the W and Z bosons
• the relative lifetimes of the W and Z bosons
• the relative probabilities for Z bosons to decay to one type of particle versus another
• the relative rates to produce different types of particles in scattering of electrons and positrons at very high energies
• the relative rates for processes involving scattering neutrinos off atoms at very low energies
• asymmetries in weak nuclear processes (ones that would be symmetric in corresponding electromagnetic processes)

and many others.

This is an enormously ambitious claim! When I began my graduate studies in 1988, we didn’t know if all these predictions would work out. But as the data from experiments came in during the 1990s and beyond, it became clear that every single one of them matched the data quite well. There were — and still are — no exceptions. And that’s why particle physicists became convinced that the Standard Model’s equations are by far the best they’ve ever found.

As an illustration, Fig. 1 shows, as a function of sin θ_w, the probabilities for Z bosons to decay to each type of particle and its anti-particle. Each individual probability is indicated by the size of the gap between one line and the line above. The total probability always adds up to 1, but the individual probabilities depend on the value of θ_w. (For instance, the width of the gap marked “muon” indicates the probability for a Z to decay to a muon and an anti-muon; it is about 5% at sin θ_w = 0, about 3% at sin θ_w = 1/2, and over 15% at sin θ_w = 1.)

As we’ll see in Fig. 3, the W and Z boson masses imply (if the Standard Model is valid) that sin θ_w is about ¹/₂. Using that measurement, we can then predict that all the various decay probabilities should be given within the green rectangle (if the Standard Model is valid). These predictions, made in the mid-1980s, proved correct in the 1990s; see Fig. 5 below.

This is what I’ll sketch in today’s post. In future posts I’ll go further, showing how this works with high precision.

The Most Important Angle in Particle Physics

Angles are a common feature of geometry and nature: 90 degree angles of right-angle triangles, the 60 degree angles of equilateral triangles, the 104.5 degree angle between the two hydrogen-oxygen bonds in a water molecule, and so many more. But some angles, more abstract, turn out to be even more important. Today I’ll tell you about θ_w , which is a shade less than 30 degrees (π/6 radians).

Note: This angle is often called “the Weinberg angle”, based on Steven Weinberg’s 1967 version of the Standard Model, but it should be called the “weak-mixing angle”, as it was first introduced seven years earlier by Sheldon Glashow, before the idea of the Higgs field.

This is the angle that lies at the heart of the Standard Model: the smallest angle of the right-angle triangle shown in Fig. 2. Two of its sides represent the strengths g₁ and g₂ of two of nature’s elementary forces: the weak-hypercharge force and the weak-isospin force. According to the Standard Model, the machinations of the Higgs field transform them into more familar forces: the electromagnetic force and the weak nuclear force. (The Standard Model is often charaterized by the code SU(3)xSU(2)xU(1); weak-isospin and weak-hypercharge are the SU(2) and U(1) parts, while SU(3) gives us the strong nuclear force).

To keep things especially simple today, let’s just say θ_w = 30 degrees, so that sin θ_w = ¹/₂. In a later post, we’ll see the angle is closer to 28.7 degrees, and this makes a difference when we’re being precise.

The Magic Angle and the W and Z Bosons

The Higgs field gives masses to the W and Z bosons, and the structure of the Standard Model predicts a very simple relation, given by the electroweak triangle as shown at the top of Fig. 3:

• 

This has the consequence shown at the top of Fig. 3, rewritten as a prediction

• 

If sin θ_w = ¹/₂ , this quantity is predicted to be 1/4 = 0.250. Measurements (m_W = 80.4 GeV/c² and m_Z = 91.2 GeV/c²) show it to be 0.223. Agreement isn’t perfect, indicating that the angle isn’t exactly 30 degrees. But it is close enough for today’s post.

Where does this formula for the W and Z masses come from? Click here for details:

Central to the Standard Model is the so-called “Higgs field”, which has been switched on since very early in the Big Bang. The Higgs field is responsible for the masses of all the known elementary particles, but in general, though we understand why the masses aren’t zero, we can’t predict their values. However, there’s one interesting exception. The ratio of the W and Z bosons’ masses is predicted.

Before the Higgs field switched on, here’s how the weak-isospin and weak-hypercharge forces were organized: there were

• 3 weak isospin fields, called W⁺, W^– and W⁰, whose particles (of the same names) had zero rest mass
• 1 weak-hypercharge field, usually called, X, whose particle (of the same name) had zero rest mass

After the Higgs field switched on by an amount v, however, these four fields were reorganized, leaving

• One, called the electromagnetic field, with particles called “photons” with zero rest mass.
• One, called Z⁰ or just Z, now has particles (of the same names) with rest mass m_Z
• Two, still called W⁺ and W^– , have particles (of the same names) with rest mass m_W

Central to this story is θ_w. In particular, the relationship between the photon and Z and the original W⁰ and X involves this angle. The picture below depicts this relation, given also as an equation

The W⁺ and W^– bosons get their masses through their interaction, via the weak-isospin force, with the Higgs field. The Z boson gets its mass in a similar way, but because the Z is a mixture of W⁰ and X, both the weak-isospin and weak-hypercharge forces play a role. And thus m_Z depends both on g₁ and g₂, while m_W depends only on g₂. Thus

where v is the “value” or strength of the switched-on Higgs field, and in the last step I have used the electroweak triangle of Fig. 2.

A slightly more complex relation relates the W boson’s lifetime t_W and the Z boson’s lifetime t_Z (this is the average time between when the particle is created and when it decays.) This is shown at the bottom of Fig. 3.

• 

This is a slightly odd-looking formula; while 81 = 9² is a simple number, 86 is a weird one. Where does it come from? We’ll see in just a moment. In any case, as seen in Fig. 3, agreement between theoretical prediction and experiment is excellent.

If the Standard Model were wrong, there would be absolutely no reason for these two predictions to be even close. So this is a step in the right direction. But it is hardly the only one. Let’s check the detailed predictions in Figure 1.

W and Z Decay Probabilities

Here’s what the Standard Model has to say about how W and Z bosons can decay.

W Decays

In this earlier post, I explained that W^– bosons can decay (oversimplifying slightly) in five ways:

• to an electron and a corresponding anti-neutrino
• to a muon and a corresponding anti-neutrino
• to a tau and a corresponding anti-neutrino
• to a down quark and an up anti-quark
• to a strange quark and a charm anti-quark

(A decay to a bottom quark and top anti-quark is forbidden by the rule of decreasing rest mass; the top quark’s rest mass is larger than the W’s, so no such decay can occur.)

These modes have simple probabilities, according to the Standard Model, and they don’t depend on sin θ_w (except through small quantum corrections which we’re ignoring here). The first three have probability 1/9. Moreover, remembering each quark comes in three varieties (called “colors”), each color of quark also occurs with probability 1/9. Altogether the predictions for the probabilities are as shown in Fig. 4, along with measurements, which agree well with the predictions. When quantum corrections (such as those discussed in this post, around its own Fig. 4) are included, agreement is much better; but that’s beyond today’s scope.

Because the W+ and W- are each others’ anti-particles, W⁺ decay probabilities are the same as those for W^–, except with all particles switched with their anti-particles.

Z Decays

Unlike W decays, Z decays are complicated and depend on sin θ_w. If sin θ_w = ¹/₂, the Standard Model predicts that the probability for a Z boson to decay to a particle/anti-particle pair, where the particle has electric charge Q and weak-isospin-component T = +1 or -1 [technically, isospin’s third component, times 2], is proportional to

• 2 (Q/2-T)² + 2(Q/2)² = 2-2TQ+Q²

where I used T² = 1 in the final expression. The fact that this answer is built from a sum of two different terms, only one of which involves T (weak-isospin), is a sign of the Higgs field’s effects, which typically marry two different types of fields in the Standard Model, only one of which has weak-isospin, to create the more familiar ones.

This implies the relative decay probabilities (remembering quarks come in three “colors”) are

• For electrons, muons and taus (Q=-1, T=-1): 1
• For each of the three neutrinos (Q=0, T=1): 2
• For down-type quarks (Q=-1/3,T=-1) : 13/9 * 3 = 13/3
• For up-type quarks (Q=2/3,T=1): 10/9 * 3 = 10/3

These are shown at left in Fig. 5.

The sum of all those numbers (remembering that there are three down-type quarks and three up-type quarks, but again top quarks can’t appear due to the rule of decreasing rest mass) is:

• 1 + 1 + 1 + 2 + 2 + 2 + 13/3 + 13/3 + 13/3 + 10/3 + 10/3 = 86/3.

And that’s where the 86 seen in the lifetime ratio (Fig. 3) comes from.

To get predictions for the actual decay probabilities (rather than just the relative probabilities), we should divide each relative probability by 86/3, so that the sum of all the probabilities together is 1. This gives us

• For electrons, muons and taus (Q=-1, T=-1): 3/86
• For each of the three neutrinos (Q=0, T=1): 6/86
• For down-type quarks (Q=-1/3, T=-1) : 13/86
• For up-type quarks (Q=2/3, T=1): 10/86

as shown on the right-hand side of Fig. 5; these are the same as those of Fig. 1 at sin θ_w = ¹/₂. Measured values are also shown in Fig. 5 for electrons, muons, taus, the combination of all three neutrinos, the bottom quark, the charm quark, and (implicitly) the sum of all five quarks. Again, they agree well with the predictions.

This is already pretty impressive. The Standard Model and its Higgs field predict that just a single angle links a mass ratio, a lifetime ratio, and the decay probabilities of the Z boson. If the Standard Model were significantly off base, some or all of the predictions would fail badly.

However, this is only the beginning. So if you’re not yet convinced, consider reading the bonus section below, which gives four additional classes of examples, or stay tune for the next post in this series, where we’ll look at how things improve with a more precise value of sin θ_w.

Bonus: Other Predictions of the Standard Model

Many other processes involving the weak nuclear force depend in some way on sin θ_w. Here are a few examples.

High-Energy Electron-Positron Collisions (click for details)

In this post I discussed the ratio of the rates for two important processes in collisions of electrons and positrons:

• electron + positron ⟶ any quark + its anti-quark
• electron + positron ⟶ muon + anti-muon

This ratio is simple at low energy (E << m_Z c²), because it involves mainly electromagnetic effects, and thus depends only on the electric charges of the particles that can be produced.

But at high energy (E >> m_Z c²) , the prediction changes, because both electromagnetic and weak nuclear forces play a role. In fact, they “interfere”, meaning that one must combine their effects in a quantum way before calculating probabilities.

[What this cryptic quantum verbiage really means is this. At low energy, if S_f is the complex number representing the effect of the photon field on this process, then the rate for the process is |S_f|². But here we have to include both S_f and S_Z, where the latter is the effect of the Z field. The total rate is not |S_f|² + |S_Z|² , a sum of two separate probabilities. Instead it is |S_f+S_Z|² , which could be larger or smaller than |S_f|² + |S_Z|² — a sign of interference.]

In the Standard Model, the answer depends on sin θ_w. The LEP 2 collider measured this ratio at energies up to 209 GeV, well above m_Z c². If we assume sin θ_w is approximately ¹/₂, data agrees with predictions. [In fact, the ratio can be calculated as a function of energy, and thus made more precise; data agrees with these more precise predictions, too.]

Low-Energy Neutrino-Nucleus Collisions (click for details)

When electrons scatter off protons and neutrons, they do so via the electromagnetic force. For electron-proton collisions, this is not surprising, since both protons and electrons carry electric charge. But it’s also true for neutrons, because even though the neutron is electrically neutral, the quarks inside it are not.

By contrast, neutrinos are electrically neutral, and so they will only scatter off protons and neutrons (and their quarks) through the weak nuclear force. More precisely, they do so through the W and Z fields (via so-called “virtual W and Z particles” [which aren’t particles.]) Oversimplifying, if one can

• obtain beams of muon neutrinos, and
• scatter them off deuterons (nuclei of heavy hydrogen, which have one proton and one neutron), or off something that similarly has equal numbers of protons and neutrons,

then simple predictions can be made for the two processes shown at the top of Fig. 7, in which the nucleus shatters (turning into multiple “hadrons” [particles made from quarks, antiquarks and gluons]) and either a neutrino or a muon emerges from the collision. (The latter can be directly observed; the former can be inferred from the non-appearance of any muon.) Analogous predictions can be made for the anti-neutrino beams, as shown at the bottom of Fig. 7.

The ratios of these four processes are predicted to depend, in a certain approximation, only on sin θ_w. Data agrees with these predictions for sin θ_w approximately ¹/₂.

More complex and detailed predictions are also possible, and these work too.

Asymmetries in Electron-Positron Collisions (click for details)

There are a number of asymmetric effects that come from the fact that the weak nuclear force is

• not “parity-invariant”, (i.e. not the same when viewed in a mirror), and
• not “charge-conjugation invariant” (i.e. not the same when all electric charges are flipped)

though it is almost symmetric under doing both, i.e. putting the world in a mirror and flipping electric charge. No such asymmetries are seen in electromagnetism, which is symmetric under both parity and charge-conjugation separately. But when the weak interactions play a role, asymmetries appear, and they all depend, yet again, on sin θ_w.

Two classes of asymmetries of great interest are:

• “Left-Right Asymmetry” (Fig. 8): The rate for electron-positron collisions to make Z bosons in collisions with positrons depends on which way the electrons are “spinning” (i.e. whether they carry angular momentum along or opposite to their direction of motion.)
• “Forward-Backward Asymmetry” (Fig. 9): The rate for electron-positron collisions to make particle-antiparticle pairs depends on whether the particles are moving roughly in the same direction as the electrons or in the same direction as the positrons.

As with the high-energy electron-positron scattering discussed above, interference between effects of the electromagnetic and Z fields, and the Z boson’s mass, causes these asymmetries to change with energy. They are particularly simple, though, both when E = m_Z c² and when E >> m_Z c².

A number of these asymmetries are measurable. Measurements of the left-right asymmetry was made at the Stanford Linear Accelerator Center (SLAC) at their Linear Collider (SLC), while I was a graduate student there. Meanwhile, measurements of the forward-backward asymmetries were made at LEP and LEP 2. All of these measurements agreed well with the Standard Model’s predictions.

A Host of Processes at the Large Hadron Collider (click for details)

Fig. 10 shows predictions (gray bands) for total rates of over seventy processes in the proton-proton collisions at the Large Hadron Collider. Also shown are measurements (colored squares) made at the CMS experiment . (A similar plot is available from the ATLAS experiment.) Many of these predictions, which are complicated as they must account for the proton’s internal structure, depend on sin θ_w .

While minor discrepancies between data and theory appear, they are of the sort that one would expect in a large number of experimental measurements. Despite the rates varying by more than a billion from most common to least common, there is not a single major discrepancy between prediction and data.

Many more measurements than just these seventy are performed at the Large Hadron Collider, not least because there are many more details in a process than just its total rate.

A Fortress

What I’ve shown you today is just a first step, and one can do better. When we look closely, especially at certain asymmetries described in the bonus section, we see that sin θ_w = ¹/₂ (i.e. θ_w = 30 degrees) isn’t a good enough approximation. (In particular, if sin θ_w were exactly ¹/₂, then the left-right asymmetry in Z production would be zero, and the forward-backward asymmetry for muon and tau production would also be zero. That rough prediction isn’t true; the asymmetries are small, only about 15%, but they are clearly not zero.)

So to really be convinced of the Standard Model’s validity, we need to be more precise about what sin θ_w is. That’s what we’ll do next time.

Nevertheless, you can already see that the Standard Model, with its Higgs field and its special triangle, works exceedingly well in predicting how particles behave in a wide range of circumstances. Over the past few decades, as it has passed one experimental test after another, it has become a fortress, extremely difficult to shake and virtually impossible to imagine tearing down. We know it can’t be the full story because there are so many questions it doesn’t answer or address. Someday it will fail, or at least require additions. But within its sphere of influence, it rules more powerfully than any theoretical idea known to our species.

Dream. I’m sitting around a table with a bunch of writer friends. I say

“You know what my biggest challenge as a writer is?”

But then someone else jumps in to say something slightly related and the conversation moves in a different direction and I’m feeling somewhat put out until someone says, Wait, Jordan was going to tell us about his biggest challenge as a writer.”

And I said, “The biggest challenge I have as a writer is conveying hope. Because there is hope, we know this, but at the same time there’s no argument for hope, and writing most naturally takes the form of an argument.”

A helpful ad from Science Made Stupid.

This past week, a paper was published from ETH Zurich in which an aluminum nitride mechanical resonator was actually used as a qubit, where the ground and first excited states of this quantum (an)harmonic oscillator represented \(|0 \rangle\) and \(|1 \rangle\).  They demonstrate actual quantum gate operations on this mechanical system (which is coupled to a more traditional transmon qubit - the setup is explained in this earlier paper).  

One key trick to being able to make a qubit out of a mechanical oscillator is to have sufficiently large anharmonicity.  An ideal, perfectly harmonic quantum oscillator has an energy spectrum given by \((n + 1/2)\hbar \omega\), where \(n\) is the number of quanta of excitations in the resonator.  In that situation, the energy difference between adjacent levels is always \(\hbar \omega\).  The problem with this from the qubit perspective is, you want to have a quantum two-level system, and how can you controllably drive transitions just between a particular pair of levels when all of the adjacent level transitions cost the same energy?  The authors of this recent paper have achieved a strong anharmonicity, basically making the "spring" of the mechanical resonator softer in one displacement direction than the other.  The result is that the energy difference between levels \(|0\rangle\) and \(|1\rangle\) is very different than the energy difference between levels \(|1\rangle\) and \(|2\rangle\), etc.  (In typical superconducting qubits, the resonance is not mechanical but an electrical \(LC\)-type, and a Josephson junction acts like a non-linear inductor, giving the desired anharmonic properties.)  This kind of mechanical anharmonicity means that you can effectively have interactions between vibrational excitations ("phonon-phonon"), analogous to what the circuit QED folks can do.  Neat stuff.

I was just on campus at Caltech, where I was surprised to learn that 30% of the undergraduate student body is composed of varsity athletes. 3% are water polo players alone! Brown, where my son goes, is 13% student athletes. At UW-Madison, just 3%.

read more

Just a brief note, in a very busy period, to alert those in the Providence, RI area that I’ll be giving a colloquium talk at the Brown University Physics Department on Monday November 18th at 4pm. Such talks are open to the public, but are geared toward people who’ve had at least one full year of physics somewhere in their education. The title is “Exploring The Foundations of our Quantum Cosmos”. Here’s a summary of what I intend to talk about:

The discovery of the Higgs boson in 2012 marked a major milestone in our understanding of the universe, and a watershed for particle physics as a discipline. What’s known about particles and fields now forms a nearly complete short story, an astonishing, counterintuitive tale of relativity and quantum physics. But it sits within a larger narrative that is riddled with unanswered questions, suggesting numerous avenues of future research into the nature of spacetime and its many fields. I’ll discuss both the science and the challenges of accurately conveying its lessons to other scientists, to students, and to the wider public.

Everything we do, we do for someone.

The simplest things we do for ourselves. We grab that chocolate bar on the table and eat it, and it makes us happier.

Unless the chocolate bar is homemade, we probably paid money for it. We do other things, working for a living, to get the money to get those chocolate bars for ourselves.

(We also get chocolate bars for our loved ones, or for people we care about. Whether this is not in a sense also getting a chocolate bar for yourself is left as an exercise to the reader.)

What we do for the money, in turn, is driven by what would make someone else happier. Sometimes this is direct: you cut someone’s hair, they enjoy the breeze, they pay you, you enjoy the chocolate.

Other times, this gets mediated. You work in HR at a haircut chain. The shareholders want more money, to buy things like chocolate bars, so they vote for a board who wants to do what the shareholders want so as not to be in breach of contract and get fewer chocolate bars, so the board tells you to do things they believe will achieve that, and you do them because that’s how you get your chocolate bars. Every so often, the shareholders take a look at how many chocolate bars they can afford and adjust.

Compared to all this, academia is weirdly un-mediated.

It gets the closest to this model with students. Students want to learn certain things because they will allow them to provide other people with better services in future, which they can use to buy chocolate bars, and other things for the sheer pleasure, a neat experience almost comparable to a chocolate bar. People running universities want more money from students so they can spend it on things like giant statues of chocolate bars, so they instruct people working in the university to teach more of the things students want. (Typically in a very indirect way, for example funding a department in the US based on number of majors rather than number of students.)

But there’s a big chunk of academics whose performance is mostly judged not by their teaching, but by their research. They are paid salaries by departments based on the past quality of their research, or paid out of grants awarded based on the expected future quality of their research. (Or to combine them, paid salaries by departments based on the expected size of their grants.)

And in principle, that introduces many layers of mediation. The research universities and grant agencies are funded by governments, which pool money together in the expectation that someday by doing so they will bring about a world where more people can eat chocolate bars.

But the potential to bring about a world of increased chocolate bars isn’t like maximizing shareholder value. Nobody can check, one year later, how much closer you are to the science-fueled chocolate bar utopia.

And so in practice, in science, people fund you because they think what you’re doing is neat. Because it scratches the chocolate-bar-shaped hole in their brains. They might have some narrative about how your work could lead to the chocolate bar utopia the government is asking for, but it’s not like they’re calculating the expected distribution of chocolate bars if they fund your project versus another. You have to convince a human being, not that you are doing something instrumentally and measurably useful…but that you are doing something cool.

And that makes us very weird people! Halfway between haircuts and HR, selling a chocolate bar that promises to be something more.

Previously: Part 7. Next: Part 9.

By this point in the course, we’ve finished the delicate work of assembling all the customary set-theoretic apparatus from the axioms, and we’ve started proving major theorems. This week, we met well ordered sets and developed all the theory we’ll need. The main results were:

• every family of well ordered sets has a least member — informally, “the well ordered sets are well ordered”;

• the Hartogs theorem: for every set $X$, there’s some well ordered set that doesn’t admit an injection into $X$;

• a very close relative of Zorn’s lemma that, nevertheless, doesn’t require the axiom of choice: for every ordered set $X$ and function $\varphi$ assigning an upper bound to each chain in $X$, there’s some chain $C$ such that $\varphi(C) \in C$.

I also included an optional chatty section on the use of transfinite recursion to strip the isolated points from any subset of $\mathbb{R}$. Am I right in understanding that this is what got Cantor started on set theory in the first place?

I was sure my next post would be about the election—the sword of Damocles hanging over the United States and civilization as a whole. Instead, I have sad news, but also news that brings memories of warmth, humor, and complexity-theoretic insight.

Steven Rudich—professor at Carnegie Mellon, central figure of theoretical computer science since the 1990s, and a kindred spirit and friend—has died at the too-early age of 63. While I interacted with him much more seldom than I wish I had, it would be no exaggeration to call him one of the biggest influences on my life and career.

I first became aware of Steve at age 17, when I read the Natural Proofs paper that he coauthored with Razborov. I was sitting in the basement computer room at Telluride House at Cornell, and still recall the feeling of awe that came over me with every page. This one paper changed my scientific worldview. It expanded my conception of what the P versus NP problem was about and what theoretical computer science could even do—showing how it could turn in on itself, explain its own difficulties in proving problems hard in terms of the truth of those same problems’ hardness, and thereby transmute defeat into victory. I may have been bowled over by the paper’s rhetoric as much as by its results: it was like, you’re allowed to write that way?

I was nearly as impressed by Steve’s PhD thesis, which was full of proofs that gave off the appearance of being handwavy, “just phoning it in,” but were in reality completely rigorous. The result that excited me the most said that, if a certain strange combinatorial conjecture was true, then there was essentially no hope of proving that P≠NP∩coNP relative to a random oracle with probability 1. I played around with the combinatorial conjecture but couldn’t make headway on it; a year or two later, I was excited when I met Clifford Smyth and he told me that he, Kahn, and Saks had just proved it. Rudich’s conjecture directly inspired me to work on what later became the Aaronson-Ambainis Conjecture, which is still unproved, but which if true, similarly implies that there’s no hope of proving P≠BQP relative to a random oracle with probability 1.

When I applied to CS PhD programs in 1999, I wrote about how I wanted to sing the ideas of theoretical computer science from the rooftops—just like Steven Rudich had done, with the celebrated Andrew’s Leap summer program that he’d started at Carnegie Mellon. (How many other models were there? Indeed, how many other models are there today?) I was then honored beyond words when Steve called me on the phone, before anyone else had, and made an hourlong pitch for me to become his student. “You’re what I call a ‘prefab’,” he said. “You already have the mindset that I try to instill in students by the end of their PhDs.” I didn’t have much self-confidence then, which is why I can still quote Steve’s words a quarter-century later. In the ensuing years, when (as often) I doubted myself, I’d think back to that phone call with Steve, and my burning desire to be what he apparently thought I was.

Alas, when I arrived in Pittsburgh for CMU’s visit weekend, I saw Steve holding court in front of a small crowd of students, dispensing wisdom and doing magic tricks. I was miffed that he never noticed or acknowledged me: had he already changed his mind about me, lost interest? It was only later that I learned that Steve was going blind at the time, and literally hadn’t seen me.

In any case, while I came within a hair of accepting CMU’s offer, in the end I chose Berkeley. I wasn’t yet 100% sure that I wanted to do quantum computing (as opposed to AI or classical complexity theory), but the lure of the Bay Area, of the storied CS theory group where Steve himself had studied, and of Steve’s academic sibling Umesh Vazirani proved too great.

Full of regrets about the road not taken, I was glad that, in the summer between undergrad and PhD, I got to attend the PCMI summer school on computational complexity at the Institute for Advanced Study in Princeton, where Steve gave a spectacular series of lectures. By that point, Steve was almost fully blind. He put transparencies up, sometimes upside-down until the audience corrected him, and then lectured about them entirely from memory. He said that doing CS theory sightless was a new, more conceptual experience for him.

Even in his new condition, Steve’s showmanship hadn’t left him; he held the audience spellbound as few academics do. And in a special lecture on “how to give talks,” he spilled his secrets.

“What the speaker imagines the audience is thinking,” read one slide. And then, inside the thought bubbles: “MORE! HARDER! FASTER! … Ahhhhh yes, QED! Truth is beauty.”

“What the audience is actually thinking,” read the next slide, below which: “When is this over? I need to pee. Can I get a date with the person next to me?” (And this was before smartphones.) And yet, Steve explained, rather than resenting the many demands on the audience’s attention, a good speaker would break through, meet people where they were, just as he was doing right then.

I listened, took mental notes, resolved to practice this stuff. I reflected that, even if my shtick only ever became 10% as funny or fluid as Steve’s, I’d still come out way ahead.

It’s possible that the last time I saw Steve was in 2007, when I visited Carnegie Mellon to give a talk about algebrization, a new barrier to solving P vs. NP (and other central problems of complexity theory) that Avi Wigderson and I had recently discovered. When I started writing the algebrization paper, I very consciously modeled it after the Natural Proofs paper; the one wouldn’t have been thinkable without the other. So you can imagine how much it meant to me when Steve liked algebrization—when, even though he couldn’t see my slides, he got enough from the spoken part of the talk to burst with “conceptual” questions and comments.

Steve not only peeled back the mystery of P vs NP insofar as anyone has. He did it with exuberance and showmanship and humor and joy and kindness. I won’t forget him.

I’ve written here only about the tiniest sliver of Steve’s life: namely, the sliver where it intersected mine. I wish that sliver were a hundred times bigger, so that there’d be a hundred times more to write. But CS theory, and CS more broadly, are communities. When I posted about Steve’s passing on Facebook, I got inundated by comments from friends of mine who (as it turned out) had taken Steve’s courses, or TA’d for him, or attended Andrew’s Leap, or otherwise knew him, and on whom he’d left a permanent impression—and I hadn’t even known any of this.

So I’ll end this post with a request: please share your Rudich stories in the comments! I’d especially love specific recollections of his jokes, advice, insights, or witticisms. We now live in a world where, even in the teeth of the likelihood that P≠NP, powerful algorithms running in massive datacenters nevertheless try to replicate the magic of human intelligence, by compressing and predicting all the text on the public Internet. I don’t know where this is going, but I can’t imagine that it would hurt for the emerging global hive-mind to know more about Steven Rudich.

Kaisa Matomäki, Maksym Radziwill, Fernando Xuancheng Shao, Joni Teräväinen, and myself have (finally) uploaded to the arXiv our paper “Higher uniformity of arithmetic functions in short intervals II. Almost all intervals“. This is a sequel to our previous paper from 2022. In that paper, discorrelation estimates such as

However, the situation is better when one is willing to consider estimates that are valid for almost all intervals, rather than all intervals, so that one now studies local higher order uniformity estimates of the form

While one can lower somewhat, there are still barriers. For instance, in the model case , that is to say prime number theorems in almost all short intervals, until very recently the best value of was , recently lowered to by Guth and Maynard (and can be lowered all the way to zero on the Density Hypothesis). Nevertheless, we are able to get some improvements at higher orders:

• For the von Mangoldt function, we can get as low as , with an arbitrary logarithmic saving in the error terms; for divisor functions, one can even get power savings in this regime.
• For the Möbius function, we can get , recovering our previous result with Tamar Ziegler, but now with type savings in the exceptional set (though not in the pointwise bound outside of the set).
• We can now also get comparable results for the divisor function.

As sample applications, we can obtain Hardy-Littlewood conjecture asymptotics for arithmetic progressions of almost all given steps , and divisor correlation estimates on arithmetic progressions for almost all .

Our proofs are rather long, but broadly follow the “contagion” strategy of Walsh, generalized from the Fourier setting to the higher order setting. Firstly, by standard Heath–Brown type decompositions, and previous results, it suffices to control “Type II” discorrelations such as

The main issue is the dependency of the polynomial on . By using a “nilsequence large sieve” introduced in our previous paper, and removing degenerate cases, we can show a functional relationship amongst the that is very roughly of the form

The day after the election, I found myself struggling with how to approach the complex analysis class I was teaching. Could I ignore the (almost literal) elephant in the room? Would my students be in the right mental state to learn math? Would I be in the right mental state to teach it?

I opened with the statement that usually in math we have the luxury of working in abstractions far removed from the real world. We are familiar with addressing mathematical problems with the (inessential) connections to the real world stripped away, leaving only the essential features to focus one’s attention. An election, for instance, might be treated as the outcome of people, each of which has a probability of voting for one candidate, and for another… and one can then try to analyze the problem from a dispassionate mathematical perspective. This type of mindset can be illuminating in many contexts. Real world events have real consequences, however, and in light of an event as consequential as the last election, a math lecture on contour integration or the central limit theorem may seem meaningless.

But there is one precious thing mathematics has, that almost no other field currently enjoys: a consensus on what the ground truth is, and how to reach it. Because of this, even the strongest differences of opinion in mathematics can eventually be resolved, and mistakes realized and corrected. This consensus is so strong, we simply take it for granted: a solution is correct or incorrect, a theorem is proved or not proved, and when a problem is solved, we simply move on to the next one. This is, sadly, not a state of affairs elsewhere. But if my students can learn from this and carry these skills— such as distinguishing an overly simple but mathematically flawed “solution” from a more complex, but accurate actual solution—to other spheres that have more contact with the real world, then my math lectures have consequence. Even—or perhaps, especially—in times like these.

Nima Arkani-Hamed thinks space-time is doomed.

That doesn’t mean he thinks it’s about to be destroyed by a supervillain. Rather, Nima, like many physicists, thinks that space and time are just approximations to a deeper reality. In order to make sense of gravity in a quantum world, seemingly fundamental ideas, like that particles move through particular places at particular times, will probably need to become more flexible.

But while most people who think space-time is doomed research quantum gravity, Nima’s path is different. Nima has been studying scattering amplitudes, formulas used by particle physicists to predict how likely particles are to collide in particular ways. He has been trying to find ways to calculate these scattering amplitudes without referring directly to particles traveling through space and time. In the long run, the hope is that knowing how to do these calculations will help suggest new theories beyond particle physics, theories that can’t be described with space and time at all.

Ten years ago, Nima figured out how to do this in a particular theory, one that doesn’t describe the real world. For that theory he was able to find a new picture of how to calculate scattering amplitudes based on a combinatorical, geometric space with no reference to particles traveling through space-time. He gave this space the catchy name “the amplituhedron“. In the years since, he found a few other “hedra” describing different theories.

Now, he’s got a new approach. The new approach doesn’t have the same kind of catchy name: people sometimes call it surfaceology, or curve integral formalism. Like the amplituhedron, it involves concepts from combinatorics and geometry. It isn’t quite as “pure” as the amplituhedron: it uses a bit more from ordinary particle physics, and while it avoids specific paths in space-time it does care about the shape of those paths. Still, it has one big advantage: unlike the amplituhedron, Nima’s new approach looks like it can work for at least a few of the theories that actually describe the real world.

The amplituhedron was mysterious. Instead of space and time, it described the world in terms of a geometric space whose meaning was unclear. Nima’s new approach also describes the world in terms of a geometric space, but this space’s meaning is a lot more clear.

The space is called “kinematic space”. That probably still sounds mysterious. “Kinematic” in physics refers to motion. In the beginning of a physics class when you study velocity and acceleration before you’ve introduced a single force, you’re studying kinematics. In particle physics, kinematic refers to the motion of the particles you detect. If you see an electron going up and to the right at a tenth the speed of light, those are its kinematics.

Kinematic space, then, is the space of observations. By saying that his approach is based on ideas in kinematic space, what Nima is saying is that it describes colliding particles not based on what they might be doing before they’re detected, but on mathematics that asks questions only about facts about the particles that can be observed.

(For the experts: this isn’t quite true, because he still needs a concept of loop momenta. He’s getting the actual integrands from his approach, rather than the dual definition he got from the amplituhedron. But he does still have to integrate one way or another.)

Quantum mechanics famously has many interpretations. In my experience, Nima’s favorite interpretation is the one known as “shut up and calculate”. Instead of arguing about the nature of an indeterminately philosophical “real world”, Nima thinks quantum physics is a tool to calculate things people can observe in experiments, and that’s the part we should care about.

From a practical perspective, I agree with him. And I think if you have this perspective, then ultimately, kinematic space is where your theories have to live. Kinematic space is nothing more or less than the space of observations, the space defined by where things land in your detectors, or if you’re a human and not a collider, in your eyes. If you want to strip away all the speculation about the nature of reality, this is all that is left over. Any theory, of any reality, will have to be described in this way. So if you think reality might need a totally new weird theory, it makes sense to approach things like Nima does, and start with the one thing that will always remain: observations.

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Election Day Update: For anyone who’s still undecided (?!?), I can’t beat this from Sam Harris.

When I think of Harris winning the presidency this week, it’s like watching a film of a car crash run in reverse: the windshield unshatters; stray objects and bits of metal converge; and defenseless human bodies are hurled into states of perfect repose. Normalcy descends out of chaos.

Important Announcement: I don’t in any way endorse voting for Jill Stein, or any other third-party candidate. But if you are a Green Party supporter who lives in a swing state, then please at least vote for Harris, and use SwapYourVote.org to arrange for two (!) people in safe states to vote for Jill Stein on your behalf. Thanks so much to friend-of-the-blog Linchuan Zhang for pointing me to this resource.

Added on Election Day: And, if you swing that way, click here to arrange to have your vote for Kamala in a swing state traded for two votes for libertarian candidate Chase Oliver in safe states. In any case, if you’re in a swing state and you haven’t yet voted (for Kamala Harris and for the norms of civilization), do!

For weeks I’d been wondering what I could say right before the election, at this momentous branch-point in the wavefunction, that could possibly do any good. Then, the other day, a Jewish voter in Pennsylvania and Shtetl-Optimized fan emailed me to ask my advice. He said that he’d read my Never-Trump From Here to Eternity FAQ and saw the problems with Trump’s autocratic tendencies, but that his Israeli friends and family wanted him to vote Trump anyway, believing him better on the narrow question of “Israel’s continued existence.” I started responding, and then realized that my response was the election-eve post I’d been looking for. So without further ado…

Thanks for writing.  Of course this is ultimately between you and your conscience (and your empirical beliefs), but I can tell you what my Israeli-American wife and I did.  We voted for Kamala, without the slightest doubt or hesitation.  We’d do it again a thousand quadrillion times.  We would’ve done the same in the swing state of Pennsylvania, where I grew up (actually in Bucks, one of the crucial swing counties).

And later this week, along with tens of millions of others, I’ll refresh the news with heart palpitations, looking for movement toward blue in Pennsylvania and Wisconsin.  I’ll be joyous and relieved if Kamala wins.  I’ll be ashen-faced if she doesn’t.  (Or if there’s a power struggle that makes the 2021 insurrection look like a dress rehearsal.)  And I’ll bet anyone, at 100:1 odds, that at the end of my life I’ll continue to believe that voting Kamala was the right decision.

I, too, have pro-Israel friends who urged me to switch to Trump, on the ground that if Kamala wins, then (they say) the Jews of Israel are all but doomed to a second Holocaust.  For, they claim, the American Hamasniks will then successfully prevail on Kamala to prevent Israel from attacking Iran’s nuclear sites, or will leave Israel to fend for itself if it does.  And therefore, Iran will finish and test nuclear weapons in the next couple years, and then it will rebuild the battered Hamas and Hezbollah under its nuclear umbrella, and then it will fulfill its stated goal since 1979, of annihilating the State of Israel, by slaughtering all the Jews who aren’t able to flee.  And, just to twist the knife, the UN diplomats and NGO officials and journalists and college students and Wikipedia editors who claimed such a slaughter was a paranoid fantasy, they’ll all cheer it when it happens, calling it “justice” and “resistance” and “intifada.”

And that, my friends say, will finally show me the liberal moral evolution of humanity since 1945, in which I’ve placed so much stock.  “See, even while they did virtually nothing to stop the first Holocaust, the American and British cultural elites didn’t literally cheer the Holocaust as it happened.  This time around, they’ll cheer.”

My friends’ argument is that, if I’m serious about “Never Again” as a moral lodestar of my life, then the one issue of Israel and Iran needs to override everything else I’ve always believed, all my moral and intellectual repugnance at Trump and everything he represents, all my knowledge of his lies, his evil, his venality, all the former generals and Republican officials who say that he’s unfit to serve and an imminent danger to the Republic.  I need to vote for this madman, this pathological liar, this bullying autocrat, because at least he’ll stand between the Jewish people and the darkness that would devour them, as it devoured them in my grandparents’ time.

My friends add that it doesn’t matter that Kamala’s husband is Jewish, that she’s mouthed all the words a thousand times about Israel’s right to defend itself, that Biden and Harris have indeed continued to ship weapons to Israel with barely a wag of their fingers (even as they’ve endured vituperation over it from their left, even as Kamala might lose the whole election over it).  Nor does it matter that a commanding majority of American Jews will vote for Kamala, or that … not most Israelis, but most of the Israelis in academia and tech who I know, would vote for Kamala if they could.  They could all be mistaken about their own interests.  But you and I, say my right-wing friends, realize that what actually matters is Iran, and what the next president will do about Iran.  Trump would unshackle Israel to do whatever it takes to prevent nuclear-armed Ayatollahs.  Kamala wouldn’t.

Anyway, I’ve considered this line of thinking.  I reject it with extreme prejudice.

To start with the obvious, I’m not a one-issue voter.  Presumably you aren’t either.  Being Jewish is a fundamental part of my humanity—if I didn’t know that before I’d witnessed the world’s reaction to October 7, then I certainly know now.  But only in the fantasies of antisemites would I vote entirely on the basis of “is this good for the Jews?”  The parts of me that care about the peaceful transfer of power, about truth, about standing up to Putin, about the basic sanity of the Commander-in-Chief in an emergency, about climate change and green energy and manufacturing, about not destroying the US economy through idiotic tariffs, about talented foreign scientists getting green cards, about the right to abortion, about RFK and his brainworm not being placed in charge of American healthcare, even about AI safety … all those parts of me are obviously for Kamala.

More interestingly, though, the Jewish part of me is also for Kamala—if possible, even more adamantly than other parts.  It’s for Kamala because…

Well, after these nine surreal years, how does one even spell out the Enlightenment case against Trump?  How does one say what hasn’t already been said a trillion times?  Now that the frog is thoroughly boiled, how does one remind people of the norms that used to prevail in America—even after Newt Gingrich and Sarah Palin and the rest had degraded them—and how those norms were what stood between us and savagery … and how laughably unthinkable is the whole concept of Trump as president, the instant you judge him according to those norms?

Kamala, whatever her faults, is basically a normal politician.  She lies, but only as normal politicians lie.  She dodges questions, changes her stances, says different things to different audiences, but only as normal politicians do.  Trump is something else entirely.  He’s one of the great flimflam artists of human history.  He believes (though “belief” isn’t quite the right word) that truth is not something external to himself, but something he creates by speaking it.  He is the ultimate postmodernist.  He’s effectively created a new religion, one of grievance and lies and vengeance against outsiders, and converted a quarter of Americans to his religion, while another quarter might vote it into power because of what they think is in it for them.

And this cult of lies … this is what you ask if Jewish people should enter into a strategic alliance with?  Do you imagine this cult is a trustworthy partner, one likely to keep its promises?

For centuries, Jews have done consistently well under cosmopolitan liberal democracies, and consistently poorly—when they remained alive at all—under nativist tyrants.  Do you expect whatever autocratic regime follows Trump, a regime of JD Vance and Tucker Carlson and the like, to be the first exception to this pattern in history?

For I take it as obvious that a second Trump term, and whatever follows it, will make the first Trump term look like a mere practice run, a Beer Hall Putsch.  Trump I was restrained by John Kelly, by thousands of civil service bureaucrats and judges, by the generals, and in the last instance, by Mike Pence.  But Trump II will be out for the blood of his enemies—he says so himself at his rallies—and will have nothing to restrain him, not even any threat of criminal prosecution.  Do you imagine this goes well for the Jews, or for pretty much anyone?

It doesn’t matter if Trump has no personal animus against Jews—excepting, of course, the majority who vote against him.  Did the idealistic Marxist intellectuals of Russia in 1917 want Stalin?  Did the idealistic Iranian students of Iran in 1979 want Khomeini?  It doesn’t matter: what matters is what they enabled.  Turn over the rock of civilization, and everything that was wriggling underneath is suddenly loosed on the world.

How much time have you spent looking at pro-Israel people on Twitter (Hen Mazzig, Haviv Rettig Gur, etc.), and then—crucially—reading their replies?  I spend at least an hour or two per day on that, angry and depressed though it makes me, perhaps because of an instinct to stare into the heart of darkness, not to look away from a genocidal evil arrayed against my family.  

Many replies are the usual: “Shut the fuck up, Zio, and stop murdering babies.”  “Two-state solution?  I have a different solution: that all you land-thieves pack your bags and go back to Poland.” But then, every time, you reach tweets like “you Jews have been hated and expelled from all the world’s countries for thousands of years, yet you never consider that the common factor is you.”  “Your Talmud commands you to kill goyim children, so that’s why you’re doing it.”  “Even while you maintain apartheid in Palestine, you cynically import millions of third-world savages to White countries, in order to destroy them.”  None of this is the way leftists talk, not even the most crazed leftists.  We’ve now gone all the way around the horseshoe.  Or, we might say, we’re no longer selecting on the left or right of politics at all, but simply on the bottom.

And then you see that these bottom-feeders often have millions of followers each.  They command armies.  The bottom-feeders—left, right, Islamic fundamentalist, and unclassifiably paranoid—are emboldened as never before.  They’re united by a common enemy, which turns out to be the same enemy they’ve always had.

Which brings us to Elon Musk.  I personally believe that Musk, like Trump, has nothing against the Jews, and is if anything a philosemite.  But it’s no longer a question of feelings.  Through his changes to Twitter, Musk has helped his new ally Trump flip over the boulder, and now all the demons that were wriggling beneath are loosed on civilization.

Should we, as Jews, tolerate the demons in exchange for Trump’s tough-guy act on Iran?  Just like the evangelicals previously turned a blind eye to Trump’s philandering, his sexual assaults, his gleeful cruelty, his spitting on everything Christianity was ever supposed to stand for, simply because he promised them the Supreme Court justices to overturn Roe v. Wade?  Faced with a man who’s never had a human relationship in his life that wasn’t entirely transactional, should we be transactional ourselves?

I’m not convinced that even if we did, we’d be getting a good bargain.  Iran is no longer alone, but part of an axis that includes China, Russia, and North Korea.  These countries prop up each other’s economies and militaries; they survive only because of each other.  As others have pointed out, the new Axis is actually more tightly integrated than the Axis powers ever were in WWII.  The new Axis has already invaded Ukraine and perhaps soon Taiwan and South Korea.  It credibly threatens to end the Pax Americana.  And to face Hamas or Hezbollah is to face Iran is to face the entire new Axis.

Now Kamala is not Winston Churchill.  But at least she doesn’t consider the tyrants of Russia, China, and North Korea to be her personal friends, trustworthy because they flatter her.  At least she, unlike Trump, realizes that the current governments of China, Russia, North Korea, and Iran do indeed form a new axis of evil, and she has the glimmers of consciousness that the founders of the United States stood for something different from what those tyrannies stand for, and that this other thing that our founders stood for was good.  If war does come, at least she’ll listen to the advice of generals, rather than clowns and lackeys.  And if Israel or America do end up in wars of survival, from the bottom of my heart she’s the one I’d rather have in charge.  For if she’s in charge, then through her, the government of the United States is still in charge.  Our ripped and tattered flag yet waves.  If Trump is in charge, who or what is at the wheel besides his own unhinged will, or that of whichever sordid fellow-gangster currently has his ear?

So, yes, as a human being and also as a Jew, this is why I voted early for Kamala, and why I hope you’ll vote for her too. If you disagree with her policies, start fighting those policies once she’s inaugurated on January 20, 2025. At least there will still be a republic, with damaged but functioning error-correcting machinery, in which you can fight.

All the best,
Scott

More Resources: Be sure to check out Scott Alexander’s election-eve post, which (just like in 2016) endorses any listed candidate other than Trump, but specifically makes the case to voters put off (as Scott is) by Democrats’ wokeness. Also check out Garry Kasparov’s epic tweet-thread on why he supports Kamala, and his essay The United States Cannot Descend Into Authoritarianism.

Time for blogging has continued to be scarce, but here are a few papers to distract (and for readers who are US citizens:  vote if you have not already done so!).

• Reaching back, this preprint by Aharonov, Collins, Popescu talks about a thought experiment in which angular momentum can seemingly be transferred from one region to another even though the probability of detecting spin-carrying particles between the two regions can be made arbitrarily low.  I've always found these kinds of discussions to be fun, even when the upshot for me is usually, "I must not really understand the subtleties of weak measurements in quantum mechanics."  This is a specific development based on the quantum Cheshire cat idea.  I know enough to understand that when one is talking about post-selection in quantum experiments, some questions are just not well-posed.  If we send a wavepacked of photons at a barrier, and we detect with a click a photon that (if it was in the middle of the incident wavepacket) seems to have therefore traversed the barrier faster than c, that doesn't mean much, since the italicized parenthetical clause above is uncheckable in principle.  
• Much more recently, this paper out last week in Nature reports the observation of superconductivity below 200 mK in a twisted bilayer of WSe2.  I believe that this is the first observation of superconductivity in a twisted bilayer of an otherwise nonsuperconducting 2D semiconductor other than graphene.  As in the graphene case, the superconductivity shows up at a particular filling of the moiré lattice, and interestingly seems to happen around zero applied vertical electric field (displacement field) in the device.  I don't have much to say here beyond that it's good to see interesting results in a broader class of materials - that suggests that there is a more general principle at work than "graphene is special".
• This preprint from last week from Klein et al. is pretty impressive.  It's been known for over 25 years (see here) that it is possible to use a single-electron transistor (SET) as a scannable charge sensor and potentiometer.  Historically, making these devices and operating them has been a real art.  They are fragile, static-sensitive, and fabricating them from evaporated metal on the tips of drawn optical fibers is touchy.  There have been advances in recent years from multiple quarters, and this paper demonstrates a particularly interesting idea: Use a single charge trap in a layer of WSe2 as the SET, and effectively put the sample of interest on the scannable tip.  This is an outgrowth of the quantum twisting microscope.

If you’re curious to know what my book is about and why it’s called “Waves in an Impossible Sea”, then watching this video is currently the quickest and most direct way to find out from me personally. It’s a public talk that I gave to a general audience at Harvard, part of the Harvard Bookstore science book series.

My intent in writing the book was to illuminate central aspects of the cosmos — and of how we humans fit into it — that are often glossed over by scientists and science writers, at least in the books and videos I’ve come across. So if you watch the lecture, I think there’s a good chance that you’ll learn something about the world that you didn’t know, perhaps about the empty space that forms the fabric of the universe, or perhaps about what “quantum” in “quantum physics” really means and why it matters so much to you and me.

The video contains 35 minutes of me presenting, plus some Q&A at the end. Feel free to ask questions of your own in the comments below, or on my book-questions page; I’ll do my best to answer them.

I have been collaborating for many years with a long-time personal friend and polymath, Tanya Klowden, on a popular book on astronomy tentatively entitled “Climbing the cosmic distance ladder“, which was initially based on a public lecture I have given on this topic for many years, but which we have found to be a far richer story than what I was aware of when I first gave those lectures. We had previously released a sample chapter on this blog back in 2020. The project had stalled for a few years for a number of reasons (including some pandemic-related issues); however, we are now trying to get it back on track. One of the steps we are taking to do so is to launch an Instagram account for the book, in which we aim to make some regular posts (tentatively on a weekly basis) on astronomical tidbits that we have encountered in the course of researching our book project. Tanya has started this series by sharing her experiences on comet-hunting in Hampstead Heath, following in the footsteps of the legendary comet-hunting 19th century astronomer Caroline Herschel (who, together with her brother William Herschel, will make a significant appearance in the “seventh rung” chapter of our book). I plan to follow up with an instagram post of my own in the next few days.

In honor of Halloween this week, let me say a bit about the spookiest term in physics: ghosts.

In particle physics, we talk about the universe in terms of quantum fields. There is an electron field for electrons, a gluon field for gluons, a Higgs field for Higgs bosons. The simplest fields, for the simplest particles, can be described in terms of just a single number at each point in space and time, a value describing how strong the field is. More complicated fields require more numbers.

Most of the fundamental forces have what we call vector fields. They’re called this because they are often described with vectors, lists of numbers that identify a direction in space and time. But these vectors actually contain too many numbers.

These extra numbers have to be tidied up in some way in order to describe vector fields in the real world, like the electromagnetic field or the gluon field of the strong nuclear force. There are a number of tricks, but the nicest is usually to add some extra particles called ghosts. Ghosts are designed to cancel out the extra numbers in a vector, leaving the right description for a vector field. They’re set up mathematically such that they can never be observed, they’re just a mathematical trick.

Mathematical tricks aren’t all that spooky (unless you’re scared of mathematics itself, anyway). But in physics, ghosts can take on a spookier role as well.

In order to do their job cancelling those numbers, ghosts need to function as a kind of opposite to a normal particle, a sort of undead particle. Normal particles have kinetic energy: as they go faster and faster, they have more and more energy. Said another way, it takes more and more energy to make them go faster. Ghosts have negative kinetic energy: the faster they go, the less energy they have.

If ghosts are just a mathematical trick, that’s fine, they’ll do their job and cancel out what they’re supposed to. But sometimes, physicists accidentally write down a theory where the ghosts aren’t just a trick cancelling something out, but real particles you could detect, without anything to hide them away.

In a theory where ghosts really exist, the universe stops making sense. The universe defaults to the lowest energy it can reach. If making a ghost particle go faster reduces its energy, then the universe will make ghost particles go faster and faster, and make more and more ghost particles, until everything is jam-packed with super-speedy ghosts unto infinity, never-ending because it’s always possible to reduce the energy by adding more ghosts.

The absence of ghosts, then, is a requirement for a sensible theory. People prove theorems showing that their new ideas don’t create ghosts. And if your theory does start seeing ghosts…well, that’s the spookiest omen of all: an omen that your theory is wrong.

On my recent trip to CERN, the lab that hosts the Large Hadron Collider, I had the opportunity to stop by the CERN control centre [CCC]. There the various particle accelerator operations are managed by accelerator experts, who make use of a host of consoles showing all sorts of data. I’d not been to the CCC in person — theoretical physicists congregate a few kilometers away on another part of CERN’s campus — although back in the LHC’s very early days, when things ran less smoothly, I used to watch some of the CCC’s monitoring screens to see how the accelerator was performing.

The atmosphere in the control room was relatively quiet, as the proton-proton collisions for the year 2024 had just come to an end the previous day. Unlike 2023, this has been a very good year. There’s a screen devoted to counting the number of collisions during the year, and things went so well in 2024 it had to be extended, for the first time, by a “1” printed on paper.

The indication “123/fb” means “123-collisions-per-femtobarn”, while one-collision-per-femtobarn corresponds to about 10¹⁴ = 100,000,000,000,000 proton-proton collisions. In other words, the year saw more than 12 million billion proton-proton collisions at each of the two large-scale experiments, ATLAS and CMS. That’s about double the best previous year, 2018.

Yes, that’s a line of bottles that you can see on the back wall in the first photo. Major events in the accelerator are often celebrated with champagne, and one of the bottles from each event is saved for posterity. Here’s one from a few weeks ago that marked the achievement of 100-collisions-per-femtobarn.

With another one and a half seasons to go in Run 3 of the LHC, running at 13.6 TeV of energy per collision (higher than the 13 TeV per collision in Run 2 from 2015 to 2018, and the 7 and 8 TeV per collision in Run 1 from 2010 to 2012), the LHC accelerator folks continue to push the envelope. Much more lies ahead in 2029 with Run 4, when the collision rate will increase by another big step.

Why not run a quantum-steampunk creative-writing course?

Quantum steampunk, as Quantum Frontiers regulars know, is the aesthetic and spirit of a growing scientific field. Steampunk is a subgenre of science fiction. In it, futuristic technologies invade Victorian-era settings: submarines, time machines, and clockwork octopodes populate La Belle Èpoque, a recently liberated Haiti, and Sherlock Holmes’s London. A similar invasion characterizes my research field, quantum thermodynamics: thermodynamics is the study of heat, work, temperature, and efficiency. The Industrial Revolution spurred the theory’s development during the 1800s. The theory’s original subject—nineteenth-century engines—were large, were massive, and contained enormous numbers of particles. Such engines obey the classical mechanics developed during the 1600s. Hence thermodynamics needs re-envisioning for quantum systems. To extend the theory’s laws and applications, quantum thermodynamicists use mathematical and experimental tools from quantum information science. Quantum information science is, in part, the understanding of quantum systems through how they store and process information. The toolkit is partially cutting-edge and partially futuristic, as full-scale quantum computers remain under construction. So applying quantum information to thermodynamics—quantum thermodynamics—strikes me as the real-world incarnation of steampunk.

But the thought of a quantum-steampunk creative-writing course had never occurred to me, and I hesitated over it. Quantum-steampunk blog posts, I could handle. A book, I could handle. Even a short-story contest, I’d handled. But a course? The idea yawned like the pitch-dark mouth of an unknown cavern in my imagination.

But the more I mulled over Edward Daschle’s suggestion, the more I warmed to it. Edward was completing a master’s degree in creative writing at the University of Maryland (UMD), specializing in science fiction. His mentor Emily Brandchaft Mitchell had sung his praises via email. In 2023, Emily had served as a judge for the Quantum-Steampunk Short-Story Contest. She works as a professor of English at UMD, writes fiction, and specializes in the study of genre. I reached out to her last spring about collaborating on a grant for quantum-inspired art, and she pointed to her protégé.

Who won me over. Edward and I are co-teaching “Writing Quantum Steampunk: Science-Fiction Workshop” during spring 2025.

The course will alternate between science and science fiction. Under Edward’s direction, we’ll read and discuss published fiction. We’ll also learn about what genres are and how they come to be. Students will try out writing styles by composing short stories themselves. Everyone will provide feedback about each other’s writing: what works, what’s confusing, and opportunities for improvement. 

The published fiction chosen will mirror the scientific subjects we’ll cover: quantum physics; quantum technologies; and thermodynamics, including quantum thermodynamics. I’ll lead this part of the course. The scientific studies will interleave with the story reading, writing, and workshopping. Students will learn about the science behind the science fiction while contributing to the growing subgenre of quantum steampunk.

We aim to attract students from across campus: physics, English, the Jiménez-Porter Writers’ House, computer science, mathematics, and engineering—plus any other departments whose students have curiosity and creativity to spare. The course already has four cross-listings—Arts and Humanities 270, Physics 299Q, Computer Science 298Q, and Mechanical Engineering 299Q—and will probably acquire a fifth (Chemistry 298Q). You can earn a Distributive Studies: Scholarship in Practice (DSSP) General Education requirement, and undergraduate and graduate students are welcome. QuICS—the Joint Center for Quantum Information and Computer Science, my home base—is paying Edward’s salary through a seed grant. Ross Angelella, the director of the Writers’ House, arranged logistics and doused us with enthusiasm. I’m proud of how organizations across the university are uniting to support the course.

The diversity we seek, though, poses a challenge. The course lacks prerequisites, so I’ll need to teach at a level comprehensible to the non-science students. I’d enjoy doing so, but I’m concerned about boring the science students. Ideally, the science students will help me teach, while the non-science students will challenge us with foundational questions that force us to rethink basic concepts. Also, I hope that non-science students will galvanize discussions about ethical and sociological implications of quantum technologies. But how can one ensure that conversation will flow?

This summer, Edward and I traded candidate stories for the syllabus. Based on his suggestions, I recommend touring science fiction under an expert’s guidance. I enjoyed, for a few hours each weekend, sinking into the worlds of Ted Chiang, Ursula K. LeGuinn, N. K. Jemison, Ken Liu, and others. My scientific background informed my reading more than I’d expected. Some authors, I could tell, had researched their subjects thoroughly. When they transitioned from science into fiction, I trusted and followed them. Other authors tossed jargon into their writing but evidenced a lack of deep understanding. One author nailed technical details about quantum computation, initially impressing me, but missed the big picture: his conflict hinged on a misunderstanding about entanglement. I see all these stories as affording opportunities for learning and teaching, in different ways.

Students can begin registering for “Writing Quantum Steampunk: Science-Fiction Workshop” on October 24. We can offer only 15 seats, due to Writers’ House standards, so secure yours as soon as you can. Part of me still wonders how the Hilbert space I came to be co-teaching a quantum-steampunk creative-writing course.¹ But I look forward to reading with you next spring!

¹A Hilbert space is a mathematical object that represents a quantum system. But you needn’t know that to succeed in the course.

Funding is available for a Doctor of Science Studentship with Dr. Matthew Leifer at the Institute for Quantum Studies, Chapman University, California, USA.  It is in Chapman’s unique interdisciplinary Math, Physics, and Philosophy (MPP) program, which emphasizes research that encompasses two or more of the three core disciplines.  This is a 3-year program that focuses on research, and students are expected to have a terminal Masters degree before they start.

This position is part of the Southern California Quantum Foundations Hub, funded by the John Templeton Foundation.  The research project must be in quantum foundations, particularly in one of the three theme areas of the grant:

1. The Nature of the Quantum State
2. Past and Future Boundary Conditions
3. Agency in Quantum Observers. 

The university also provides other scholarships for the MPP program.  Please apply before January 15, 2025, to receive full consideration for the available funding.

Please follow the “Graduate Application” link on the MPP website to apply.

For informal inquiries about the position and research projects, please get in touch with me.

Yes, he’s the guy from The Elegant Universe book and TV series. Our conversation is 1 hour 40 minutes; as usual I strongly recommend listening at 2x speed. The topics, chosen by Brian, include quantum computing (algorithms, hardware, error-correction … the works), my childhood, the interpretation of quantum mechanics, the current state of AI, the future of sentient life in the cosmos, and mathematical Platonism. I’m happy with how it turned out; in particular, my verbal infelicities seem to have been at a minimum this time. I recommend skipping the YouTube comments if you want to stay sane, but do share your questions and reactions in the comments here. Thanks to Brian and his team for doing this. Enjoy!

Update (Oct. 28): If that’s not enough Scott Aaronson video content for you, please enjoy another quantum computing podcast interview, this one with Ayush Prakash and shorter (clocking in at 45 minutes). Ayush pitched this podcast to me as an opportunity to explain quantum computing to Gen Z. Thus, I considered peppering my explanations of interference and entanglement with such phrases as ‘fo-shizzle’ and ‘da bomb,’ but I desisted after reflecting that whatever youth slang I knew was probably already outdated whenever I’d picked it up, back in the twentieth century.

In 2020, many of us logged experiences that we’d never anticipated. I wrote a nonfiction book and got married outside the Harvard Faculty Club (because nobody was around to shoo us away). Equally unexpectedly, I received an invitation to collaborate with a professional artist. One Bruce Rosenbaum emailed me out of the blue:

I watched your video on Quantum Steampunk: Quantum Information Meets Thermodynamics. [ . . . ] I’d like to explore collaborating with you on bringing together the fusion of Quantum physics and Thermodynamics into the real world with functional Steampunk art and design.

This Bruce Rosenbaum, I reasoned, had probably seen some colloquium of mine that a university had recorded and posted online. I’d presented a few departmental talks about how quantum thermodynamics is the real-world incarnation of steampunk.

I looked Bruce up online. Wired Magazine had called the Massachusetts native “the steampunk evangelist,” and The Wall Street Journal had called him “the steampunk guru.” He created sculptures for museums and hotels, in addition to running workshops that riffed on the acronym STEAM (science, technology, engineering, art, and mathematics). MTV’s Extreme Cribs had spotlighted his renovation of a Victorian-era church into a home and workshop.

All right, I replied, I’m game. But research fills my work week, so can you talk at an unusual time?

We Zoomed on a Saturday afternoon. Bruce Zooms from precisely the room that you’d hope to find a steampunk artist in: a workshop filled with brass bits and bobs spread across antique-looking furniture. Something intricate is usually spinning atop a table behind him. And no, none of it belongs to a virtual background. Far from an overwrought inventor, though, Bruce exudes a vibe as casual as the T-shirt he often wears—when not interviewing in costume. A Boston-area accent completed the feeling of chatting with a neighbor.

Bruce proposed building a quantum-steampunk sculpture. I’d never dreamed of the prospect, but it sounded like an adventure, so I agreed. We settled on a sculpture centered on a quantum engine. Classical engines inspired the development of thermodynamics around the time of the Industrial Revolution. One of the simplest engines—the heat engine—interacts with two environments, or reservoirs: one cold and one hot. Heat—the energy of random atomic motion—flows from the hot to the cold. The engine siphons off part of the heat, converting it into work—coordinated energy that can, say, turn a turbine. 

Can a quantum system convert random heat into useful work? Yes, quantum thermodynamicists have shown. Bell Labs scientists designed a quantum engine formed from one atom, during the 1950s and 1960s. Since then, physicists have co-opted superconducting qubits, trapped ions, and more into quantum engines. Entanglement can enhance quantum engines, which can both suffer and benefit from quantum coherences (wave-like properties, in the spirit of wave–particle duality). Experimentalists have realized quantum engines in labs. So Bruce and I placed (an artistic depiction of) a quantum engine at our sculpture’s center. The engine consists of a trapped ion—a specialty of Maryland, where I accepted a permanent position that spring.

Bruce engaged an illustrator, Jim Su, to draw the sculpture. We iterated through draft after draft, altering shapes and fixing scientific content. Versions from the cutting-room floor now adorn the Maryland Quantum-Thermodynamics Hub’s website.

Designing the sculpture was a lark. Finding funding to build it has required more grit. During the process, our team grew to include scientific-computing expert Alfredo Nava-Tudelo, physicist Bill Phillips, senior faculty specialist Daniel Serrano, and Quantum Frontiers gatekeeper Spiros Michalakis. We secured a grant from the University of Maryland’s Arts for All program this spring. The program is promoting quantum-inspired art this year, in honor of the UN’s designation of 2025 as the International Year of Quantum Science and Technology. 

Through the end of 2024, we’re building a tabletop version of the sculpture. We were expecting a 3D-printout version to consume our modest grant. But quantum steampunk captured the imagination of Empire Group, the design-engineering company hired by Bruce to create and deploy technical drawings. Empire now plans to include metal and moving parts in the sculpture. 

Empire will create CAD (computer-aided–design) drawings this November, in dialogue with the scientific team and Bruce. The company will fabricate the sculpture in December. The scientists will create educational materials that explain the thermodynamics and quantum physics represented in the sculpture. Starting in 2025, we’ll exhibit the sculpture everywhere possible. Plans include the American Physical Society’s Global Physics Summit (March Meeting), the quantum-steampunk creative-writing course I’m co-teaching next spring, and the Quantum World Congress. Bruce will incorporate the sculpture into his STEAMpunk workshops. Drop us a line if you want the Quantum-Steampunk Engine sculpture at an event as a centerpiece or teaching tool. And stay tuned for updates on the sculpture’s creation process and outreach journey.

Our team’s schemes extend beyond the tabletop sculpture: we aim to build an 8’-by-8’-by-8’ version. The full shebang will contain period antiques, lasers, touchscreens, and moving and interactive parts. We hope that a company, university, or individual will request the full-size version upon seeing its potential in the tabletop.

After all, what are steampunk and science for, if not dreaming?

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Traditionally, mathematics research projects are conducted by a small number (typically one to five) of expert mathematicians, each of which are familiar enough with all aspects of the project that they can verify each other’s contributions. It has been challenging to organize mathematical projects at larger scales, and particularly those that involve contributions from the general public, due to the need to verify all of the contributions; a single error in one component of a mathematical argument could invalidate the entire project. Furthermore, the sophistication of a typical math project is such that it would not be realistic to expect a member of the public, with say an undergraduate level of mathematics education, to contribute in a meaningful way to many such projects.

For related reasons, it is also challenging to incorporate assistance from modern AI tools into a research project, as these tools can “hallucinate” plausible-looking, but nonsensical arguments, which therefore need additional verification before they could be added into the project.

Proof assistant languages, such as Lean, provide a potential way to overcome these obstacles, and allow for large-scale collaborations involving professional mathematicians, the broader public, and/or AI tools to all contribute to a complex project, provided that it can be broken up in a modular fashion into smaller pieces that can be attacked without necessarily understanding all aspects of the project as a whole. Projects to formalize an existing mathematical result (such as the formalization of the recent proof of the PFR conjecture of Marton, discussed in this previous blog post) are currently the main examples of such large-scale collaborations that are enabled via proof assistants. At present, these formalizations are mostly crowdsourced by human contributors (which include both professional mathematicians and interested members of the general public), but there are also some nascent efforts to incorporate more automated tools (either “good old-fashioned” automated theorem provers, or more modern AI-based tools) to assist with the (still quite tedious) task of formalization.

However, I believe that this sort of paradigm can also be used to explore new mathematics, as opposed to formalizing existing mathematics. The online collaborative “Polymath” projects that several people including myself organized in the past are one example of this; but as they did not incorporate proof assistants into the workflow, the contributions had to be managed and verified by the human moderators of the project, which was quite a time-consuming responsibility, and one which limited the ability to scale these projects up further. But I am hoping that the addition of proof assistants will remove this bottleneck.

I am particularly interested in the possibility of using these modern tools to explore a class of many mathematical problems at once, as opposed to the current approach of focusing on only one or two problems at a time. This seems like an inherently modularizable and repetitive task, which could particularly benefit from both crowdsourcing and automated tools, if given the right platform to rigorously coordinate all the contributions; and it is a type of mathematics that previous methods usually could not scale up to (except perhaps over a period of many years, as individual papers slowly explore the class one data point at a time until a reasonable intuition about the class is attained). Among other things, having a large data set of problems to work on could be helpful for benchmarking various automated tools and compare the efficacy of different workflows.

One recent example of such a project was the Busy Beaver Challenge, which showed this July that the fifth Busy Beaver number was equal to . Some older crowdsourced computational projects, such as the Great Internet Mersenne Prime Search (GIMPS), are also somewhat similar in spirit to this type of project (though using more traditional proof of work certificates instead of proof assistants). I would be interested in hearing of any other extant examples of crowdsourced projects exploring a mathematical space, and whether there are lessons from those examples that could be relevant for the project I propose here.

More specifically I would like to propose the following (admittedly artificial) project as a pilot to further test out this paradigm, which was inspired by a MathOverflow question from last year, and discussed somewhat further on my Mastodon account shortly afterwards.

The problem is in the field of universal algebra, and concerns the (medium-scale) exploration of simple equational theories for magmas. A magma is nothing more than a set equipped with a binary operation . Initially, no additional axioms on this operation are imposed, and as such magmas by themselves are somewhat boring objects. Of course, with additional axioms, such as the identity axiom or the associative axiom, one can get more familiar mathematical objects such as groups, semigroups, or monoids. Here we will be interested in (constant-free) equational axioms, which are axioms of equality involving expressions built from the operation and one or more indeterminate variables in . Two familiar examples of such axioms are the commutative axiom

To illustrate the project I have in mind, let me first introduce eleven examples of equational axioms for magmas:

• Equation1:
• Equation2:
• Equation3:
• Equation4:
• Equation5:
• Equation6:
• Equation7:
• Equation8:
• Equation9:
• Equation10:
• Equation11:

One can then ask which axioms imply which others. For instance, Equation1 implies all the other axioms in this list, which in turn imply Equation11. Equation8 implies Equation9 as a special case, which in turn implies Equation10 as a special case. The full poset of implications can be depicted by the following Hasse diagram:

This in particular answers the MathOverflow question of whether there were equational axioms intermediate between the constant axiom Equation1 and the associative axiom Equation10.

Most of the implications here are quite easy to prove, but there is one non-trivial one, obtained in this answer to a MathOverflow post closely related to the preceding one:

Proposition 1 Equation4 implies Equation7.

Proof: Suppose that obeys Equation4, thus

A formalization of the above argument in Lean can be found here.

I will remark that the general question of determining whether one set of equational axioms determines another is undecidable; see Theorem 14 of this paper of Perkins. (This is similar in spirit to the more well known undecidability of various word problems.) So, the situation here is somewhat similar to the Busy Beaver Challenge, in that past a certain point of complexity, we would necessarily encounter unsolvable problems; but hopefully there would be interesting problems and phenomena to discover before we reach that threshold.

The above Hasse diagram does not just assert implications between the listed equational axioms; it also asserts non-implications between the axioms. For instance, as seen in the diagram, the commutative axiom Equation7 does not imply the Equation4 axiom

• Equation2 does not imply Equation3.
• Equation3 does not imply Equation5.
• Equation3 does not imply Equation7.
• Equation5 does not imply Equation6.
• Equation5 does not imply Equation7.
• Equation6 does not imply Equation7.
• Equation6 does not imply Equation10.
• Equation7 does not imply Equation6.
• Equation7 does not imply Equation10.
• Equation9 does not imply Equation8.
• Equation10 does not imply Equation9.
• Equation10 does not imply Equation6.

As one can see, it is already somewhat tedious to compute the Hasse diagram of just eleven equations. The project I propose is to try to expand this Hasse diagram by a couple orders of magnitude, covering a significantly larger set of equations. The set I propose is the set of equations that use the magma operation at most four times, up to relabeling and the reflexive and symmetric axioms of equality; this includes the eleven equations above, but also many more. How many more? Recall that the Catalan number is the number of ways one can form an expression out of applications of a binary operation (applied to placeholder variables); and, given a string of placeholder variables, the Bell number is the number of ways (up to relabeling) to assign names to each of these variables, where some of the placeholders are allowed to be assigned the same name. As a consequence, ignoring symmetry, the number of equations that involve at most four operations is

It is not clear to me at all what the geometry of will look like. Will most equations be incomparable with each other? Will it stratify into layers of “strong” and “weak” axioms? Will there be a lot of equivalent axioms? It might be interesting to record now any speculations as what the structure of this poset, and compare these predictions with the outcome of the project afterwards.

A brute force computation of the poset would then require comparisons, which looks rather daunting; but of course due to the axioms of a partial order, one could presumably identify the poset by a much smaller number of comparisons. I am thinking that it should be possible to crowdsource the exploration of this poset in the form of submissions to a central repository (such as the github repository I just created) of proofs in Lean of implications or non-implications between various equations, which could be validated in Lean, and also checked against some file recording the current status (true, false, or open) of all the comparisons, to avoid redundant effort. Most submissions could be handled automatically, with relatively little human moderation required; and the status of the poset could be updated after each such submission.

I would imagine that there is some “low-hanging fruit” that could establish a large number of implications (or anti-implications) quite easily. For instance, laws such as Equation2 or Equation3 more or less completely describe the binary operation , and it should be quite easy to check which of the laws are implied by either of these two laws. The poset has a reflection symmetry associated to replacing the binary operator by its reflection , which in principle cuts down the total work by a factor of about two. Specific examples of magmas, such as the natural numbers with the addition operation, obey some set of equations in but not others, and so could be used to generate a large number of anti-implications. Some existing automated proving tools for equational logic, such as Prover9 and Mace4 (for obtaining implications and anti-implications respectively), could then be used to handle most of the remaining “easy” cases (though some work may be needed to convert the outputs of such tools into Lean). The remaining “hard” cases could then be targeted by some combination of human contributors and more advanced AI tools.

Perhaps, in analogy with formalization projects, we could have a semi-formal “blueprint” evolving in parallel with the formal Lean component of the project. This way, the project could accept human-written proofs by contributors who do not necessarily have any proficiency in Lean, as well as contributions from automated tools (such as the aforementioned Prover9 and Mace4), whose output is in some other format than Lean. The task of converting these semi-formal proofs into Lean could then be done by other humans or automated tools; in particular I imagine modern AI tools could be particularly valuable for this portion of the workflow. I am not quite sure though if existing blueprint software can scale to handle the large number of individual proofs that would be generated by this project; and as this portion would not be formally verified, a significant amount of human moderation might also be needed here, and this also might not scale properly. Perhaps the semi-formal portion of the project could instead be coordinated on a forum such as this blog, in a similar spirit to past Polymath projects.

It would be nice to be able to integrate such a project with some sort of graph visualization software that can take an incomplete determination of the poset as input (in which each potential comparison in is marked as either true, false, or open), completes the graph as much as possible using the axioms of partial order, and then presents the partially known poset in a visually appealing way. If anyone knows of such a software package, I would be happy to hear of it in the comments.

Anyway, I would be happy to receive any feedback on this project; in addition to the previous requests, I would be interested in any suggestions for improving the project, as well as gauging whether there is sufficient interest in participating to actually launch it. (I am imagining running it vaguely along the lines of a Polymath project, though perhaps not formally labeled as such.)

UPDATE, Sep 30 2024: The project is up and running (and highly active), with the main page being this Github repository. See also the Lean Zulip chat for some (also very active) discussion on the project.

In the waning minutes before the Orioles postseason begins I ought to mark down, as is my habit, some notes on games I saw; by complete chance I was visiting Harvard CMSA (where I talked with Mike Freedman about parallel parking) the same week the Orioles were at Fenway, so I caught a couple of games. Tiny notes:

• No amount of upgrades can make Fenway not feel old. The building itself, its shape and bones, makes watching a game feel like watching a game felt when I was a kid in the 80s. I guess if I ever go to Wrigley I might feel the same. But now that the Oakland Coliseum is done hosting baseball, those might be the only options.
• Adley Rutschman has been pressing for the whole second half; in the first game, with the bases loaded and two outs, had two great takes on close pitches to get to 3-2 and then poked an opposite field single for what turned out to be the winning margin. That’s the hitter he can be. Good to see.
• I got there a little late and missed a classic cheap Fenway HR by Cedric Mullins, 334 feet to the pole, but then he was kind enough to hit another one, a real one, shortly after I got there.
• Albert Suárez, the story of the year, journeyman plucked out of the minors turned key part of the rotation after Bradish, Means, Rodriguez all went down. He never looked dominant, a lot of guys squared up on him for loud outs, but he also induced a bunch of really ugly, lost-looking swings. I don’t know enough about the Red Sox to know whether this was Suárez or whether the Red Sox just swing like that.
• Went to the first game by myself; made me nostalgic for grad school, when I used to come out by myself to see the Orioles a lot. CJ came up to meet me for the second game, his first time in Fenway. Orioles looked limp the whole time but Kremer kept it close, 7 K in 7 innings, left with a 2-1 deficit which Tony Santander promptly erased with a 400-foot shot. Then Keegan Akin came in to close out a lead in the 10th and Orioles bullpen things happened.

Orioles-Royals starts in half an hour. I hope I do not have to say “Orioles bullpen things happened.” The lineup is basically back to full strength at just the right time. We only have Burnes and Kremer left out of our opening day starting rotation, but if things go well, Burnes, Kremer, Eflin and Suárez is enough starting pitching to win a playoff series.

Hopefully, you didn’t notice, but Golem V has been replaced. Superficially, the new machine looks pretty much like the old.

It’s another Mac Mini, with an (8-core) Apple Silicon M2 chip (instead of a quad-core Intel Core i7), 24 GB of RAM (instead of 16), dual 10Gbase-T NICs (instead of 1Gbase-T), a 1TB internal SSD and a 2TB external SSD (TimeMachine backup).

The transition was anything but smooth.

The first step involved retrieving the external HD, which contained a clone of the internal System drive, from UDC and running Migration Assistant to transfer the data to the new machine.

Except … Migration Assistant refused to touch the external HD. It (like the System drive of Golem V) was formatted with a case-sensitive filesystem. Ten years ago, that was perfectly OK, and seemed like the wave of the future. But the filesystem (specifically, the Data Volume) for current versions of Macos is case-insensitive and there is no way to format it as case-sensitive. Since transferring data from a case-sensitive to a case-insensitive filesystem is potentially lossy, Migration Assistant refused to even try.

The solution turned out to be:

• Format a new drive as case-insensitive.
• Use rsync to copy the old (case-sensitive) drive onto the new one.
• rsync complained about a handful of files, but none were of any consequence.
• Run Migration Assistant on the new case-insensitive drive.

And that was just Day 1. Recompiling/reinstalling a whole mess ‘o software occupied the next several weeks, with similar hurdles to overcome.

For instance, installing Perl XS modules, using cpan consistently failed with a

fatal error: 'EXTERN.h' file not found

error. Googling the failures led me to perlmonks.org, where a post sagely opined

First, do not use the system Perl on MacOS. As Corion says, that is for Apple, not for you.

This is nonsense. The system Perl is the one Apple intends you to use. But … if you’re gonna do development on Macos (and installing Perl XS modules apparently constitutes development), you need to use the Macos SDK. And cpan doesn’t seem to be smart enough to do that. The Makefile it generates says

PERL_INC = /System/Library/Perl/5.34/darwin-thread-multi-2level/CORE

Edit that by hand to read

PERL_INC = /Library/Developer/CommandLineTools/SDKs/MacOSX14.sdk/System/Library/Perl/5.34/darwin-thread-multi-2level/CORE

and everything compiles and installs just fine.

And don’t even get me started on the woeful state of the once-marvelous Fink Package Manager.

One odder bit of breakage does deserve a mention. sysctl is used to set (or read) various Kernel parameters (including one that I very much need for my setup: net.inet.ip.forwarding=1). And there’s a file /etc/sysctl.conf where you can store these settings, so that they persist across reboots. Unnoticed by me, Migration Assistant didn’t copy that file to the new Golem, which was the source of much puzzlement and consternation when the new Golem booted up for the first time at UDC, and the networking wasn’t working right.

When I realized what was going on, I just thought, “Aha! I’ll recreate that file and all will be good.” Imagine my surprise when I rebooted the machine a couple of days later and, again, the networking wasn’t working right. Turns out that, unlike every other Unix system I have seen (and unlike the previous Golem), the current version(s) of Macos completely ignore /etc/sysctl.conf. If you want to persist those settings between reboots, you have to do that in a cron-job (or launchd script or whatever.)

Anyway, enough complaining. The new Golem seems to be working now, in no small part thanks to the amazing support (and boundless patience) of Chris Murphy, Andrew Manhein and the rest of the crew at UDC. Thanks guys!

The other morning I noticed there was a dead mouse in our yard right by the door, and I didn’t feel like dealing with it, and Dr. Mrs. Q didn’t feel like dealing with it, and that night we didn’t feel like dealing with it either, and we said, knowing we were just procrastinating, “well maybe some animal will carry it off in the night but if not we’ll deal with it in the morning.” But some animal did carry it off in the night. I was happy not to have to bag and trash a dead mouse, but also felt it was bad for my moral constitution for my procrastination to have been rewarded in this way.

Building Quantum Computers: A Practical Introduction by Shayan Majidy, Christopher Wilson, and Raymond Laflamme has been published by Cambridge University Press and will be released in the US on September 30. The authors invited me to write a Foreword for the book, which I was happy to do. The publisher kindly granted permission for me to post the Foreword here on Quantum Frontiers.

Foreword

The principles of quantum mechanics, which as far as we know govern all natural phenomena, were discovered in 1925. For 99 years we have built on that achievement to reach a comprehensive understanding of much of the physical world, from molecules to materials to elementary particles and much more. No comparably revolutionary advance in fundamental science has occurred since 1925. But a new revolution is in the offing.

Up until now, most of what we have learned about the quantum world has resulted from considering the behavior of individual particles — for example a single electron propagating as a wave through a crystal, unfazed by barriers that seem to stand in its way. Understanding that single-particle physics has enabled us to explore nature in unprecedented ways, and to build information technologies that have profoundly transformed our lives.

What’s happening now is we’re learning how to instruct particles to evolve in coordinated ways that can’t be accurately described in terms of the behavior of one particle at a time. The particles, as we like to say, can become entangled. Many particles, like electrons or photons or atoms, when highly entangled, exhibit an extraordinary complexity that we can’t capture with the most powerful of today’s supercomputers, or with our current theories of how nature works. That opens extraordinary opportunities for new discoveries and new applications.

Most temptingly, we anticipate that by building and operating large-scale quantum computers, which control the evolution of very complex entangled quantum systems, we will be able to solve some computational problems that are far beyond the reach of today’s digital computers. The concept of a quantum computer was proposed over 40 years ago, and the task of building quantum computing hardware has been pursued in earnest since the 1990s. After decades of steady progress, quantum information processors with hundreds of qubits have become feasible and are scientifically valuable. But we may need quantum processors with millions of qubits to realize practical applications of broad interest. There is still a long way to go.

Why is it taking so long? A conventional computer processes bits, where each bit could be, say, a switch which is either on or off. To build highly complex entangled quantum states, the fundamental information-carrying component of a quantum computer must be what we call a “qubit” rather than a bit. The trouble is that qubits are much more fragile than bits — when a qubit interacts with its environment, the information it carries is irreversibly damaged, a process called decoherence. To perform reliable logical operations on qubits, we need to prevent decoherence by keeping the qubits nearly perfectly isolated from their environment. That’s very hard to do. And because a qubit, unlike a bit, can change continuously, precisely controlling a qubit is a further challenge, even when decoherence is in check.

While theorists may find it convenient to regard a qubit (or a bit) as an abstract object, in an actual processor a qubit needs to be encoded in a particular physical system. There are many options. It might, for example, be encoded in a single atom which can be in either one of two long-lived internal states. Or the spin of a single atomic nucleus or electron which points either up or down along some axis. Or a single photon that occupies either one of two possible optical modes. These are all remarkable encodings, because the qubit resides in a very simple single quantum system, yet, thanks to technical advances over several decades, we have learned to control such qubits reasonably well. Alternatively, the qubit could be encoded in a more complex system, like a circuit conducting electricity without resistance at very low temperature. This is also remarkable, because although the qubit involves the collective motion of billions of pairs of electrons, we have learned to make it behave as though it were a single atom.

To run a quantum computer, we need to manipulate individual qubits and perform entangling operations on pairs of qubits. Once we can perform such single-qubit and two-qubit “quantum gates” with sufficient accuracy, and measure and initialize the qubits as well, then in principle we can perform any conceivable quantum computation by assembling sufficiently many qubits and executing sufficiently many gates.

It’s a daunting engineering challenge to build and operate a quantum system of sufficient complexity to solve very hard computation problems. That systems engineering task, and the potential practical applications of such a machine, are both beyond the scope of Building Quantum Computers. Instead the focus is on the computer’s elementary constituents for four different qubit modalities: nuclear spins, photons, trapped atomic ions, and superconducting circuits. Each type of qubit has its own fascinating story, told here expertly and with admirable clarity.

For each modality a crucial question must be addressed: how to produce well-controlled entangling interactions between two qubits. Answers vary. Spins have interactions that are always on, and can be “refocused” by applying suitable pulses. Photons hardly interact with one another at all, but such interactions can be mocked up using appropriate measurements. Because of their Coulomb repulsion, trapped ions have shared normal modes of vibration that can be manipulated to generate entanglement. Couplings and frequencies of superconducting qubits can be tuned to turn interactions on and off. The physics underlying each scheme is instructive, with valuable lessons for the quantum informationists to heed.

Various proposed quantum information processing platforms have characteristic strengths and weaknesses, which are clearly delineated in this book. For now it is important to pursue a variety of hardware approaches in parallel, because we don’t know for sure which ones have the best long term prospects. Furthermore, different qubit technologies might be best suited for different applications, or a hybrid of different technologies might be the best choice in some settings. The truth is that we are still in the early stages of developing quantum computing systems, and there is plenty of potential for surprises that could dramatically alter the outlook.

Building large-scale quantum computers is a grand challenge facing 21st-century science and technology. And we’re just getting started. The qubits and quantum gates of the distant future may look very different from what is described in this book, but the authors have made wise choices in selecting material that is likely to have enduring value. Beyond that, the book is highly accessible and fun to read. As quantum technology grows ever more sophisticated, I expect the study and control of highly complex many-particle systems to become an increasingly central theme of physical science. If so, Building Quantum Computers will be treasured reading for years to come.

John Preskill
Pasadena, California

I’ve been asked, innumerable times, to explain quantum entanglement to some lay audience. Most of the elementary explanations that I have seen (heck, maybe all of them) fail to draw any meaningful distinction between “entanglement” and mere “(classical) correlation.”

This drives me up the wall, so each time I am asked, I strive to come up with an elementary explanation of the difference. Rather than keep reinventing the wheel, let me herewith record my latest attempt.

“Entanglement” is a bit tricky to explain, versus “correlation” — which has a perfectly classical interpretation.

Say I tear a page of paper in two, crumple up the two pieces into balls and (at random) hand one to Adam and the other to Betty. They then go their separate ways and — sometime later — Adam unfolds his piece of paper. There’s a 50% chance that he got the top half, and 50% that he got the bottom half. But if he got the top half, we know for certain that Betty got the bottom half (and vice versa).

That’s correlation.

In this regard, the entangled state behaves exactly the same way. What distinguishes the entangled state from the merely correlated is something that doesn’t have a classical analogue. So let me shift from pieces of paper to photons.

You’re probably familiar with the polaroid filters in good sunglasses. They absorb light polarized along the horizontal axis, but transmit light polarized along the vertical axis.

Say, instead of crumpled pieces of paper, I send Adam and Betty a pair of photons.

In the correlated state, one photon is polarized horizontally, and one photon is polarized vertically, and there’s a 50% chance that Adam got the first while Betty got the second and a 50% chance that it’s the other way around.

Adam and Betty send their photons through polaroid filters, both aligned vertically. If Adam’s photon makes it through the filter, we can be certain that Betty’s gets absorbed and vice versa. Same is true if they both align their filters horizontally.

Say Adam aligns his filter horizontally, while Betty aligns hers vertically. Then either both photons make it though (with 50% probability) or both get absorbed (also with 50% probability).

All of the above statements are also true in the entangled state.

The tricky thing, the thing that makes the entangled state different from the correlated state, is what happens if both Adam and Betty align their filters at a 45° angle. Now there’s a 50% chance that Adam’s photon makes it through his filter, and a 50% chance that Betty’s photon makes it through her filter.

(You can check this yourself, if you’re willing to sacrifice an old pair of sunglasses. Polarize a beam of light with one sunglass lens, and view it through the other sunglass lens. As you rotate the second lens, the intensity varies from 100% (when the lenses are aligned) to 0 (when they are at 90°). The intensity is 50% when the second lens is at 45°.)

So what is the probability that both Adam and Betty’s photons make it through? Well, if there’s a 50% chance that his made it through and a 50% chance that hers made it through, then you might surmise that there’s a 25% chance that both made it through.

That’s indeed the correct answer in the correlated state.

In fact, in the correlated state, each of the 4 possible outcomes (both photons made it through, Adam’s made it through but Betty’s got absorbed, Adam’s got absorbed but Betty’s made it through or both got absorbed) has a 25% chance of taking place.

But, in the entangled state, things are different.

In the entangled state, the probability that both photons made it through is 50% – the same as the probability that one made it through. In other words, if Adam’s photon made it through the 45° filter, then we can be certain that Betty’s made it through. And if Adam’s was absorbed, so was Betty’s. There’s zero chance that one of their photons made it through while the other got absorbed.

Unfortunately, while it’s fairly easy to create the correlated state with classical tools (polaroid filters, half-silvered mirrors, …), creating the entangled state requires some quantum mechanical ingredients. So you’ll just have to believe me that quantum mechanics allows for a state of two photons with all of the aforementioned properties.

Sorry if this explanation was a bit convoluted; I told you that entanglement is subtle…

I met Mike Freedman last week at CMSA and I learned a great metaphor about an old favorite subject of mine, random walks on groups.

The Heisenberg group is the group of upper triangular matrices with 1’s on the diagonal:

You can take a walk on the integral or Z/pZ points of the Heisenberg group using the standard generators

and their inverses. How do you get a central element

with these generators? The way that presents itself most immediately is that the commutator [x,y] is the central element with 1 in the upper right-hand corner. So the matrix above is [x,y]^c, a word of length 4c. But you can do better! If m is an integer of size about , then [x^m, y^m] is central with an m^2 in the upper right-hand corner; then you can multiply by another or so copies of [x,y] to get the desired element in about steps.

Mike F. likes to think of the continuous version of this walk. This is a funny process; the Heisenberg group over R is a 3-manifold, but you only have two infinitesimal directions in which you’re allowed to move. The cost of moving infinitesimally in the c-direction above is infinite! (One way of thinking of this: by the above argument, it costs to travel in that direction, but when is close to 0, the cost per unit of distance goes to infinity!

This is what’s called a subriemannian structure. It’s like a Riemannian metric, but in each tangent space there’s a proper subspace in which you’re allowed to move at nonzero speed. But the brackets between things are substantial enough that you can still get anywhere on the manifold, you just have to shimmy a bit.

That was not a very clean definition, and I’m not going to give you one, but I will give you a very useful metaphor Mike explained to me. It’s like parallel parking! The state of your car is described by a point on a three-manifold R^2 x S^1, where the first two coordinates govern the car’s position on the street and the last the direction of the wheels. (OK this is probably an interval in S^1, not the whole S^1, unless you have a very cool car, but set that aside.) And at any point you have the ability to turn the wheels sideways, or you can inch forward or backwards, but you can’t hop the car to the side! So there’s a two-dimensional subspace of the tangent space in which you can move at finite costs. But if you need to move your car two feet in a direction orthogonal to the direction you’re facing — that is, if you need to parallel park — you can do it. You just have to shimmy a bit, where “shimmy” is the parallel parking term for “commutator” — this is really the same thing as [x,y] giving you a positive amount of motion in the infinite-cost direction.

I’d been meaning to write a plug for my paper A self-dual integral form for the moonshine module on this blog for almost 7 years, but never got around to it until now. It turns out that sometimes, if you wait long enough, someone else will do your work for you. In this case, I recently noticed that Lieven Le Bruyn wrote up a nice summary of the result in 2021. I thought I’d add a little history of my own interaction with the problem.

I met boatloads of physicists as a master’s student at the Perimeter Institute for Theoretical Physics in Waterloo, Canada. Researchers pass through Perimeter like diplomats through my current neighborhood—the Washington, DC area—except that Perimeter’s visitors speak math instead of legalese and hardly any of them wear ties. But Nilanjana Datta, a mathematician at the University of Cambridge, stood out. She was one of the sharpest, most on-the-ball thinkers I’d ever encountered. Also, she presented two academic talks in a little black dress.

The academic year had nearly ended, and I was undertaking research at the intersection of thermodynamics and quantum information theory for the first time. My mentors and I were applying a mathematical toolkit then in vogue, thanks to Nilanjana and colleagues of hers: one-shot quantum information theory. To explain one-shot information theory, I should review ordinary information theory. Information theory is the study of how efficiently we can perform information-processing tasks, such as sending messages over a channel. 

Say I want to send you copies of a message. Into how few bits (units of information) can I compress the copies? First, suppose that the message is classical, such that a telephone could convey it. The average number of bits needed per copy equals the message’s Shannon entropy, a measure of your uncertainty about which message I’m sending. Now, suppose that the message is quantum. The average number of quantum bits needed per copy is the von Neumann entropy, now a measure of your uncertainty. At least, the answer is the Shannon or von Neumann entropy in the limit as approaches infinity. This limit appears disconnected from reality, as the universe seems not to contain an infinite amount of anything, let alone telephone messages. Yet the limit simplifies the mathematics involved and approximates some real-world problems.

But the limit doesn’t approximate every real-world problem. What if I want to send only one copy of my message—one shot? One-shot information theory concerns how efficiently we can process finite amounts of information. Nilanjana and colleagues had defined entropies beyond Shannon’s and von Neumann’s, as well as proving properties of those entropies. The field’s cofounders also showed that these entropies quantify the optimal rates at which we can process finite amounts of information.

My mentors and I were applying one-shot information theory to quantum thermodynamics. I’d read papers of Nilanjana’s and spoken with her virtually (we probably used Skype back then). When I learned that she’d visit Waterloo in June, I was a kitten looking forward to a saucer of cream.

Nilanjana didn’t disappoint. First, she presented a seminar at Perimeter. I recall her discussing a resource theory (a simple information-theoretic model) for entanglement manipulation. One often models entanglement manipulators as experimentalists who can perform local operations and classical communications: each experimentalist can poke and prod the quantum system in their lab, as well as link their labs via telephone. We abbreviate the set of local operations and classical communications as LOCC. Nilanjana broadened my view to the superset SEP, the operations that map every separable (unentangled) state to a separable state.

Then, because she eats seminars for breakfast, Nilanjana presented an even more distinguished talk the same day: a colloquium. It took place at the University of Waterloo’s Institute for Quantum Computing (IQC), a nearly half-hour walk from Perimeter. Would I be willing to escort Nilanjana between the two institutes? I most certainly would.

Nilanjana and I arrived at the IQC auditorium before anyone else except the colloquium’s host, Debbie Leung. Debbie is a University of Waterloo professor and another of the most rigorous quantum information theorists I know. I sat a little behind the two of them and marveled. Here were two of the scions of the science I was joining. Pinch me.

My relationship with Nilanjana deepened over the years. The first year of my PhD, she hosted a seminar by me at the University of Cambridge (although I didn’t present a colloquium later that day). Afterward, I wrote a Quantum Frontiers post about her research with PhD student Felix Leditzky. The two of them introduced me to second-order asymptotics. Second-order asymptotics dictate the rate at which one-shot entropies approach standard entropies as (the number of copies of a message I’m compressing, say) grows large. 

The following year, Nilanjana and colleagues hosted me at “Beyond i.i.d. in Information Theory,” an annual conference dedicated to one-shot information theory. We convened in the mountains of Banff, Canada, about which I wrote another blog post. Come to think of it, Nilanjana lies behind many of my blog posts, as she lies behind many of my papers.

But I haven’t explained about the little black dress. Nilanjana wore one when presenting at Perimeter and the IQC. That year, I concluded that pants and shorts caused me so much discomfort, I’d wear only skirts and dresses. So I stuck out in physics gatherings like a theorem in a newspaper. My mother had schooled me in the historical and socioeconomic significance of the little black dress. Coco Chanel invented the slim, simple, elegant dress style during the 1920s. It helped free women from stifling, time-consuming petticoats and corsets: a few decades beforehand, dressing could last much of the morning—and then one would change clothes for the afternoon and then for the evening. The little black dress offered women freedom of movement, improved health, and control over their schedules. Better, the little black dress could suit most activities, from office work to dinner with friends.

Yet I didn’t recall ever having seen anyone present physics in a little black dress.

I almost never use this verb, but Nilanjana rocked that little black dress. She imbued it with all the professionalism and competence ever associated with it. Also, Nilanjana had long, dark hair, like mine (although I’ve never achieved her hair’s length); and she wore it loose, as I liked to. I recall admiring the hair hanging down her back after she received a question during the IQC colloquium. She’d whirled around to write the answer on the board, in the rapid-fire manner characteristic of her intellect. If one of the most incisive scientists I knew could wear dresses and long hair, then so could I.

Felix is now an assistant professor at the University of Illinois in Urbana-Champaign. I recently spoke with him and Mark Wilde, another one-shot information theorist and a guest blogger on Quantum Frontiers. The conversation led me to reminisce about the day I met Nilanjana. I haven’t visited Cambridge in years, and my research has expanded from one-shot thermodynamics into many-body physics. But one never forgets the classics.

George Field, brilliant theoretical astrophysicist and truly great human being, passed away on the morning of July 31. He was my Ph.D. thesis advisor and one of my favorite people in the world. I often tell my own students that the two most important people in your life who you will (consensually) choose are your spouse and your Ph.D. advisor. With George, I got incredibly lucky.

I am not the person to recount George’s many accomplishments as a scientist and a scientific citizen. He was a much more mainstream astrophysicist than I ever was, doing foundational work on the physics of the interstellar and intergalactic medium, astrophysical magnetic fields, star formation, thermal instability, accretion disks, and more. One of my favorite pieces of work he did was establishing that you could use spectral lines of hydrogen to determine the temperature of an ambient cosmic radiation field. This was before the discovery of the Cosmic Microwave Background, although George’s method became a popular way of measuring the CMB temperature once it was discovered. (George once told me that he had practically proven that there must be an anisotropic microwave radiation component in the universe, using this kind of reasoning — but his thesis advisor told him it was too speculative, so he never published it.)

At the height of his scientific career, as a professor at Berkeley, along came a unique opportunity: the Harvard College Observatory and the Smithsonian Astrophysical Observatory were considering merging into a single unit, and they needed a visionary leader to be the first director. After some negotiations, George became the founding director of the Harvard-Smithsonian Center for Astrophysics in 1973. He guided it to great success before stepping down a decade later. During those years he focused more on developing CfA and being a leader in astronomy than on doing research, including leading an influential Decadal Survey in Astronomy for the National Academy of Sciences (the “Field Report”). He never stopped advocating for good science, including in 2016 helping to draft an open letter in support of climate research.

I remember in 1989, when I was still a beginning grad student, hearing that George had just been elected to the National Academy of Sciences. I congratulated him, and he smiled and graciously thanked me. Talking to one of the other local scientists, they expressed surprise that he hadn’t been elected long before, which did indeed seem strange to me. Eventually I learned that he had been elected long before — but turned it down. That is extremely rare, and I wondered why. George explained that it had been a combination of him thinking the Academy hadn’t taken a strong enough stance against the Vietnam War, and that they wouldn’t let in a friend of his for personality reasons rather than scientific ones. By 1989 those reasons had become moot, so he was happy to accept.

It was complete luck that I ended up with George as my advisor. I was interested in particle physics and gravity, which is really physics more than astronomy, but the Harvard physics department didn’t accept me, while the astronomy department did. Sadly Harvard didn’t have any professors working on those topics, but I was randomly assigned to George as one of the few members of the theory group. Particle physics was not his expertise, but he had noticed that it was becoming important to cosmology, so thought it would be good to learn about it a bit. In typical fashion, he attended a summer school in particle physics as a student — not something most famous senior scientists tend to do. At the school he heard lectures by MIT theorist Roman Jackiw, who at the time was thinking about gravity and electromagnetism in 2+1 spacetime dimensions. This is noticeably different than the 3+1 dimensions in which we actually live — a tiny detail that modern particle theorists have learned to look past, but one that rubbed George’s astrophysicist heart the wrong way. So George wondered whether you could do similar things as in Roman’s theory, but in the real world. Roman said no, because that would violate Lorentz invariance — there would be a preferred frame of reference. Between the two of them they eventually thought to ask, so what if that were actually true? That’s where I arrived on the scene, with very little knowledge but a good amount of enthusiasm and willingness to learn. Eventually we wrote “Limits on a Lorentz- and Parity-Violating Modification of Electrodynamics,” which spelled out the theoretical basis of the idea and also suggested experimental tests, most effectively the prediction of cosmic birefringence (a rotation of the plane of polarization of photons traveling through the universe).

Both George and I were a little dubious that violating Lorentz invariance was the way to make a serious contribution to particle physics. To our surprise, the paper turned out to be quite influential. In retrospect, we had shown how to do something interesting: violate Lorentz invariance by coupling to a field with a Lorentz-violating expectation value in a gauge-invariant way. There turn out to be many other ways to do that, and correspondingly many experimental tests to be investigated. And later I realized that a time-evolving dark energy field could do the same thing — and now there is an ongoing program to search for such an effect. There’s a lesson there: wild ideas are well worth investigating if they can be directly tied to experimental constraints.

Despite being assigned to each other somewhat arbitrarily, George and I hit it off right away (or at least once I stopped being intimidated). He was unmatched in both his pure delight at learning new things about the universe, and his absolute integrity in doing science the right way. Although he was not an expert in quantum field theory or general relativity, he wanted to know more about them, and we learned together. But simply by being an example of what a scientist should be, I learned far more from him. (He once co-taught a cosmology course with Terry Walker, and one day came to class more bedraggled than usual. Terry later explained to us that George had been looking into how to derive the spectrum of the cosmic microwave background, was unsatisfied with the usual treatment, and stayed up all night re-doing it himself.)

I was also blessed to become George’s personal friend, as well as getting to know his wonderful wife Susan. I would visit them while they were vacationing, and George would have been perfectly happy to talk about science the entire time, but Susan kept us all more grounded. He also had hidden talents. I remember once taking a small rowboat into a lake, but it was extremely windy. Being the younger person (George must have been in his 70s at the time), I gallantly volunteered to do the rowing. But the wind was more persistent than I was, and after a few minutes I began to despair of making much headway. George gently suggested that he give it a try, and bip-bip-bip just like that we were in the middle of the lake. Turns out he had rowed for a crew team as an undergraduate at MIT, and never lost his skills.

George remained passionate about science to the very end, even as his health began to noticeably fail. For the last couple of years we worked hard to finish a paper on axions and cosmic magnetic fields. (The current version is a bit muddled, I need to get our updated version onto the arxiv.) It breaks my heart that we won’t be able to write any more papers together. A tremendous loss.

• AIP Oral History Interview — July 1980, part 1
• AIP Oral History Interview — July 1980, part 2
• AIP Oral History Interview — December 2007
• Autobiographical Essay — “Wondering About Things”