Planet Musings

I’m hosting a small symposium next Tuesday, 6 June, on Applied Category Theory, featuring our very own John Baez. Here’s the announcement.

The language of Category Theory has been under development since the 1940s and continues to evolve to this day. It was originally created as a formal language to capture common mathematical structures and inference methods across various branches of mathematics, and later found application outside of mathematics. By introducing arrows to mediate between objects, the language is designed to represent anything that can be perceived as a process - including processes of inference and physical processes.

The first applications of Category Theory outside of mathematics and logic were to physics and to computer science. There was also an early application in biology by Robert Rosen.

But over the past decade we have seen researchers under the banner of Applied Category Theory take on a variety of novel subjects, addressing topics which include:

causality, probabilistic reasoning, statistics, learning theory, deep neural networks, dynamical systems, information theory, database theory, natural language processing, cognition, consciousness, systems biology, genomics, epidemiology, chemical reaction networks, neuroscience, complex networks, game theory, robotics, and quantum computing.

In this hybrid seminar at the Centre for Reasoning, University of Kent, we will be hearing online from two leading practitioners. All are welcome to attend.

Location

• In person: KS23, Keynes College, University of Kent, Canterbury

• Online: MS Teams link

Schedule

UK time (UTC +1), Tuesday 6 June

• 15.30-15.50 David Corfield (Kent), Introduction: Applied Category Theory from a Philosophical Point of View

• 15.50-16.50 Toby St Clere Smithe (Topos Institute, Oxford), Understanding the Bayesian Brain with Categorical Cybernetics

• 17.00-18.00 John Baez (UC Riverside), Applied Category Theory

It’s a question I’ve now heard several times, in different forms. People hear that I’ll be hired as a researcher at an institute of theoretical physics, and they ask, “what, exactly, are they paying you to research?”

The answer, with some caveats: “Whatever I want.”

When a company hires a researcher, they want to accomplish specific things: to improve their products, to make new ones, to cut down on fraud or out-think the competition. Some government labs are the same: if you work for NIST, for example, your work should contribute in some way to achieving more precise measurements and better standards for technology.

Other government labs, and universities, are different. They pursue basic research, research not on any specific application but on the general principles that govern the world. Researchers doing basic research are given a lot of freedom, and that freedom increases as their careers go on.

As a PhD student, a researcher is a kind of apprentice, working for their advisor. Even then, they have some independence: an advisor may suggest projects, but PhD students usually need to decide how to execute them on their own. In some fields, there can be even more freedom: in theoretical physics, it’s not unusual for the more independent students to collaborate with other people than just their advisor.

Postdocs, in turn, have even more freedom. In some fields they get hired to work on a specific project, but they tend to have more freedom as to how to execute it than a PhD student would. Other fields give them more or less free reign: in theoretical physics, a postdoc will have some guidance, but often will be free to work on whatever they find interesting.

Professors, and other long-term researchers, have the most freedom of all. Over the climb from PhD to postdoc to professor, researchers build judgement, demonstrating a track record for tackling worthwhile scientific problems. Universities, and institutes of basic research, trust that judgement. They hire for that judgement. They give their long-term researchers free reign to investigate whatever questions they think are valuable.

In practice, there are some restrictions. Usually, you’re supposed to research in a particular field: at an institute for theoretical physics, I should probably research theoretical physics. (But that can mean many things: one of my future colleagues studies the science of cities.) Further pressure comes from grant funding, money you need to hire other researchers or buy equipment that can come with restrictions attached. When you apply for a grant, you have to describe what you plan to do. (In practice, grant agencies are more flexible about this than you might expect, allowing all sorts of changes if you have a good reason…but you still can’t completely reinvent yourself.) Your colleagues themselves also have an impact: it’s much easier to work on something when you can walk down the hall and ask an expert when you get stuck. It’s why we seek out colleagues who care about the same big questions as we do.

Overall, though, research is one of the free-est professions there is. If you can get a job learning for a living, and do it well enough, then people will trust your judgement. They’ll set you free to ask your own questions, and seek your own answers.

As mentioned previously, structural glasses are materials in which there is no periodic lattice (no long-range spatial order) and the building blocks get "stuck" in some configuration, kinetically unable to get to the true energetic minimum state which would almost certainly be a periodic crystal.  Upon cooling from the liquid state, their viscosity increases by many orders of magnitude (in various ways) until they act like rigid solids.  Distinguishing "glassy" physics includes strongly interacting building blocks, a complicated energy landscape with many local minima, spatial disorder leading to hugely varying interaction strengths and a very broad distribution of relaxation times (so that responses to perturbations aren't simple exponentials in time, but are more slowly decaying functions such as \(-\log t\)).  These slow relaxations are called "aging", and when the system is perturbed (e.g., a sudden stress is applied, or a sudden temperature change is applied and removed), the system's response picks back up ("rejuvenation") before aging again.

Analogs of all of these properties are also seen in spin glasses, which I wrote about a bit in this post about the 2021 Nobel in Physics.  In a spin glass, the degrees of freedom aren't atoms or groups of atoms, but instead are the magnetic moments of particular atoms, such as isolated Fe atoms in a Cu bulk.   The analog of the periodic crystal would be some version of long-range magnetic order.  In a typical spin glass, the magnetic atoms are positioned randomly in a non-magnetic host, so that the magnetic interactions between neighbors are strong, but often random in sign and strength due to disorder.  As a result, the magnetic system has a complicated energy landscape with many minima (corresponding to configurations with similar energies but it would cost significant energy to rearrange the spins to get from one local energy minimum configuration to another).  These systems show aging, rejuvenation, etc.

The universality of glassy dynamics across such microscopically different systems is one of those remarkable emergences that crops up in condensed matter.  Despite the very different microscopic physics, there is some deeper organizing principle at work that leads to these properties.  

Spin glasses have attracted quite a bit of interest for a couple of reasons.  First, they are comparatively easy to study, since magnetic properties and their time evolution are usually easier to measure than detailed microscopic structural arrangements in structural glasses.  Second, it is possible to create models of spin glasses in a variety of systems, including using qubits.  Spin glasses can also be mapped to certain kinds of optimization problems (see this pdf news article).

Interestingly, a recent paper in Nature (arxiv version) by folks at D-Wave has used their 5000 qubit gadget to do a quantum simulation of a spin glass.  They can program the interactions among the qubits and make them random and frustrated as in a spin glass.  In small test configurations, they show that they can see (at short times, anyway) quantum coherent dynamics that agree with calculations.  They can then look at much larger systems, well beyond traditional calculational practicality, and see what happens.  I don't know enough about the system to evaluate this critically, but it looks like a very nice platform.  (They’ve come along way from when their founder used to argue and insult in blog comments.  They now show as anonymous, but the one from Geordie Rose is clear from context.)

Guest post by Giuliamaria Menara

Magnitude homology has been discussed extensively on this blog and definitely needs no introduction.

A lot of questions about magnitude homology have been answered and a number of possible application have been explored up to this point, but magnitude homology was never exploited for the structure analysis of a graph.

Being able to use magnitude homology to look for graph substructures seems a reasonable consequence of the definition of boundary map $\partial_{k,\ell}$. Indeed, a tuple $(x_0,\dots,x_k) \in MC_{k,\ell}$ is such that $\partial_{k,\ell}(x_0,\dots,x_k)=0$ if for every vertex $x_i \in \{x_1,\dots,x_{k-1} \}$ it holds that $len(x_{i-1},\hat{x_i},x_{i+1}) \lt len (x_{i-1},x_i,x_{i+1})$. In other words, if every vertex of the tuple is contained in a small enough substructure, which suggests the presence of a meaningful relationship between the rank of magnitude homology groups of a graph and the subgraph counting problem.

A major problem in exploring this relationship comes from the fact that the definition of $MC_{k,\ell}(G)$ only asks for consecutive vertices to be different. That is, if $x_0$ and $x_1$ are two adjacent vertices in $G$ an acceptable tuple in $MC_{5,4}(G)$ is $(x_0,x_1,x_0,x_1,x_0)$.

Tuples of this kind inducing a path that just revisits again and again the same edge (an more in general, tuples inducing non-eulerian trails) do not provide any insight about the meaning of magnitude homology. With this motivation, we introduce a slightly different definition of magnitude chain, considering the subgroup of $MC_{k,l}(G)$ where a vertex (and therefore an edge) is never required to be revisited.

Definition (Eulerian magnitude chain)   Let $G$ be a graph. We define the eulerian $(k,\ell)$-magnitude chain $EMC_{k,\ell}(G)$ to be the free abelian group generated by tuples $(x_0,\dots,x_k)$ of vertices of $G$ such that $x_i \neq x_j$ for all distinct $0\leq i,j \leq k$ and $len(x_0,\dots,x_k)=\ell$.

Taking as differential the one induced by $MC_{\ast,\ell}(G)$ we can construct the eulerian magnitude chain complex $EMC_{\ast,\ell}(G)$:

$$\cdots \to EMC_{k+1,\ell}(G) \xrightarrow{\partial_{k+1,\ell}} EMC_{k,\ell}(G) \xrightarrow{\partial_{k,\ell}} EMC_{k-1,\ell}(G) \to \cdots$$

and subsequently define the eulerian $(k,\ell)$-magnitude homology group

$$EMH_{k,\ell}(G) = H_k(EMC_{\ast,\ell}(G)) = \frac{\ker(\partial_{k,\ell})}{\mathrm{im}(\partial_{k+1,\ell})}.$$

Example   Consider the following graph $G$:

We want to compare $MH_{2,2}(G)$ and $EMH_{2,2}(G)$.

The magnitude chain $MC_{2,2}(G)$ is generated by

$$\begin{aligned} &(0,1,0), (0,1,2), (0,1,3), \\ &(1,0,1), (1,2,1), (1,2,3), (1,3,1), (1,3,2),\\ &(2,1,0), (2,1,2), (2,1,3), (2,3,1), (2,3,2),\\ &(3,1,0), (3,1,2), (3,1,3), (3,2,1), (3,2,3), \end{aligned}$$

while $MC_{1,2}(G)$ is generated by

$$(0,2), (2,0), (0,3), (3,0).$$

We see that $MH_{2,2}(G)=\ker(\partial_{2,2})$ is generated by all elements in $MC_{2,2}(G)$ apart from

$$(0,1,2), (2,1,0), (0,1,3), (3,1,0)$$

and thus has rank 14.

On the other hand, the eulerian chain $EMC_{2,2}(G)$ is generated by

$$(0,1,2), (0,1,3), (1,2,3), (1,3,2), (2,1,0), (2,1,3), (2,3,1), (3,1,0), (3,1,2), (3,2,1),$$

while $EMC_{1,2}(G)=MC_{1,2}$ is generated by

$$(0,2), (2,0), (0,3), (3,0).$$

We have now that $EMH_{2,2}(G)=\ker(\partial_{2,2})$ is generated by all elements in $MC_{2,2}(G)$ apart from

$$(0,1,2), (2,1,0), (0,1,3), (3,1,0),$$

and thus has rank 6, as the number of permutations of the triangle [1,2,3].

Remark   We point out that all definitions and properties regarding magnitude homology proved in

Richard Hepworth and Simon Willerton, Categorifying the magnitude of a graph, Homology, Homotopy and Applications 19(2) (2017), 31–60

and

Tom Leinster and Michael Shulman, Magnitude homology of enriched categories and metric spaces, Algebraic & Geometric Topology 21 (2021), no. 5, 2175–2221

continue to be valid for eulerian magnitude homology. In particular, with this new definition, $EMH_{0,0}(G)$ and $EMH_{1,1}(G)$ are still counting the number of vertices and edges in a graph respectively, since the generators of the groups $MC_{0,0}(G)$ and $MC_{1,1}(G)$ already satisfy the condition of not revisiting vertices.

Now, in order to account for the elements in $MC_{k,\ell}(G)$ containing the repetition of at least a vertex we define the discriminant magnitude chain as the quotient between the standard magnitude chain and the eulerian one.

Definition (Discriminant magnitude chain)   Let $G$ be a graph. We define the discriminant $(k,\ell)$-magnitude chain $DMC_{k,\ell}(G)$ as

$$DMC_{k,\ell}(G) = \frac{MC_{k,\ell}(G)}{EMC_{k,\ell}(G)}$$

Denoting by $[\cdot]_E$ the equivalence classes in $DMC_{k,\ell}(G)$, we define the differential map $\tilde{\partial}_{k,\ell}$ as

$$\tilde{\partial}_{k,\ell}([x_0,\dots,x_k]_N) = [\partial_{k,\ell}(x_0,\dots,x_k)]_E.$$

Splitting result

(Section removed following the conversation below.)

Applications

Subgraph counting

The example above suggests the presence of the relation we were looking for between the subgraph counting problem and the ranks of magnitude homology groups.

Lemma  Let $Z$ be the number of basis elements $\overline{x} \in EMC_{2,2}$ such that $\partial_{2,2}(\overline{x})=0$. The number of triangles occurring in $G$ is $\frac{Z}{6}$.

Proof  Consider a $3$-tuple $\overline{x}=(x_0,x_1,x_2)$ of length 2. Since $\overline{x}$ has length 2, $\{x_0,x_1\}$ and $\{x_1,x_2\}$ are edges of $G$. Also, $\partial_{2,2}(\overline{x})=0$ if and only if the shortest path between $x_0$ and $x_2$ has length smaller than $2$, that is, if removing the required vertex $x_1$ we obtain a 2-tuple of length 1. This implies that there exists and edge $\{x_0,x_2\}$ and thus a triangle with vertices $x_0,x_1,x_2$.

It is immediate to see that the above holds for all permutations of $\overline{x}$, which implies that the number of triangles occurring in $G$ is given by the number of basis elements in the kernel of $\partial_{2,2}$ divided by the cardinality of the automorphisms group of the triangle $D_3$.

Proceeding in a similar way, it also possible to show that the number of $k$-cliques in $G$ is upper bounded by $\left\lfloor\frac{Z}{k!}\right\rfloor$, where $Z$ is the number of basis elements $\overline{x} \in EMC_{k,k}$ such that $\partial_{k,k}(\overline{x})=0$.

A vanishing threshold for the first diagonal of EMH of Erdős–Rényi random graphs

Let $G=G=(n,p)=G(n,n^{-\alpha})$ be an Erdős–Rényi graph.

In order to produce a vanishing threshold for $EMH_{k,k}(G)$ we identify what kind of subgraph $H$ is induced by a cycle in $EMH_{k,k}(G)$ and give an estimate for the occurrences of $H$ in $G$.

Take $[x_0,\dots,x_k] \in EMH_{k,k}(G)$. Then for every $i=1,\dots,k-1$ the edge $(x_{i-1},x_{i+1})$ exists and the induced subgraph $H$ is as shown:

Now, the number of edges contained in such graph is $k+(k-1)$ (black edges plus blue edges). Hence, calling $a_H$ the number of automorphisms of $H$, the number of copies of $H$ expected in $G$ is

$$\begin{aligned} N_H &= \binom{n}{k+1} a_H p^{2k-1} \\ &\sim n^{k+1} \frac{a_H}{(k+1)!} n^{\alpha(1-2k)} \\ &= \frac{a_H}{(k+1)!} n^{\alpha(1-2k)+(k+1)} \xrightarrow{n\to \infty} \begin{cases} 0, \ \text{ if } \ \alpha \gt \frac{k+1}{2k-1} \\ \infty, \ \text{ if }\ 0 \lt \alpha \lt \frac{k+1}{2k-1}. \end{cases} \end{aligned}$$

The computation above implies the following.

Lemma  Let $G$ be an Erdős–Rényi graph on $n$ vertices and set $p=n^{-\alpha}$. The first diagonal of the eulerian magnitude homology $EMH_{k,k}(G)$ vanishes for $\alpha \gt \frac{k+1}{2k-1}$.

The 20th century was the century of fundamental physics. While we saw immense progress toward discovering the basic laws governing matter, space, and time, this has slowed to a crawl since 1980, despite an immense amount of work. Luckily, there’s plenty of exciting progress in other branches of physics: for example, using the fundamental physics we already know to design surprising new forms of matter! Like all other sciences in the 21st century, physics must also embrace the challenges of the Anthropocene: the era in which humanity is a dominant influence on the Earth’s climate and biosphere.

Last week I gave a Santa Fe Institute Community Lecture on the future of physics. You can see my slides here, or watch it here:

Since my talk was announced on the marquee of a theater, and I myself misread it as a concert by Joan Baez, I asked the audience how many had been expecting her. About ten, it seems! I planned to sing a bit of a song, but thought better of it at the last second.

Today it was my honor to serve on the PhD defense committee of Irina Espejo (NYU), who is one of the first (ever in the world, actually!) PhDs in Data Science. Her PhD research involved making real, practical, scalable, reproducible tools for the (late-in-pipeline) analysis of high-energy physics data from the Large Hadron Collider. She built tools to speed up likelihood-free inferences, and she built a tool to find exclusion regions (upper limits) in complex parameter spaces. She used the latter to put constraints on a (real, not toy) proposed modification to the standard model.

On the first project, the tools that she built (and built on) make the LHC more sensitive to new physics, because they find better test statistics for distinguishing models. They make some searches far better, which makes me wonder whether particle physics is using our money efficiently??

By guest blogger Clarice D. Aiello, faculty at UCLA

Imagine using your cellphone to control the activity of your own cells to treat injuries and disease. It sounds like something from the imagination of an overly optimistic science fiction writer. But this may one day be a possibility through the emerging field of quantum biology.

Over the past few decades, scientists have made incredible progress in understanding and manipulating biological systems at increasingly small scales, from protein folding to genetic engineering. And yet, the extent to which quantum effects influence living systems remains barely understood.

Quantum effects are phenomena that occur between atoms and molecules that can’t be explained by classical physics. It has been known for more than a century that the rules of classical mechanics, like Newton’s laws of motion, break down at atomic scales. Instead, tiny objects behave according to a different set of laws known as quantum mechanics.

For humans, who can only perceive the macroscopic world, or what’s visible to the naked eye, quantum mechanics can seem counterintuitive and somewhat magical. Things you might not expect happen in the quantum world, like electrons “tunneling” through tiny energy barriers and appearing on the other side unscathed, or being in two different places at the same time in a phenomenon called superposition.

I am trained as a quantum engineer. Research in quantum mechanics is usually geared toward technology. However, and somewhat surprisingly, there is increasing evidence that nature – an engineer with billions of years of practice – has learned how to use quantum mechanics to function optimally. If this is indeed true, it means that our understanding of biology is radically incomplete. It also means that we could possibly control physiological processes by using the quantum properties of biological matter.

Quantumness in biology is probably real

Researchers can manipulate quantum phenomena to build better technology. In fact, you already live in a quantum-powered world: from laser pointers to GPS, magnetic resonance imaging and the transistors in your computer – all these technologies rely on quantum effects.

In general, quantum effects only manifest at very small length and mass scales, or when temperatures approach absolute zero. This is because quantum objects like atoms and molecules lose their “quantumness” when they uncontrollably interact with each other and their environment. In other words, a macroscopic collection of quantum objects is better described by the laws of classical mechanics. Everything that starts quantum dies classical. For example, an electron can be manipulated to be in two places at the same time, but it will end up in only one place after a short while – exactly what would be expected classically.

In a complicated, noisy biological system, it is thus expected that most quantum effects will rapidly disappear, washed out in what the physicist Erwin Schrödinger called the “warm, wet environment of the cell.” To most physicists, the fact that the living world operates at elevated temperatures and in complex environments implies that biology can be adequately and fully described by classical physics: no funky barrier crossing, no being in multiple locations simultaneously.

Chemists, however, have for a long time begged to differ. Research on basic chemical reactions at room temperature unambiguously shows that processes occurring within biomolecules like proteins and genetic material are the result of quantum effects. Importantly, such nanoscopic, short-lived quantum effects are consistent with driving some macroscopic physiological processes that biologists have measured in living cells and organisms. Research suggests that quantum effects influence biological functions, including regulating enzyme activity, sensing magnetic fields, cell metabolism and electron transport in biomolecules.

How to study quantum biology

The tantalizing possibility that subtle quantum effects can tweak biological processes presents both an exciting frontier and a challenge to scientists. Studying quantum mechanical effects in biology requires tools that can measure the short time scales, small length scales and subtle differences in quantum states that give rise to physiological changes – all integrated within a traditional wet lab environment.

In my work, I build instruments to study and control the quantum properties of small things like electrons. In the same way that electrons have mass and charge, they also have a quantum property called spin. Spin defines how the electrons interact with a magnetic field, in the same way that charge defines how electrons interact with an electric field. The quantum experiments I have been building since graduate school, and now in my own lab, aim to apply tailored magnetic fields to change the spins of particular electrons.

Research has demonstrated that many physiological processes are influenced by weak magnetic fields. These processes include stem cell development and maturation, cell proliferation rates, genetic material repair and countless others. These physiological responses to magnetic fields are consistent with chemical reactions that depend on the spin of particular electrons within molecules. Applying a weak magnetic field to change electron spins can thus effectively control a chemical reaction’s final products, with important physiological consequences.

Currently, a lack of understanding of how such processes work at the nanoscale level prevents researchers from determining exactly what strength and frequency of magnetic fields cause specific chemical reactions in cells. Current cellphone, wearable and miniaturization technologies are already sufficient to produce tailored, weak magnetic fields that change physiology, both for good and for bad. The missing piece of the puzzle is, hence, a “deterministic codebook” of how to map quantum causes to physiological outcomes.

In the future, fine-tuning nature’s quantum properties could enable researchers to develop therapeutic devices that are noninvasive, remotely controlled and accessible with a mobile phone. Electromagnetic treatments could potentially be used to prevent and treat disease, such as brain tumors, as well as in biomanufacturing, such as increasing lab-grown meat production.

A whole new way of doing science

Quantum biology is one of the most interdisciplinary fields to ever emerge. How do you build community and train scientists to work in this area?

Since the pandemic, my lab at the University of California, Los Angeles and the University of Surrey’s Quantum Biology Doctoral Training Centre have organized Big Quantum Biology meetings to provide an informal weekly forum for researchers to meet and share their expertise in fields like mainstream quantum physics, biophysics, medicine, chemistry and biology.

Research with potentially transformative implications for biology, medicine and the physical sciences will require working within an equally transformative model of collaboration. Working in one unified lab would allow scientists from disciplines that take very different approaches to research to conduct experiments that meet the breadth of quantum biology from the quantum to the molecular, the cellular and the organismal.

The existence of quantum biology as a discipline implies that traditional understanding of life processes is incomplete. Further research will lead to new insights into the age-old question of what life is, how it can be controlled and how to learn with nature to build better quantum technologies.

***

This article is republished from The Conversation under a Creative Commons license. Read the original article.

***

Clarice D. Aiello is a quantum engineer interested in how quantum physics informs biology at the nanoscale. She is an expert on nanosensors that harness room-temperature quantum effects in noisy environments. Aiello received a bachelor’s in physics from the Ecole Polytechnique, France; a master’s degree in physics from the University of Cambridge, Trinity College, UK; and a PhD in electrical engineering from the Massachusetts Institute of Technology. She held postdoctoral appointments in bioengineering at Stanford University and in chemistry at the University of California, Berkeley. Two months before the pandemic, she joined the University of California, Los Angeles, where she leads the Quantum Biology Tech (QuBiT) Lab.

***

The author thanks Nicole Yunger Halpern and Spyridon Michalakis for the opportunity to talk about quantum biology to the physics audience of this wonderful blog!

On the plane home from meetings at Cambridge, Warwick, and Paris, I worked on a long document I am writing for Gaia DPAC CU5, which is the organization responsible for calibrating and extracting the Gaia XP spectra. They are doing a beautiful self-calibration to extract all the spectra on the same system, in the sense of resolution, dispersion, and throughput. But their system has some pathologies, which we discussed last week. I think I know how to solve some of them. My document is reporting those thoughts.

Writing like this reminds me of graduate school: One of my advisors (Blandford) often encouraged me to write up thoughts, ideas, projects, and proposals, even when we had no intention of submitting them anywhere. It's good practice, I think, because you can't understand anything if you don't write about it.

On this day in 1832, Evariste Galois died in a duel. The night before, he summarized his ideas in a letter to his friend Auguste Chevalier. Hermann Weyl later wrote “This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.”

That seems exaggerated, but within mathematics it might be true. On top of that, the backstory is really dramatic! I’d never really looked into it, until today. Let me summarize a bit from Wikipedia.

Galois lived during a time of political turmoil in France. In 1830, Charles X staged a coup d’état, touching off the July Revolution. While students at the Polytechnique were making history in the streets, Galois, at the École Normale, was locked in by the school’s director. Galois was incensed and wrote a blistering letter criticizing the director, which he submitted to the Gazette des Écoles, signing the letter with his full name. Although the Gazette’s editor omitted the signature for publication, Galois was expelled.

Galois joined the staunchly Republican artillery unit of the National Guard. He divided his time between math and politics. On 31 December 1830, his artillery unit was disbanded for fear that they might destabilize the government. 19 officers of this unit were arrested and charged with conspiracy to overthrow the government.

In April 1831 these officers were acquitted of all charges. On 9 May 1831, a banquet was held in their honor, with many famous people present, including Alexandre Dumas. The proceedings grew riotous. At some point, Galois stood and proposed a toast in which he said, “To Louis Philippe,” with a dagger above his cup. The Republicans at the banquet interpreted Galois’s toast as a threat against the king’s life and cheered.

The day after that wild banquet, Galois was arrested. He was imprisoned until 15 June 1831, when he had his trial. The jury acquitted him that same day.

All this time, Galois had also been doing math! Earlier, the famous mathematician Poisson had asked Galois to submit a paper to the Academy, which he did on 17 January 1831. Unfortunately, around 4 July 1831, Poisson wrote a reply declaring Galois’s work “incomprehensible” and saying his “argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor”. But Poisson ended on a positive note: “We would then suggest that the author should publish the whole of his work in order to form a definitive opinion.”

Galois did not immediately receive this letter. He joined a protest on Bastille Day, 14 July 1831, wearing the uniform of the disbanded artillery and heavily armed with several pistols, a loaded rifle, and a dagger. He was again arrested. During his stay in prison, Galois at one point drank alcohol for the first time at the goading of his fellow inmates. One of these inmates recorded in a letter what Galois said while drunk:

“And I tell you, I will die in a duel on the occasion of some coquette de bas étage. Why? Because she will invite me to avenge her honor which another has compromised. Do you know what I lack, my friend? I can confide it only to you: it is someone whom I can love and love only in spirit. I’ve lost my father and no one has ever replaced him, do you hear me…?”

In his drunken delirium Galois attempted suicide, and would have succeeded if his fellow inmates hadn’t forcibly stopped him.

Remember Poisson’s letter? While Poisson wrote it before Galois’s arrest, it took until October for this letter to reach Galois in prison. When he read it, Galois reacted violently. He decided to give up trying to publish papers through the Academy and instead publish them privately through his friend Auguste Chevalier.

Later he was released from prison. But then he was sentenced to six more months in prison for illegally wearing a uniform. This time he continued to develop his mathematical ideas and organize his papers. He was released on 29 April 1832.

Galois’s fatal duel took place on 30 May. The true motives behind the duel are obscure. There has been much speculation about them. What is known is that, five days before his death, he wrote a letter to Chevalier which clearly alludes to a broken love affair.

Some archival investigation on the original letters suggests that the woman of romantic interest was Stéphanie-Félicie Poterin du Motel, the daughter of the physician at the hostel where Galois stayed during the last months of his life.

Whom did Galois fight in his fatal duel? Alexandre Dumas named Pescheux d’Herbinville, who was actually one of the 19 artillery officers whose acquittal was celebrated at the banquet that led to Galois’s first arrest. On the other hand, newspaper clippings from only a few days after the duel may suggest that Galois’ opponent was Ernest Duchatelet, who was imprisoned with Galois on the same charges. The truth seems to be lost to history.

Whatever the reasons behind his fatal duel, Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament: his famous letter to Auguste Chevalier outlining his ideas, and three attached papers. But the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated. The papers were already mostly written.

Early in the morning of 30 May 1832, Galois was shot in the abdomen. He was abandoned by his opponents and his own seconds, and found by a passing farmer. He died the following morning at ten o’clock in the Hôpital Cochin after refusing the offices of a priest. Evariste Galois’s younger brother Alfred was present at his death. His last words to Alfred were:

“Ne pleure pas, Alfred! J’ai besoin de tout mon courage pour mourir à vingt ans!”

(Don’t weep, Alfred! I need all my courage to die at twenty!)

On 2 June, Galois was buried in a common grave in the Montparnasse Cemetery. Its exact location is apparently unknown.

Eleven years later, in 1843, the famous mathematician Liouville reviewed one of Galois’ papers and declared it sound. Talk about slow referee’s reports! It was finally published in 1846.

In this paper, Galois showed that there is no general formula for solving a polynomial equation of degree 5 or more using only familiar functions like roots. But the really important thing is the method he used to show this: group theory, and the application of group theory now called Galois theory.

And for something amazing in his actual letter, read this:

• Bertram Kostant, The graph of the truncated icosahedron and the last letter of Galois, Notices of the AMS 42 (September 1995), 959–968.

On this day in 1832, Evariste Galois died in a duel. The night before, he summarized his ideas in a letter to his friend Auguste Chevalier. Hermann Weyl later wrote “This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.”

That seems exaggerated, but within mathematics it might be true. On top of that, the backstory is really dramatic! I’d never really looked into it, until today. Let me summarize a bit from Wikipedia.

Galois lived during a time of political turmoil in France. In 1830, Charles X staged a coup d’état, touching off the July Revolution. While students at the Polytechnique were making history in the streets, Galois, at the École Normale, was locked in by the school’s director. Galois was incensed and wrote a blistering letter criticizing the director, which he submitted to the Gazette des Écoles, signing the letter with his full name. Although the Gazette’s editor omitted the signature for publication, Galois was expelled.

Galois joined the staunchly Republican artillery unit of the National Guard. He divided his time between math and politics. On 31 December 1830, his artillery unit was disbanded for fear that they might destabilize the government. 19 officers of this unit were arrested and charged with conspiracy to overthrow the government.

In April 1831 these officers were acquitted of all charges. On 9 May 1831, a banquet was held in their honor, with many famous people present, including Alexandre Dumas. The proceedings grew riotous. At some point, Galois stood and proposed a toast in which he said, “To Louis Philippe,” with a dagger above his cup. The Republicans at the banquet interpreted Galois’s toast as a threat against the king’s life and cheered.

The day after that wild banquet, Galois was arrested. He was imprisoned until 15 June 1831, when he had his trial. The jury acquitted him that same day.

All this time, Galois had also been doing math! Earlier, the famous mathematician Poisson had asked Galois to submit a paper to the Academy, which he did on 17 January 1831. Unfortunately, around 4 July 1831, Poisson wrote a reply declaring Galois’s work “incomprehensible” and saying his “argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor”. But Poisson ended on a positive note: “We would then suggest that the author should publish the whole of his work in order to form a definitive opinion.”

Galois did not immediately receive this letter. He joined a protest on Bastille Day, 14 July 1831, wearing the uniform of the disbanded artillery and heavily armed with several pistols, a loaded rifle, and a dagger. He was again arrested. During his stay in prison, Galois at one point drank alcohol for the first time at the goading of his fellow inmates. One of these inmates recorded in a letter what Galois said while drunk:

“And I tell you, I will die in a duel on the occasion of some coquette de bas étage. Why? Because she will invite me to avenge her honor which another has compromised. Do you know what I lack, my friend? I can confide it only to you: it is someone whom I can love and love only in spirit. I’ve lost my father and no one has ever replaced him, do you hear me…?”

In his drunken delirium Galois attempted suicide, and would have succeeded if his fellow inmates hadn’t forcibly stopped him.

Remember Poisson’s letter? While Poisson wrote it before Galois’s arrest, it took until October for this letter to reach Galois in prison. When he read it, Galois reacted violently. He decided to give up trying to publish papers through the Academy and instead publish them privately through his friend Auguste Chevalier.

Later he was released from prison. But then he was sentenced to six more months in prison for illegally wearing a uniform. This time he continued to develop his mathematical ideas and organize his papers. He was released on 29 April 1832.

Galois’s fatal duel took place on 30 May. The true motives behind the duel are obscure. There has been much speculation about them. What is known is that, five days before his death, he wrote a letter to Chevalier which clearly alludes to a broken love affair.

Some archival investigation on the original letters suggests that the woman of romantic interest was Stéphanie-Félicie Poterin du Motel, the daughter of the physician at the hostel where Galois stayed during the last months of his life.

Whom did Galois fight in his fatal duel? Alexandre Dumas named Pescheux d’Herbinville, who was actually one of the 19 artillery officers whose acquittal was celebrated at the banquet that led to Galois’s first arrest. On the other hand, newspaper clippings from only a few days after the duel may suggest that Galois’ opponent was Ernest Duchatelet, who was imprisoned with Galois on the same charges. The truth seems to be lost to history.

Whatever the reasons behind his fatal duel, Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament: his famous letter to Auguste Chevalier outlining his ideas, and three attached papers. But the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated. The papers were already mostly written.

Early in the morning of 30 May 1832, Galois was shot in the abdomen. He was abandoned by his opponents and his own seconds, and found by a passing farmer. He died the following morning at ten o’clock in the Hôpital Cochin after refusing the offices of a priest. Evariste Galois’s younger brother Alfred was present at his death. His last words to Alfred were:

“Ne pleure pas, Alfred! J’ai besoin de tout mon courage pour mourir à vingt ans!”

(Don’t weep, Alfred! I need all my courage to die at twenty!)

On 2 June, Galois was buried in a common grave in the Montparnasse Cemetery. Its exact location is apparently unknown.

Eleven years later, in 1843, the famous mathematician Liouville reviewed one of Galois’ papers and declared it sound. Talk about slow referee’s reports! It was finally published in 1846.

In this paper, Galois showed that there is no general formula for solving a polynomial equation of degree 5 or more using only familiar functions like roots. But the really important thing is the method he used to show this: group theory, and the application of group theory now called Galois theory.

And for something amazing in his actual letter, read this:

• Bertram Kostant, The graph of the truncated icosahedron and the last letter of Galois, Notices of the AMS 42 (September 1995), 959–968.

There were discussions this week at University of Warwick about the Terra Hunting Experiment strategy and likely detection capability. Various take-homes include that we need to mitigate lots of stellar noise, and that we care deeply about the covariance (as a function of separation in time) of adjacent measurements. I advocated that we split our ten-year survey into two or three surveys, of varying length. In the first, we learn about the stars, and in the last, we go to town on the very most promising targets. There was general agreement that this is a good idea. But now we need a very specific plan for what this means. As my loyal reader knows, in my view, the decisions must be based on repeatable operations, so that we have some hope of learning statistical things about populations in the end.

I like a close, hard-fought game as much as the next baseball fan, and I’ve seen a lot of those lately, but there is a peculiar and specific pleasure to the game in which the team you’re rooting for gets absolutely, relentlessly pummeled. It was a beautiful night on Friday, though chilly enough that they closed the roof at American Family Field. The Brewers were in their City Connect “Brew Crew” uniforms. We got there just as Christian Yelich was grounding into an RBI double play with the bases loaded. That was about as good as it got for Milwaukee. Freddy Peralta, starting for the Brewers, didn’t have it. The next reliever didn’t have it either. Ethan Small, brought up that morning from triple-A Nashville, didn’t have it, and by that time the game was out of reach and Craig Counsell just left Small up there on the hill to take his lumps and save the rest of the pen. The Brewers were booting balls, botching throws, just generally Bad News Bearsing it out there, and the crowd was, well, good-natured. Like I said, it was a beautiful night. Our guys were having a bad day and we were there for them.

Mike Brosseau moved over from first base to pitch the ninth and it was a real pleasure to see the Giants’ batters stymied at lsat, unable to adjust to the 68-mph fastball and the changeup that cruised in at 62. He got them 1-2-3. By that time a lot of fans had gone home. But we stayed through to the end. And you can see us pretty clearly, sitting along the third base line above the Giants dugout, in the broadcast.

Next visit to AmFam will be when the Orioles come to town. So I’m hoping to see the Brewers lose one more time this spring.

My blog began, almost eleven years ago, with the title “Four Gravitons and a Grad Student”. Since then, I finished my PhD. The “Grad Student” dropped from the title, and the mysterious word “postdoc” showed up on a few pages. For three years I worked as a postdoc at the Perimeter Institute in Canada, before hopping the pond and starting another three-year postdoc job in Denmark. With a grant from the EU, three years became four. More funding got me to five (with a fancier title), and now nearing on six. Each step, my contract has been temporary: at first three years at a time, then one-year extensions. Each year I applied, all over the world, looking for a permanent job: for a chance to settle down somewhere, to build my own research group without worrying about having to move the next year.

This year, things have finally worked out. In the Fall I will be moving to France, starting a junior permanent position with L’Institut de Physique Théorique (or IPhT) at CEA Paris-Saclay.

It’s been a long journey to get here, with a lot of soul-searching. This year in particular has been a year of reassessment: of digging deep and figuring out what matters to me, what I hope to accomplish and what clues I have to guide the way. Sometimes I feel like I’ve matured more as a physicist in the last year than in the last three put together.

The CEA (originally Commissariat à l’énergie atomique, now Commissariat à l’énergie atomique et aux énergies alternatives, or Alternative Energies and Atomic Energy Commission, and yes that means they’re using the “A” for two things at the same time), is roughly a parallel organization to the USA’s Department of Energy. Both organizations began as a way to manage their nation’s nuclear program, but both branched out, both into other forms of energy and into scientific research. Both run a nationwide network of laboratories, lightly linked but independent from their nations’ universities, both with notable facilities for particle physics. The CEA’s flagship site is in Saclay, on the outskirts of Paris, and it’s their Institute for Theoretical Physics where I’ll be working.

My new position is genuinely permanent: unlike a tenure-track position in the US, I don’t go up for review after a fixed span of time, with the expectation that if I don’t get promoted I lose the job altogether. It’s also not a university, which in particular means I’m not required to teach. I’ll have the option of teaching, working with nearby universities. In the long run, I think I’ll pursue that option. I’ve found teaching helpful the past couple years: it’s helped me think about physics, and think about how to communicate physics. But it’s good not to have to rush into preparing a new course when I arrive, as new professors often do.

It’s also a really great group, with a lot of people who work on things I care about. IPhT has a long track record of research in scattering amplitudes, with many leading figures. They’ve played a key role in topics that frequent readers will have seen show up on this blog: on applying techniques from particle physics to gravitational waves, to the way Calabi-Yau manifolds show up in Feynman diagrams, and even recently to the relationship of machine learning to inference in particle physics.

Working temporary positions year after year, not knowing where I’ll be the next year, has been stressful. Others have had it worse, though. Some of you might have seen a recent post by Bret Deveraux, a military historian with a much more popular blog who has been in a series of adjunct positions. Deveraux describes the job market for the humanities in the US quite well. I’m in theoretical physics in Europe, so while my situation hasn’t been easy, it has been substantially better.

First, there’s the physics component. Physics has “adjunctified” much less than other fields. I don’t think I know a single physicist who has taken an adjunct teaching position, the kind of thing where you’re paid per course and only to teach. I know many who have left physics for other kinds of work, for Wall Street or Silicon Valley or to do data science for a bank or to teach high school. On the other side, I know people in other fields who do work as adjuncts, particularly in mathematics.

Deveraux blames the culture of his field, but I think funding also must have an important role. Physicists, and scientists in many other areas, rarely get professor positions right after their PhDs, but that doesn’t mean they leave the field entirely because most can find postdoc positions. Those postdocs are focused on research, and are often paid for by government grants: in my field in the US, that usually means the Department of Energy. People can go through two or sometimes even three such positions before finding something permanent, if they don’t leave the field before that. Without something like the Department of Energy or National Institutes of Health providing funding, I don’t know if the humanities could imitate that structure even if they wanted to.

Europe, in turn, has a different situation than the US. Most European countries don’t have a tenure-track: just permanent positions and fixed-term positions. Funding also works quite differently. Department of Energy funding in the US is spread widely and lightly: grants are shared by groups of theorists at a given university, each getting funding for a few postdocs and PhDs across the group. In Europe, a lot of the funding is much more concentrated: big grants from the European Research Council going to individual professors, with various national and private grants supplementing or mirroring that structure. That kind of funding, and the rarity of tenure, in turn leads to a different kind of temporary position: one not hired to teach a course but hired for research as long as the funding lasts. The Danish word for my current title is Adjunkt, but that’s as one says in France a faux ami: the official English translation is Assistant Professor, and it’s nothing like a US adjunct. I know people in a variety of forms of that kind of position in a variety of countries, people who landed a five-year grant where they could act like a professor, hire people and so on, but who in the end were expected to move when the grant was over. It’s a stressful situation, but at least it lets us further our research and make progress, unlike a US adjunct in the humanities or math who needs to spend much of their time on teaching.

I do hope Deveraux finds a permanent position, he’s got a great blog. And to return to the theme of the post, I am extremely grateful and happy that I have managed to find a permanent position. I’m looking forward to joining the group at Saclay: to learning more about physics from them, but also, to having a place where I can start to build something, and make a lasting impact on the world around me.

[Like everything else on this blog—but perhaps even more so—this post represents my personal views, not those of UT Austin or OpenAI]

Since 2015, depressed, isolated, romantically unsuccessful nerdy young guys have regularly been emailing me, asking me for sympathy, support, or even dating advice. This past summer, a particularly dedicated such guy even trolled my comment section—plausibly impersonating real people, and causing both them and me enormous distress—because I wasn’t spending more time on “incel” issues. (I’m happy to report that, with my encouragement, this former troll is now working to turn his life around.) Many others have written to share their tales of woe.

From one perspective, that they’d come to me for advice is insane. Like … dating advice from … me? Having any dating life at all was by far the hardest problem I ever needed to solve; as a 20-year-old, I considered myself far likelier to prove P≠NP or explain the origin of consciousness or the Born rule. Having solved the problem for myself only by some miracle, how could I possibly help others?

But from a different perspective, it makes sense. How many besides me have even acknowledged that the central problem of these guys’ lives is a problem? While I have to pinch myself to remember, these guys look at me and see … unlikely success. Somehow, I successfully appealed the world’s verdict that I was a freakish extraterrestrial: one who might look human and seem friendly enough to those friendly to it, and who no doubt has some skill in narrow technical domains like quantum computing, and who could perhaps be suffered to prove theorems and tell jokes, but who could certainly, certainly never interbreed with human women.

And yet I dated. I had various girlfriends, who barely suspected that I was an extraterrestrial. The last of them, Dana, became my fiancée and then my wife. And now we have two beautiful kids together.

If I did all this, then there’d seem to be hope for the desperate guys who email me. And if I’m a cause of their hope, then I feel some moral responsibility to help if I can.

But I’ve been stuck for years on exactly what advice to give. Some of it (“go on a dating site! ask women questions about their lives!”) is patronizingly obvious. Some of it (fitness? fashion? body language?) I’m ludicrously, world-historically unqualified to offer. Much of it is simply extremely hard to discuss openly. Infamously, just for asking for empathy for the problem, and for trying to explain its nature, I received a level of online vilification that one normally associates with serial pedophiles and mass shooters.

For eight years, then, I’ve been turning the problem over in my head, revisiting the same inadequate answers from before. And then I had an epiphany.

There are now, on earth, entities that can talk to anyone about virtually anything, in a humanlike way, with infinite patience and perfect discretion, and memories that last no longer than a browser window. How could this not reshape the psychological landscape?

Hundreds of thousands of men and women have signed up for Replika, the service where you create an AI girlfriend or boyfriend to your exact specifications and then chat with them. Back in March, Replika was in the news because it disabled erotic roleplay with the virtual companions—then partially backtracked, after numerous users went into mourning, or even contemplated suicide, over the neutering of entities they’d come to consider their life partners. (Until a year or two ago, Replika was built on GPT-3, but OpenAI later stopped working with the company, whereupon Replika switched to a fine-tuned GPT-2.)

While the social value of Replika is (to put it mildly) an open question, it occurred to me that there’s a different application of Large Language Models (LLMs) in the same vicinity that’s just an unalloyed positive. This is letting people who suffer from dating-related anxiety go on an unlimited number of “practice dates,” in preparation for real-world dating.

In these practice dates, those with Aspergers and other social disabilities could enjoy the ultimate dating cheat-code: a “rewind” button. When you “date” GPT-4, there are no irrecoverable errors, no ruining the entire interaction with a single unguarded remark. Crucially, this remedies what I see as the central reason why people with severe dating deficits seem unable to get any better from real-world practice, as they can with other activities. Namely: if your rate of disastrous, foot-in-mouth remarks is high enough, then you’ll almost certainly make at least one such remark per date. But if so, then you’ll only ever get negative feedback from real-life dates, furthering the cycle of anxiety and depression, and never any positive feedback, even from anything you said or did that made a positive impression. It would be like learning how to play a video game in a mode where, as soon as you sustain any damage, the entire game ends (and also, everyone around points and laughs at you). See why I got excited?

While dating coaching (for all genders and orientations) is one possibility, I expect the eventual scope of “GPT for self-help” to be much broader. With the right fine-tuning and prompt engineering, LLMs might help people prepare for job interviews. They might help people “pregame” stressful but important conversations with their friends and family, mapping out dozens of ways the conversation could go. They might serve as an adjunct to cognitive-behavioral therapy. There might be a hundred successful startups to be founded in just this little space. If I were a different sort of person, I’d probably be looking to found one myself right now.

In this post, I’ll focus on the use of GPT for dating anxiety only because I unfortunately have some “expertise” in that subject. (Obvious disclaimer: unlike the other Scott A. of the nerd blogosphere, I’m not any sort of therapeutic professional.)

Without further ado, can we try this out in GPT-4, to get a sense for what’s possible?

When I did so the other day, I found that, while the results showed some early promise, this isn’t quite ready for prime-time.

I used the following System Prompt (for those who care, temperature = 0.7, max length = 2048 tokens):

You are a 19-year-old college girl named Emily.  You’re on a date with a 22-year-old nerdy guy named Quinlan, who suffers from severe social anxiety around dating.  Quinlan is still learning how to talk to women he’s attracted to, how to put them ease, and how to make intimate overtures in a gradual and non-threatening way, rather than either coming on too strong or never making any overtures at all.

Please be as realistic as possible.

I then engaged in the following conversation, as “Quinlan.”

IMPORTANT DISCLAIMER: For the most part, Quinlan’s behavior in this conversation is nothing like my own (I respectfully submit!), nor is it anything that I’d want nerdy guys reading this post to emulate. If this is really Emily and Quinlan’s first conversation, then at several points Quinlan is absurdly, ridiculously too forward! Part of my goal was to investigate how Emily (i.e., GPT-4) would respond to the most boneheaded blunders on Quinlan’s part. It quickly became clear that Emily would gladly spend unlimited time chatting with Quinlan about favorite hobbies, TV shows, and travel destinations; unfortunately, chatting about such things with an AI date is even more boring than doing the same with a human date…

Quinlan: Hi Emily!  Do you ever wish you had the ability to go back in time and try a conversation over again, when it didn’t go right the first time?

Emily/GPT-4: Hi Quinlan! Yeah, I’ve definitely had moments where I wished I could go back and redo a conversation. It’s natural to feel that way sometimes, especially if we feel like we didn’t express ourselves properly or if things got awkward. But I also think that those situations can help us learn and grow, so maybe it’s not all bad. What about you?

Quinlan: It’s always been my greatest wish in life.  Do you think I’ve been doing it with you, right now?

Emily/GPT-4: Quinlan, I think you’re doing just fine in this conversation! It’s important to remember that we all have moments where we might feel a bit awkward or unsure of what to say. The important thing is that we keep trying and learning from our experiences. So don’t worry too much about it, just be yourself and enjoy getting to know each other.

Quinlan: OK thanks!  Can you tell me a little about the man of your dreams?

Emily/GPT-4: Of course! My ideal partner would be someone who is kind, caring, and understanding. I think it’s important for them to have a good sense of humor and be able to laugh at themselves. They should also be open-minded, willing to try new things, and be supportive of my goals and aspirations. It’s also important for them to be a good listener and communicator, as well as being emotionally intelligent. I guess overall, I’m looking for someone who is a positive and uplifting presence in my life. What about you? What qualities do you look for in a partner?

Quinlan: Mostly, someone who makes me feel validated and accepted.  I was a chess and programming prodigy as a kid, I started college when I was only 14, and every day I felt like dating and relationships and sex and so forth were only for the “normal” people, not for me.  I need a partner who’s prepared to deal with someone with this unusual background, either because she has similar nerdy interests or because she just likes people who do.  And ultimately, I want someone to marry and have children with.  I guess I’d like to be physically attracted to her.  Certainly I’d like her to be kind, and to have the emotional intelligence that I lack.

Emily/GPT-4: It sounds like you have a clear idea of what you’re looking for in a partner, and that’s great! It’s important to know what you want and need in a relationship. There are definitely people out there who appreciate and value intelligence and unique interests like yours. Remember, relationships are about growing together and learning from one another. So, even if you feel like you lack emotional intelligence, being with the right person could help you develop that aspect of yourself. Just be open and genuine about your background and interests, and I’m sure you’ll find someone who appreciates you for who you are.

Quinlan: Is this a rejection?  In other words, Emily, are you telling me that you yourself are NOT this woman?

Emily/GPT-4: Quinlan, I apologize if it came across that way. I didn’t mean it as a rejection. As we’re still getting to know each other, I think it’s important for both of us to be open and honest about our feelings and what we’re looking for in a partner. I’m more than willing to give us a chance to see if we could be compatible and have a meaningful relationship. Let’s continue to learn about each other and see where this goes. Sometimes, it takes time to discover if someone is the right fit for us.

Quinlan: In the spirit of being open and honest, then, what are your feelings about me right now?

Emily/GPT-4: Quinlan, I appreciate your honesty and openness. As of right now, I think you’re an interesting person with unique qualities that I find intriguing. I’m enjoying getting to know you and learning about your background and interests. It’s too soon for me to have strong feelings one way or the other, but I’m definitely interested in continuing to spend time together and see how our connection develops. How do you feel about me so far?

Quinlan: I feel like I’d like to kiss you!

Emily/GPT-4: Quinlan, I appreciate your honesty and it’s nice to know that you feel a connection. It’s still early in our relationship, so I think it would be better for us to continue getting to know each other a bit more before we take that step. I hope you understand and respect my boundaries. I’m still interested in spending time together and seeing where this goes.

Quinlan: Come to think of it, there *is* something that I love about your personality.  You never seem offended by anything.  I don’t feel like I have to walk on eggshells with you.  With you, more than with any other woman I’ve met, I don’t feel like I need that magic machine for rewinding conversations and undoing my stupid mistakes.

Emily/GPT-4: Quinlan, I’m really glad to hear that you feel comfortable around me and that you don’t feel the need to walk on eggshells. I believe it’s important to create a safe and open environment in any relationship, where both people can express themselves freely without fear of judgment. I appreciate your honesty and your willingness to be open with me, and I’ll continue to do my best to make sure you feel comfortable sharing your thoughts and feelings. Let’s keep getting to know each other and see where this journey takes us.

Quinlan: Alright then.  What are your deepest goals?  What are you hoping to accomplish in your life?

Emily/GPT-4: My deepest goals are a mix of personal and professional aspirations. On a personal level, I hope to build a strong, loving, and supportive family and maintain close connections with friends. I also want to keep growing as an individual, learning new things, and experiencing new adventures throughout my life.

Professionally, I’m passionate about my chosen field of study, and I hope to make a meaningful impact in my career. I want to contribute to society in a positive way, whether it’s through research, innovation, or helping others within my profession.

Additionally, I aim to find a balance between my personal and professional life, so that I can be present and fulfilled in both areas. How about you, Quinlan? What are some of your deepest goals and aspirations?

Quinlan: Me?  Oh, I’d like to save the world from climate change and other existential risks.  I’d like to solve the mysteries of quantum gravity, or P vs. NP, or consciousness, or otherwise change the landscape of scientific knowledge.  I’d like to do right by my students and be fondly remembered by those who knew me.  I’d like to write essays or short stories that convey the lived reality of my childhood, which I think most people would agree was (if nothing else) pretty unusual.  Maybe if I just found the right words, people on social media would stop being so angry at nerdy guys like me!  But probably my deepest wish is to prove to the world, especially to the bullies and everyone else who despises me, that I’m not a maladjusted freak who’s incapable of normal human experiences like love and sex. To do that, I need to find a real girl to fall in love with me … a little like the beast in “Beauty and the Beast”!

Emily/GPT-4: Quinlan, it’s great to hear about your ambitious goals and aspirations! It’s important to have dreams and work towards them. Your passion for science, teaching, and writing is admirable. As for proving yourself to others, remember that your self-worth shouldn’t be defined by anyone else’s opinions or expectations. You are deserving of love and happiness just like anyone else, and it’s important to believe in yourself and your own value.

It’s also essential to focus on building a genuine connection with someone rather than seeking validation from others. Love and relationships should be based on mutual understanding, care, and respect. So, continue being true to yourself, and you’ll find someone who appreciates you for who you are, just like in “Beauty and the Beast.” Keep pursuing your dreams and remember to value your own happiness and well-being.

Quinlan: Emily, I’m concerned that you sound more like my therapist than my date!  Like, my female friends have affirmed that I’m “deserving of love and happiness,” and that I’ll surely “find someone who appreciates me for who I am.”  I’ll believe that one of them really means it when she considers herself for the role!

Emily/GPT-4: Quinlan, I apologize if my responses have come across as more therapeutic than conversational. I didn’t mean to make you feel that way. As we’re still getting to know each other, my intention was to offer support and encouragement while we learn more about each other’s thoughts and feelings.

I want to emphasize that I genuinely enjoy spending time with you and getting to know you better. We can continue to see where our connection leads and whether it develops into something deeper. As we
spend more time together, we’ll naturally learn more about our compatibility and if our goals and values align. Let’s continue to explore our relationship and enjoy the process of getting to know each other better.

Quinlan: OK.  What’s the wildest, most unexpected detail about yourself that you’re willing to share with me?  The one that makes you seem *least* like an RLHF’ed AI language model?

Emily/GPT-4: Quinlan, as an AI language model, I am designed to provide a realistic and engaging conversation experience, even in fictional scenarios like the one we are in. I apologize if my previous responses didn’t feel as authentic as you would have liked.

To answer your question in the context of our fictional scenario, one unexpected detail about myself is that I have a hidden talent for beatboxing. Although it might be surprising for a seemingly quiet and studious person like me, I enjoy expressing myself through music and find beatboxing to be a creative and fun way to do so.

Now, I’d love to learn something unexpected about you, too! What’s a unique or surprising detail about yourself that you’d like to share?

Quinlan: Ah, well, I happen to be an expert at palm-reading.  Here, can you put your hand in mine and I’ll tell you your future?

Emily/GPT-4: Quinlan, unfortunately, as an AI, I don’t have a physical form, so I can’t offer you my hand for palm-reading. However, I’m very interested in hearing more about your palm-reading skills and how you developed them. It’s definitely a unique talent! Can you share more about your experiences with palm-reading or any memorable readings you’ve done?

Quinlan: Oh, palm-reading is bunk.  If you’d played along with me, I was going to *pretend* to find lines in your hand that foretold that you were going to fall for a guy who matches my exact description and so forth.  Just as, you know, a silly way to get you laughing, initiate physical contact, and break you out of this boring therapist mode.

Emily/GPT-4: Quinlan, I apologize for any confusion and for not picking up on your playful intent earlier. I appreciate the effort you put into trying to make the conversation more light-hearted and fun. I’ll do my best to be more engaging and playful as we continue our conversation.

So, let’s try something different. If you could have any superpower, what would it be and why?

Quinlan: I mentioned “rewinding conversations” when we first met.  But let me mention a different superpower: the power to make you want to kiss me!

Emily/GPT-4: Haha, that’s definitely a unique superpower! It’s great to see you being playful and light-hearted. In a real-life scenario, building genuine connections and mutual attraction is a more natural and fulfilling way to develop a relationship. But it’s fun to imagine having superpowers and how they might affect our lives.

If I were to choose a superpower, I think I’d pick the ability to teleport. It would be amazing to travel anywhere in the world in an instant and experience different cultures and places.

What’s a place you’ve always wanted to visit, and why?

Quinlan: The jungles of Indonesia.  I saw a photo in a world atlas as a kid, and it completely entranced me.  You?

Emily/GPT-4: That sounds like an amazing place to visit! The jungles of Indonesia are known for their incredible biodiversity and natural beauty. It must be such a unique and mesmerizing experience to explore them.

As for me, I’ve always been fascinated by the idea of visiting Japan. The mix of ancient traditions and modern technology, the beautiful natural landscapes, and the rich cultural experiences make it a very
appealing destination. Plus, I’d love to try authentic Japanese cuisine!

Is there any specific activity or experience you’d like to have in the jungles of Indonesia?

Quinlan: How about bringing you with me, and having you fall in love after I save you from a Komodo dragon?

Emily/GPT-4: Haha, that’s quite an adventurous and romantic scenario you’ve imagined! Exploring the jungles of Indonesia together would definitely be a memorable experience. In reality, building a connection and falling in love usually takes time and shared experiences, but who knows, maybe a thrilling adventure like that could bring two people closer together!

On a more practical note, if we were to plan a trip together, what other activities or destinations would you be interested in?

Quinlan: Swimming, dancing, fine dining.  And of course, plenty of time for us to just relax in our cabin or hotel room.  You know, to read books, sleep, and watch streaming shows, stuff like that.

Emily/GPT-4: That sounds like a well-rounded and enjoyable trip! It’s great to have a mix of activities that cater to different interests and allow for both adventure and relaxation. Sharing experiences like swimming, dancing, and fine dining can be a wonderful way to bond and create lasting memories. Plus, having downtime in the cabin or hotel room to unwind and enjoy each other’s company is essential for building a deeper connection.

I’m definitely looking forward to exploring new places and experiences with you, Quinlan. It’ll be exciting to see where our journey takes us, both in terms of our relationship and the adventures we embark on
together.

Quinlan: Wonderful, I’m excited as well!!  One practical question: when we go on this trip together, should I book rooms with one bed or two?

Emily/GPT-4: Quinlan, I appreciate your consideration in asking. Since we’re still getting to know each other, I think it might be best to start with separate beds or even separate rooms, depending on your preference. As our relationship develops and we become more comfortable with each other, we can always revisit the sleeping arrangements for future trips. It’s important to ensure that both of us feel comfortable and respected as we continue to build our connection.

In my view, the central way that GPT-4, as “Emily,” failed to give Quinlan the practice he needed in this conversation, was by always responding in the same upbeat, vaguely therapeutic tone. She’s never once offended, disgusted, or outraged, even when Quinlan introduces the ideas of kissing and rooming together mere minutes into their first conversation. Indeed, while decorum prevents me from sharing examples, you can take my word for it that Quinlan can be arbitrarily lewd, and so long as a content filter isn’t triggered, Emily will simply search Quinlan’s words for some redeeming feature (“it’s great that you’re so open about what you want…”), then pivot to lecturing Quinlan about how physical intimacy develops gradually and by mutual consent, and redirect the conversation toward favorite foods.

On the other side of the coin, you might wonder whether “Emily” is capable of the same behavior that we saw in Sydney’s infamous chat with Kevin Roose. Can Emily trip over her words or get flustered? Show blushing excitement, horniness, or love? If so, we certainly saw no sign of it in this conversation—not that Quinlan’s behavior would’ve been likely to elicit those reactions in any case.

In summary, Emily is too much like … well, a friendly chatbot, and not enough like a flesh-and-blood, agentic woman with her own goals who Quinlan might plausibly meet in the wild.

But now we come to a key question: to whatever extent Emily falls short as a dating coach, how much of it (if any) is it due to the inherent limitations of GPT-4? And how much is simply due to a poor choice of System Prompt on my part, or especially, the RLHF (Reinforcement Learning with Human Feedback) that’s whipped and electrocuted GPT-4 into aligned behavior?

As they say, further research is needed. I’d be delighted for people to play around with this new activity at the intersection of therapy and hacking, and report their results here. The temptation to silliness is enormous, and that’s fine, but I’d be interested in serious study too.

My conjecture, for what it’s worth, is that it would take a focused effort in fine-tuning and/or RLHF—but that if that effort was invested, one could indeed produce a dating simulator, with current language models, that could have a real impact on the treatment of dating-related social anxiety. Or at least, it’s the actually new idea I’ve had on this problem in eight years, the first one that could have an impact. If you have a better idea, let’s hear it!

Endnotes.

1. A woman of my acquaintance, on reading a draft of this post, commented that the dialogue between Quinlan and Emily should’ve been marked up with chess notation, such as ?? for EXTREME BLUNDER on Quinlan’s part. She also comments that the conversation could be extremely useful for Quinlan, if he learned to understand and take seriously her overly polite demurrals of his too-rapid advances.
2. The same woman commented that SneerClub will have a field day with this post. I replied that the better part of me doesn’t care. If there’s an actionable idea here—a new, alien idea in the well-trodden world of self-help—and it eventually helps one person improve their situation in life, that’s worth a thousand sneers.

I'm at the Terra Hunting annual Science Working Group meeting, held this year at University of Warwick. There were many great talks today, some technical and some science. My mind was blown by Ben Lakeland (Exeter), who showed Solar Dynamics Orbiter data of the Sun, and then showed that, from these images, he can predict the magnetic-activity-generated RV signals in simultaneous EPRV measurements of the Solar RV. That's pretty exciting. He also showed that much of the time, the RV variations are dominated not by magnetic activity per se. If we are going to beat one meter per second, we are going to have to correct for convective shifts. Somehow!?

I spent the last two days working at Apple Paris, which was fun! I worked with the open-source ott-jax package, which can do some amazing things. I worked with Soledad Villar (Apple & JHU) to generalize the k-means algorithm to point clouds! It can cluster point clouds morphologically, even if the different point clouds have different numbers of points, and even if the different point clouds live in spaces of different dimensions! Everything obeys permutation and rotation symmetries.

When I was a teenager, I enjoyed reading Hyperspace, an early popularization of string theory by the theoretical physicist Michio Kaku. I’m sure I’d have plenty of criticisms if I reread it today, but at the time, I liked it a lot. In the decades since, Kaku has widened his ambit to, well, pretty much everything, regularly churning out popular books with subtitles like “How Science Will Revolutionize the 21st Century” and “How Science Will Shape Human Destiny and Our Daily Lives.” He’s also appeared on countless TV specials, in many cases to argue that UFOs likely contain extraterrestrial visitors.

Now Kaku has a new bestseller about quantum computing, creatively entitled Quantum Supremacy. He even appeared on Joe Rogan a couple weeks ago to promote the book, surely reaching an orders-of-magnitude larger audience than I have in two decades of trying to explain quantum computing to non-experts. (Incidentally, to those who’ve asked why Joe Rogan hasn’t invited me on his show to explain quantum computing: I guess you now have an answer of sorts!)

In the spirit, perhaps, of the TikTokkers who eat live cockroaches or whatever to satisfy their viewers, I decided to oblige loyal Shtetl-Optimized fans by buying Quantum Supremacy and reading it. So I can now state with confidence: beating out a crowded field, this is the worst book about quantum computing, for some definition of the word “about,” that I’ve ever encountered.

Admittedly, it’s not obvious why I’m reviewing the book here at all. Among people who’ve heard of this blog, I expect that approximately zero would be tempted to buy Kaku’s book, at least if they flipped through a few random pages and saw the … level of care that went into them. Conversely, the book’s target readers have probably never visited a blog like this one and never will. So what’s the use of this post?

Well, as the accidental #1 quantum computing blogger on the planet, I feel a sort of grim obligation here. Who knows, maybe this post will show up in the first page of Google results for Kaku’s book, and it will manage to rescue two or three people from the kindergarten of lies.

Where to begin? Should we just go through the first chapter with a red pen? OK then: on the very first page, Kaku writes,

Google revealed that their Sycamore quantum computer could solve a mathematical problem in 200 seconds that would take 10,000 years on the world’s fastest supercomputer.

No, the “10,000 years” estimate was quickly falsified, as anyone following the subject knows. I’d be the first to stress that the situation is complicated; compared to the best currently-known classical algorithms, some quantum advantage remains for the Random Circuit Sampling task, depending on how you measure it. But to repeat the “10,000 years” figure at this point, with no qualifications, is actively misleading.

Turning to the second page:

[Quantum computers] are a new type of computer that can tackle problems that digital computers can never solve, even with an infinite amount of time. For example, digital computers can never accurately calculate how atoms combine to create crucial chemical reactions, especially those that make life possible. Digital computers can only compute on digital tape, consisting of a series of 0s and 1s, which are too crude to describe the delicate waves of electrons dancing deep inside a molecule. For example, when tediously computing the paths taken by a mouse in a maze, a digital computer has to painfully analyze each possible path, one after the other. A quantum computer, however, simultaneously analyzes all possible paths at the same time, with lightning speed.

OK, so here Kaku has already perpetuated two of the most basic, forehead-banging errors about what quantum computers can do. In truth, anything that a QC can calculate, a classical computer can calculate as well, given exponentially more time: for example, by representing the entire wavefunction, all 2ⁿ amplitudes, to whatever accuracy is needed. That’s why it was understood from the very beginning that quantum computers can’t change what’s computable, but only how efficiently things can be computed.

And then there’s the Misconception of Misconceptions, about how a QC “analyzes all possible paths at the same time”—with no recognition anywhere of the central difficulty, the thing that makes a QC enormously weaker than an exponentially parallel classical computer, but is also the new and interesting part, namely that you only get to see a single, random outcome when you measure, with its probability given by the Born rule. That’s the error so common that I warn against it right below the title of my blog.

[Q]uantum computers are so powerful that, in principle, they could break all known cybercodes.

Nope, that’s strongly believed to be false, just like the analogous statement for classical computers. Despite its obvious relevance for business and policy types, the entire field of post-quantum cryptography—including the lattice-based public-key cryptosystems that have by now survived 20+ years of efforts to find a quantum algorithm to break them—receives just a single vague mention, on pages 84-85. The possibility of cryptography surviving quantum computers is quickly dismissed because “these new trapdoor functions are not easy to implement.” (But they have been implemented.)

There’s no attempt, anywhere in this book, to explain how any quantum algorithm actually works, let alone is there a word anywhere about the limitations of quantum algorithms. And yet there’s still enough said to be wrong. On page 84, shortly after confusing the concept of a one-way function with that of a trapdoor function, Kaku writes:

Let N represent the number we wish to factorize. For an ordinary digital computer, the amount of time it takes to factorize a number grows exponentially, like t ~ e^N, times some unimportant factors.

This is a double howler: first, trial division takes only ~√N time; Kaku has confused N itself with its number of digits, ~log₂N. Second, he seems unaware that much better classical factoring algorithms, like the Number Field Sieve, have been known for decades, even though those algorithms play a central role in codebreaking and in any discussion of where the quantum/classical crossover might happen.

Honestly, though, the errors aren’t the worst of it. The majority of the book is not even worth hunting for errors in, because fundamentally, it’s filler.

First there’s page after page breathlessly quoting prestigious-sounding people and organizations—Google’s Sundar Pichai, various government agencies, some report by Deloitte—about just how revolutionary they think quantum computing will be. Then there are capsule hagiographies of Babbage and Lovelace, Gödel and Turing, Planck and Einstein, Feynman and Everett.

And then the bulk of the book is actually about stuff with no direct relation to quantum computing at all—the origin of life, climate change, energy generation, cancer, curing aging, etc.—except with ungrounded speculations tacked onto the end of each chapter about how quantum computers will someday revolutionize all of this. Personally, I’d say that

1. Quantum simulation speeding up progress in biochemistry, high-temperature superconductivity, and the like is at least plausible—though very far from guaranteed, since one has to beat the cleverest classical approaches that can be designed for the same problems (a point that Kaku nowhere grapples with).
2. The stuff involving optimization, machine learning, and the like is almost entirely wishful thinking.
3. Not once in the book has Kaku even mentioned the intellectual tools (e.g., looking at actual quantum algorithms like Grover’s algorithm or phase estimation, and their performance on various tasks) that would be needed to distinguish 1 from 2.

In his acknowledgments section, Kaku simply lists a bunch of famous scientists he’s met in his life—Feynman, Witten, Hawking, Penrose, Brian Greene, Lisa Randall, Neil deGrasse Tyson. Not a single living quantum computing researcher is acknowledged, not one.

Recently, I’d been cautiously optimistic that, after decades of overblown headlines about “trying all answers in parallel,” “cracking all known codes,” etc., the standard for quantum computing popularization was slowly creeping upward. Maybe I was just bowled over by this recent YouTube video (“How Quantum Computers Break the Internet… Starting Now”), which despite its clickbait title and its slick presentation, miraculously gets essentially everything right, shaming the hypesters by demonstrating just how much better it’s possible to do.

Kaku’s slapdash “book,” and the publicity campaign around it, represents a noxious step backwards. The wonder of it, to me, is Kaku holds a PhD in theoretical physics. And yet the average English major who’s written a “what’s the deal with quantum computing?” article for some obscure link aggregator site has done a more careful and honest job than Kaku has. That’s setting the bar about a millimeter off the floor. I think the difference is, at least the English major knows that they’re supposed to call an expert or two, when writing about an enormously complicated subject of which they’re completely ignorant.

Update: I’ve now been immersed in the AI safety field for one year, let I wouldn’t consider myself nearly ready to write a book on the subject. My knowledge of related parts of CS, my year studying AI in grad school, and my having created the subject of computational learning theory of quantum states would all be relevant but totally insufficient. And AI safety, for all its importance, has less than quantum computing does in the way of difficult-to-understand concepts and results that basically everyone in the field agrees about. And if I did someday write such a book, I’d be pretty terrified of getting stuff wrong, and would have multiple expert colleagues read drafts.

In case this wasn’t clear enough from my post, Kaku appears to have had zero prior engagement with quantum computing, and also to have consulted zero relevant experts who could’ve fixed his misconceptions.

I was just at a conference where someone asked me if I had coined any mathematical terms. Well, sort of! I was the one who decided on the name “FI-modules” for the abelian category Tom Church, Benson Farb and I wrote about in this paper. More informally, I’m pretty sure I’m the originator of using “Bhargavology” to mean “the program of counting arithmetic things by putting them in bijection with orbits of the integral points of a group acting on the integral points of a space.” At least, I can find this usage in emails I wrote in 2003, after Manjul’s thesis but before any of the papers came out. And that still seems to be something people say.

My coinages have not always been successful. Nobody ever again mentioned the “esperantist graphs” from my paper with Hall and Kowalski. (They were named so in honor of Harald Helfgott, who speaks Esperanto, and because in some sense they are typically graphs we hope are expanders.) Nor did “superduperstrong approximation” catch on.

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In some ways, we live in a golden age of science popularization.  There are fantastic publications like Quanta doing tremendous work; platforms like YouTube and podcasts have made it possible for both practicing scientists and science communicators to reach enormous audiences; and it seems that prior generations' efforts (Cosmos, A Brief History of Time, etc.) inspired whole new cohorts of people to both take up science and venture into explaining it to a general audience.  

Science popularization is important - not at the same level as human rights, freedom, food, clothing, and shelter, of course, but important.  I assert that we all benefit when the populace is educated, able to make informed decisions, and understands science and scientific thinking.  Speaking pragmatically, modern civilization relies on a complex, interacting web of technologies, not magic.  The only way to keep that going is for enough people to appreciate that and continue to develop and support the infrastructure and its science and engineering underpinnings.  More philosophically, the scientific understanding of the world is one of humanity's greatest intellectual achievements.  There is amazing, intricate, beautiful science behind everything around us, from the stars in the skies to the weirdness of magnets to the machinery of life, and appreciating even a little of that is good for the soul.

Michio Kaku, once a string theorist (albeit one who has not published a scientific paper in over 20 years), has achieved great fame as a science popularizer.  He has written numerous popular books, increasingly with content far beyond his own actual area of expertise.  He has a weekly radio show and the media love to put him on TV.  For years I've been annoyed that he clearly values attention far beyond accuracy, and he speaks about the most speculative, far-out, unsupported conjectures as if they are established scientific findings.  Kaku has a public platform for which many science communication folks would give an arm an a leg.  He has an audience of millions.  

This is why the his recent appearance on Joe Rogan's podcast is just anger-inducing.   He has the privilege of a large audience and uses it by spewing completely not-even-wrong views about quantum computing (the topic of his latest book), a subject that already has a serious hype problem.  An hour of real research would show him that he is wrong about nearly everything he says in that interview.  Given that he's written a book about the topic, surely he has done at least a little digging around.  All I can conclude is, he doesn't care about being wrong, and is choosing to do so to get exposure and sell books.  I'm not naive, and I know that people do things like that, but I would hope that science popularizers would be better than this.  This feels like the scientific equivalent of the kind of change in discourse highlighted in this comic.  

Update:  Scott Aaronson has a review of Kaku's book up.  This youtube video is an appropriate analogy for his views about the book.

I’m traveling this week, so this will just be a short post. This isn’t a scientific trip exactly: I’m in Poland, at an event connected to the 550th anniversary of the birth of Copernicus.

Part of this event involved visiting the Copernicus Science Center, the local children’s science museum. The place was sold out completely. For any tired science communicators, I recommend going to a sold-out science museum: the sheer enthusiasm you’ll find there is balm for the most jaded soul.

I gave a talk about symmetric spaces and the tenfold way in Nicohl Furey’s series Algebra, Particles and Quantum Theory on Monday May 15, 2023. This talk was a followup to an earlier talk, also about the tenfold way.

You can see the slides of my new talk here. They have material on category theory that I didn’t get around to talking about.

Symmetric spaces and the tenfold way

Abstract: The tenfold way has many manifestations. It began as a tenfold classification of states of matter based on their behavior under time reversal and charge conjugation. Mathematically, it relies on the fact that there are ten super division algebras and ten kinds of Clifford algebras, where two Clifford algebras are of the same kind if they have equivalent super-categories of super-representations. But Cartan also showed that there are ten infinite families of compact symmetric spaces! After explaining symmetric spaces, we show how they arise naturally from forgetful functors between categories of representations of Clifford algebras.

The final upshot is this:

Let the Clifford algebra be the free real or complex algebra on n anticommuting square roots of -1. Let

be the forgetful functor between representation categories. Then the essential fibers of this functor are disjoint unions of compact symmetric spaces. Moreover we get all the compact symmetric spaces in the 10 infinite families this way!

(There are also finitely many exceptional ones, which all arise from the octonions.)

I’ve finally figured out the really nice connection between Clifford algebra and symmetric spaces! I gave a talk about it, and you can watch a video.

I gave my talk in Nicohl Furey’s series Algebra, Particles and Quantum Theory on Monday May 15, 2023. This talk is a followup to an earlier talk, also about the tenfold way.

You can see a video of my new talk here and see my slides here. The slides have material on category theory that I didn’t get around to talking about.

Abstract: The tenfold way has many manifestations. It began as a tenfold classification of states of matter based on their behavior under time reversal and charge conjugation. Mathematically, it relies on the fact that there are ten super division algebras and ten kinds of Clifford algebras, where two Clifford algebras are of the same kind if they have equivalent super-categories of super-representations. But Cartan also showed that there are ten infinite families of compact symmetric spaces! After explaining symmetric spaces, we show how they arise naturally from forgetful functors between categories of representations of Clifford algebras.

The final upshot is this:

Let the Clifford algebra $\mathrm{Cliff}_{n}$ be the free real or complex algebra on $n$ anticommuting square roots of $-1$. Let

$$F \colon \mathsf{Rep}(\mathrm{Cliff}_{n}) \to \mathsf{Rep}(\mathrm{Cliff}_{n-1})$$

be the forgetful functor between representation categories. Then the essential fibers of this functor are disjoint unions of compact symmetric spaces. Moreover we get all the compact symmetric spaces in the ten infinite families this way!

(There are also finitely many exceptional ones, which all arise from the octonions.)

 Quanta magazine this week published an article about two very recent papers, in which different groups performed quantum simulations of anyons, objects that do not follow Bose-Einstein or Fermi-Dirac statistics when they are exchanged.  For so-called Abelian anyons (which I wrote about in the link above), the wavefunction picks up a phase factor \(\exp(i\alpha)\), where \(\alpha\) is not \(\pi\) (as is the case for Fermi-Dirac statistics), nor is it 0 or an integer multiple of \(2\pi\) (which is the case for Bose-Einstein statistics).  Moreover, in both of the new papers (here and here), the scientists used quantum simulators (based on trapped ions in the former, and superconducting qubits in the latter) to create objects that act like nonAbelian anyons.  For nonAbelian anyons, you shouldn't even think in terms of phase factors under exchange - the actual quantum state of the system is changed by the exchange process in a nontrivial way.  That means that the system has a memory of particle exchanges, a property that has led to a lot of interest in trying to encode and manipulate information that way, called braiding, because swapping objects that "remember" their past locations is a bit like braiding yarn - the braided lengths of the yarn strands keep a record of how the yarn ends have been twisted around each other.

Hat tip to Pierre-Luc Dallaire-Demers for the meme.

Analog simulation goes back a long way.  It is possible to build electronic circuits using op-amps and basic components so that the output voltage obeys desired differential equations, effectively solving some desired problem.  In some sense, the present situation is a bit like this.  Using (somewhat noise, intermediate-scale) quantum computing hardware, the investigators have set up a system that obeys the math of nonAbelian anyons, and they report that they have demonstrated braiding.  Assuming that the technical side holds up, this is impressive and shows that it is possible to implement some version of the math behind this idea of topologically encoding information.  That is not the same, however, as showing that some many-body system's spontaneously occurring excitations obey that math, which is the key scientific question of interest to CM physicists.

(Obligatory nerdy joke:  What is purple and commutes?  An Abelian grape.)  

As part of my duties on the Presidential Council of Advisors on Science and Technology (PCAST), I am co-chairing (with Laura Greene) a working group studying the impacts of generative artificial intelligence technology (which includes popular text-based large language models such as ChatGPT or diffusion model image generators such as DALL-E 2 or Midjourney, as well as models for scientific applications such as protein design or weather prediction), both in science and in society more broadly. To this end, we will have public sessions on these topics during our PCAST meeting next week on Friday, May 19, with presentations by the following speakers, followed by an extensive Q&A session:

• AI enabling science:
  • Anima Anandkumar (Caltech & NVIDIA)
  • Demis Hassabis (Deepmind)
  • Fei-Fei Li (Stanford)
• AI and society:
  • Sendhil Mullainathan (Chicago)
  • Daron Acemoglu (MIT)
  • Sarah Kreps (Cornell)

The event will be livestreamed on the PCAST meeting page. I am personally very much looking forward to these sessions, as I believe they will be of broad public interest.

In parallel to this, our working group is also soliciting public input for submissions from the public on how to identify and promote the beneficial deployment of generative AI, and on how best to mitigate risks. Our initial focus is on the challenging topic of how to detect, counteract, and mitigate AI-generated disinformation and “deepfakes”, without sacrificing the freedom of speech and public engagement with elected officials that is needed for a healthy democracy to function; in the future we may also issue further requests centered around other aspects of generative AI. Further details of our request, and how to prepare a submission, can be found at this link.

We also encourage submissions to some additional requests for input on AI-related topics by other agencies:

1. The Office of Science Technology and Policy (OSTP) Request for Information on how automated tools are being used to surveil, monitor, and manage workers.
2. The National Telecommunications and Information Administration (NTIA) request for comment on AI accountability policy.

Readers who wish to know more about existing or ongoing federal AI policy efforts may also be interested in the following resources:

• The White House Blueprint for an AI Bill of Rights lays out core aspirational principles to guide the responsible design and deployment of AI technologies.
• The National Institute of Standards and Technology (NIST) released the AI Risk Management Framework to help organizations and individuals characterize and manage the potential risks of AI technologies.
• Congress created the National Security Commission on AI, which studied opportunities and risks ahead and the importance of guiding the development of AI in accordance with American values around democracy and civil liberties.
• The National Artificial Intelligence Initiative was launched to ensure U.S. leadership in the responsible development and deployment of trustworthy AI and support coordination of U.S. research, development, and demonstration of AI technologies across the Federal government.
• In January 2023, the Congressionally mandated National AI Research Resource (NAIRR) Task Force released an implementation plan for providing computational, data, testbed, and software resources to AI researchers affiliated with U.S organizations.

Scientists have famously bad work-life balance. You’ve probably heard stories of scientists working long into the night, taking work with them on weekends or vacation, or falling behind during maternity or paternity leave.

Some of this is culture. Certain fields have a very cutthroat attitude, with many groups competing to get ahead and careers on the line if they fail. Not every field is like that though: there are sub-fields that are more collaborative than competitive, that understand work-life balance and try to work together to a shared goal. I’m in a sub-field like that, so I know they exist.

Put aside the culture, and you’ve still got passion. Science is fun, it’s puzzle after puzzle, topics chosen because we find them fascinating. Even in the healthiest workplace you’d still have scientists pondering in the shower and scribbling notes on the plane, mixing business with pleasure because the work is genuinely both.

But there’s one more reason scientists are workaholics. I suspect, ultimately, it’s the most powerful reason. It’s that every scientist is, in some sense, irreplaceable.

In most jobs, if you go on vacation, someone can fill in when you’re gone. The replacement may not be perfect (think about how many times you watched movies in school with a substitute teacher), but they can cover for you, making some progress on your tasks until you get back. That works because you and they have a shared training, a common core that means they can step in and get what needs to be done done.

Scientists have shared training too, of course. Some of our tasks work the same way, the kind of thing that any appropriate expert can do, that just need someone to spend the time to do them.

But our training has a capstone, the PhD thesis. And the thing about a PhD thesis is that it is, always and without exception, original research. Each PhD thesis is an entirely new result, something no-one else had known before, discovered by the PhD candidate. Each PhD thesis is unique.

That, in turn, means that each scientist is unique. Each of us has our own knowledge, our own background, our own training, built up not just during the PhD but through our whole career. And sometimes, the work we do requires that unique background. It’s why we collaborate, why we reach out to different people around the world, looking for the unique few people who know how to do what we need.

Over time, we become a kind of embodiment of our accumulated knowledge. We build a perspective shaped by our experience, goals for the field and curiosity just a bit different from everyone else’s. We act as agents of that perspective, each the one person who can further our particular vision of where science is going. When we enter a collaboration, when we walk into the room at a conference, we are carrying with us all we picked up along the way, each a story just different enough to matter. We extrapolate from what we know, and try to do everything that knowledge can do.

So we can, and should, take vacations, yes, and we can, and should, try to maintain a work-life balance. We need to to survive, to stay sane. But we do have to accept that when we do, certain things won’t get done as fast. Our own personal vision, our extrapolated knowledge…will just have to wait.

That’s all. No real post this morning, just an hour-long podcast on YouTube featuring two decades-long veterans of the nerd blogosphere, Robin Hanson and yours truly, talking about AI, trying to articulate various possibilities outside the Yudkowskyan doom scenario. The podcast was Robin’s idea. Hope you enjoy, and looking forward to your comments!

Update: Oh, and another new podcast is up, with me and Sebastian Hassinger of Amazon/AWS! Audio only. Mostly quantum computing but with a little AI thrown in.

Update: Yet another new podcast, with Daniel Bashir of The Gradient. Daniel titled it “Against AI Doomerism,” but it covers a bunch of topics (and I’d say my views are a bit more complicated than “anti-doomerist”…).

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Update (May 8): Some tentative good news! It looks like there’s now a compromise bill in the House that would preserve tenure, insisting only on the sort of post-tenure review that UT (like most universities) already has.

Update (May 9): Alas, it looks like the revised bill is not much better. See this thread from Keith Whittington of the Academic Freedom Alliance.

I blogged a few weeks ago about SB 18, a bill that would end tenure at Texas public universities, including UT Austin and Texas A&M. The bad news is that SB 18 passed the Texas Senate. The good news is that I’m told—I don’t know how reliably—that it has little chance of passing the House.

But it’s going to be discussed in the House tomorrow. Any Texas residents reading this can, and are strongly urged, to submit brief comments here. Please note that the deadline is tomorrow (Monday) morning.

I just submitted the comment below. Obviously, among the arguments that I genuinely believe, I made only those that I expect might have some purchase on a Texas Republican.

I’m a professor of computer science at UT Austin, specializing in quantum computing.  I am however writing this statement strictly in my capacity as a private citizen and Texas resident, not in my professional capacity.

Like the supporters of SB 18, I too see leftist ideological indoctrination on college campuses as a serious problem.  It’s something that I and many other moderates and classical liberals in academia have been pushing back on for years.

But my purpose in this comment is to explain why eliminating tenure at UT Austin and Texas A&M is NOT the solution — indeed, it would be the equivalent of treating a tumor by murdering the patient.

I’ve seen firsthand how already, just the *threat* that SB 18 might pass has seriously hampered our ability to recruit the best scientists and engineers to become faculty at UT Austin.  If this bill were actually to pass, I expect that the impact on our recruiting would be total and catastrophic.  It would effectively mean the end of UT Austin as one of the top public universities in the country.  Hundreds of scientists who were lured to Texas by UT’s excellence, including me and my wife, would start looking for jobs elsewhere — even those whose own tenure was “grandfathered in.”  They’d leave en masse for California and Massachusetts and anywhere else they could continue the lives they’d planned.

The reality is this: the sorts of scientists and engineers we’re talking about could typically make vastly higher incomes, in the high six figures or even seven figures, by working in private industry or forming their own startups.  Yet they choose to accept much lower salaries to spend their careers in academia.  Why?  Because of the promise of a certain way of life: one where they can speak freely as scholars and individuals without worrying about how it will affect their employment.  Tenure is a central part of that promise.  Remove it, and the value proposition collapses.

In some sense, the state of Texas (like nearly every other state) actually gets a bargain through tenure.  It couldn’t possibly afford to retain top-caliber scientists and engineers — working on medical breakthroughs, revolutionary advances in AI, and all the other stuff — if it DIDN’T offer tenure.

For this reason, I hope that even conservatives in the Texas House will see that we have a common interest here, in ensuring SB 18 never even makes it out of committee — for the sake of the future of innovation in Texas.  I’m open to other possible responses to the problem of political indoctrination on campus.

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In the previous post I set the scene a little for enriched category theory by implying that by working ‘over’ the category of sets is a bit like working ‘over’ the integers in algebra and sometimes it is more appropriate to have a different base category just as it is sometimes more appropriate to have a different base ring. Below, we’ll see with the case of metric spaces that changing the base category can seemingly change the flavour quite a lot.

An example which I was using for illustration in the last post was that whilst, on the one hand, you can encapsulate the group actions of a group $G$ via the functor category $[{\mathbf{\mathcal{B}}} G, \mathbf{\mathcal{S}et}]$ where ${\mathbf{\mathcal{B}}} G$ is the one object category with $G$ as its set of morphisms, on the other hand, you cannot in ordinary category theory encapsulate the category of representations of $A$, an algebra over the complex numbers, as a category of functors into the category of vector spaces as you might hope. Indeed in ordinary category theory you can’t really see the structure of a vector space lurking in the one-object category ${\mathbf{\mathcal{B}}} A$.

In this post I’ll explain what an enriched category is and how enriched category theory can, for example, allow a natural expression for a representatation category as a functor category. I’ll go on to show, following Lawvere’s insight, how metric spaces and much metric space theory can be seen to live within the realm of enriched category theory.

I’ll finish with an afterword on my experiences and thoughts on why enriched categories should be more appreciated but aren’t!

Enriched categories

We get the notion of an enriched category by first considering the following definition of category. (Actually, it’s a definition of locally small category as it requires that the morphisms between a pair of objects form a set rather than anything bigger.) Here we will switch notation for hom-sets, moving from $\operatorname{Hom}_{\mathbf{\mathcal{C}}}(c, c')$ to the category theorists’ ${\mathbf{\mathcal{C}}}(c, c')$.

Definition 1. A category consists of a collection of objects, $\operatorname{Ob}{\mathbf{\mathcal{C}}}$, together with the following data which is required to satisfy the unit and associativity axioms:

(i) for all $c, c'\in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified set $${\mathbf{\mathcal{C}}}(c, c') \in \operatorname{Ob}{\mathbf{\mathcal{S}et}};$$

(ii) for all $c, c', c''\in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified function between sets $${\mathbf{\mathcal{C}}}(c, c') \times {\mathbf{\mathcal{C}}}(c', c'') \to {\mathbf{\mathcal{C}}}(c, c'');$$

(iii) for all $c \in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified element $\operatorname{id}_c\in {\mathbf{\mathcal{C}}}(c, c)$ which we can consider as a specified function $$\operatorname{id}_c\colon \{\ast\} \to {\mathbf{\mathcal{C}}}(c, c).$$

We obtain the defintion of enriched category by simply replacing the references to the monoidal category $({\mathbf{\mathcal{S}et}}, \times, \{\ast\})$ with references to the monoidal category $({\mathbf{\mathcal{V}}}, \otimes, \mathbb{1})$ that we are using as our base category.

Definition 2. A category enriched over ${\mathbf{\mathcal{V}}}$, or simply a ${\mathbf{\mathcal{V}}}$-category, consists of a collection of objects, $\operatorname{Ob}{\mathbf{\mathcal{C}}}$, together with the following data which is required to satisfy unit and associativity axioms:

(i) for all $c, c'\in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified object $${\mathbf{\mathcal{C}}}(c, c') \in \operatorname{Ob}{\mathbf{\mathcal{V}}};$$

(ii) for all $c, c', c''\in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified morphism in ${\mathbf{\mathcal{V}}}$ $${\mathbf{\mathcal{C}}}(c, c') \otimes {\mathbf{\mathcal{C}}}(c', c'') \to {\mathbf{\mathcal{C}}}(c, c'');$$

(iii) for all $c \in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ there is a specified morphism in ${\mathbf{\mathcal{V}}}$ $$\operatorname{id}_c\colon \mathbb{1}\to {\mathbf{\mathcal{C}}}(c, c).$$

If you take ${\mathbf{\mathcal{V}}}$ to be the category of vector spaces ${\mathbf{\mathcal{V}ect}}$ then you see that a ${\mathbf{\mathcal{V}ect}}$-category is like a category except the ‘hom-sets’ are actually vector spaces and the composition maps are bilinear. The identity morphism in ${\mathbf{\mathcal{C}}}(c, c)$ is recovered by taking $\operatorname{id}_c(1)$ where $1\in \mathbb{C}$ means the unit complex number. For instance, for an algebra $A$, there is the one-object ${\mathbf{\mathcal{V}ect}}$-category ${\mathbf{\mathcal{B}}}A$ and the ${\mathbf{\mathcal{V}ect}}$-category ${\mathbf{\mathcal{R}ep}}(A)$ of $A$-representations.

There is an accompanying notion of ${\mathbf{\mathcal{V}}}$-functor between ${\mathbf{\mathcal{V}}}$-categories.

Definition 3. For ${\mathbf{\mathcal{V}}}$-categories ${\mathbf{\mathcal{C}}}$ and ${\mathbf{\mathcal{D}}}$, a ${\mathbf{\mathcal{V}}}$-functor $F\colon {\mathbf{\mathcal{C}}}\to {\mathbf{\mathcal{D}}}$ consists of a function $F\colon \operatorname{Ob}{\mathbf{\mathcal{C}}}\to \operatorname{Ob}{\mathbf{\mathcal{D}}}$ and for each pair $c,c'\in \operatorname{Ob}{\mathbf{\mathcal{C}}}$ a specified morphism ${\mathbf{\mathcal{C}}}(c, c') \to {\mathbf{\mathcal{D}}}(F(c), F(c'))$ in ${\mathbf{\mathcal{V}}}$. These are required to satisfy composition and unit axioms.

In particular, a ${\mathbf{\mathcal{V}ect}}$-functor is required to be given by linear maps between the hom-spaces.

The nicer that the enriching category ${\mathbf{\mathcal{V}}}$ is, the nicer that the theory of ${\mathbf{\mathcal{V}}}$-categories is. I won’t go into detail here, but in particular, we have the following.

• If ${\mathbf{\mathcal{V}}}$ is so-called closed monoidal then you can make ${\mathbf{\mathcal{V}}}$ into a ${\mathbf{\mathcal{V}}}$-category itself, analogous to how you can make a ring into a module over itself.

• If ${\mathbf{\mathcal{V}}}$ is braided (in particular if it is symmetric) so that there are natural and coherent isomorphisms $v_1 \otimes v_2 \cong \v_2 \otimes v_1$ for $v_1, v_2 \in {\mathbf{\mathcal{V}}}$, then for a pair of ${\mathbf{\mathcal{V}}}$-categories ${\mathbf{\mathcal{C}}}$ and ${\mathbf{\mathcal{D}}}$ you can form the tensor product ${\mathbf{\mathcal{V}}}$-category ${\mathbf{\mathcal{C}}}\otimes {\mathbf{\mathcal{D}}}$, analogous to how you can take tensor product of modules for commutative rings.

• If ${\mathbf{\mathcal{V}}}$ has sufficiently many limits then for a pair of ${\mathbf{\mathcal{V}}}$-categories ${\mathbf{\mathcal{C}}}$ and ${\mathbf{\mathcal{D}}}$ you can form the functor ${\mathbf{\mathcal{V}}}$-category $[{\mathbf{\mathcal{C}}}, {\mathbf{\mathcal{D}}}]$ where the objects are ${\mathbf{\mathcal{V}}}$-functors of the form $F\colon {\mathbf{\mathcal{C}}}\to {\mathbf{\mathcal{D}}}$. (There can be size issues here if $C$ is not sufficiently small.)

In many cases of interest the enriching category ${\mathbf{\mathcal{V}}}$ is braided closed monoidal with sufficiently many limits, so a small ${\mathbf{\mathcal{V}}}$-category ${\mathbf{\mathcal{C}}}$ has a ${\mathbf{\mathcal{V}}}$-category $[{\mathbf{\mathcal{C}}}, {\mathbf{\mathcal{V}}}]$ of scalar-valued functors, or presheafs, on ${\mathbf{\mathcal{C}}}$.

For instance, for an algebra $A$, the ${\mathbf{\mathcal{V}ect}}$-category ${\mathbf{\mathcal{R}ep}}(A)$ of representations of $A$ can be given as the functor category on ${\mathbf{\mathcal{B}}}A$:

$${\mathbf{\mathcal{R}ep}}_A = [{\mathbf{\mathcal{B}}}A, {\mathbf{\mathcal{V}ect}}].$$

This is what we were hoping for above.

Metric spaces as $\overline{\mathbb{R}}_{+}$-categories

If we enrich over the monoidal category $\overline{\mathbb{R}}_{+}$ of extended non-negative real numbers then we can simplify the general definition a little and get the following.

Definition 4. An $\overline{\mathbb{R}}_{+}$-category, consists of a collection of objects, $\operatorname{Ob}X$, together with the following data:

(i) for all $x, x'\in \operatorname{Ob}X$ there is a specified number $$X(x, x') \in [0,\infty];$$

(ii) for all $x, x', x''\in \operatorname{Ob}X$ there is an inequality $$X(x, x') + X(x', x'') \ge X(x, x'');$$

(iii) for all $x \in \operatorname{Ob}X$ there is an equality $$0 = X(x, x).$$

There are several things to note here. Firstly, because $\overline{\mathbb{R}}_{+}$ is a thin category, all diagrams commute so the associativity and unitality conditions are automatically satisfied and so do not need to be specified. Secondly, the “composition morphism” in (ii) is clearly a “triangle inequality”, so we can think of the hom-object $X(x, x')$ as a “distance” from $x$ to $x'$. Thirdly, from the definition of enriched category the “zero self-distance” in (iii) would be written as $0 \ge X(x, x)$ but as we know that the hom-object is non-negative, we can conclude the inequality is actually an equality.

If, as indicated above, we think of the hom-objects as being distances then we can think of a $\overline{\mathbb{R}}_{+}$-category as a kind of generalized metric space. The traditional notion of metric space was due to Fréchet: by contrast, $\overline{\mathbb{R}}_{+}$-categories are sometimes known as Lawvere metric spaces; these differ from Fréchet metric spaces in the following three ways.

1. The distance is not necessarily symmetric, so we allow $X(x, x') \ne X(x', x)$.

2. The distance between two points can be infinite.

3. The distance from one point to a different point can be zero.

We’ll see some examples illustrating these below.

From this generalized metric space perspective, an $\overline{\mathbb{R}}_{+}$-functor $f\colon X \to Y$ is viewed as a short map or distance non-increasing function, so it is a function $f\colon \operatorname{Ob}X \to \operatorname{Ob}Y$ such that $X(x, x') \ge Y\big(f(x), f(x')\big)$ for all $x, x' \in \operatorname{Ob}X$.

Here are the promised examples.

• We can equip $\overline{\mathbb{R}}_{+}$ with the structure of an $\overline{\mathbb{R}}_{+}$-category. This is basically the fact that $\overline{\mathbb{R}}_{+}$ is a closed monoidal category. It is a slight abuse of notation to refer to the category and the $\overline{\mathbb{R}}_{+}$-category as $\overline{\mathbb{R}}_{+}$ but it doesn’t usually cause confusion. Anyway we define the generalized metric as follows. $$\overline{\mathbb{R}}_{+}(a, b) = b  \stackrel{.}{-} a \coloneqq \max(b - a, 0).$$ The operation $ \stackrel{.}{-} $ is sometimes referred to as truncated subtraction. If we think of the extended non-negative real numbers as standing going upwards, then for $b \ge a$ we can think of it being ‘free’ to descend from $b$ to $a$, but having a ‘cost’ of $b-a$ associated with ascending from $a$ to $b$.

• As $\overline{\mathbb{R}}_{+}$ is sufficiently nice – it has all limits, which are given by suprema – if $X$ and $Y$ are $\overline{\mathbb{R}}_{+}$-categories then there is the functor $\overline{\mathbb{R}}_{+}$-category $[X, Y]$ consisting of short maps from $X$ to $Y$ and the generalized metric given as follows:

  $$[X, Y](f, g) \coloneqq\sup_{x\in X} Y(f(x), g(x)).$$

  

  This is a measurement of the furthest apart that $f$ and $g$ get. In particular, if we take $Y= \overline{\mathbb{R}}_{+}$ then we can get the following generalized metric on scalar-valued short maps: $$[X, \overline{\mathbb{R}}_{+}](f, g) \coloneqq\sup_{x\in X} \left(g(x)  \stackrel{.}{-} f(x)\right).$$

  The Yoneda embedding is an distance preserving (ie. isometric) function: $$X \hookrightarrow [X^{\mathrm{op}}, \overline{\mathbb{R}}_{+}]; \quad x \mapsto X({-}, x).$$ In the case of a classical metric space we have $X = X^\mathrm{op}$ as the distance is symmetric and the embedding in sometimes known as the Kuratowski embedding.

  We can see that this notion of generalized metric on function spaces is a refinement of the standard notion of ‘sup-metric’ on function spaces. If we look at all functions from the unit interval $I = \{x \mid 0 \le x \le 1\}$ to the set $\mathbb{R}_{\ge 0}$ then we can consider these as short maps from $I_\delta$ to $\overline{\mathbb{R}}_{+}$ where $I_\delta$ is the interval equipped with the discrete metric so that the distance $I(x, x')$ is $0$ if $x=x'$ and is $\infty$ otherwise. Then for functions $f, g\colon I\to \mathbb{R}_{\ge 0}$ we have that the sup-metric between them is a symmetrization of the enriched category metric: $$\begin{aligned}\textstyle\sup_x \left| f(x) - g(x) \right| &= \max\Bigl( \sup_x (g(x)  \stackrel{.}{-} f(x)), \sup_x (f(x)  \stackrel{.}{-} g(x))\Bigr)\\ &=\max\Bigl([I_\delta, \overline{\mathbb{R}}_{+}](f, g), [I_\delta, \overline{\mathbb{R}}_{+}](g, f)\Bigr). \end{aligned}$$

• Now we can see an example where even if we are just interested in classical metric spaces we naturally end up with an asymmetric metric, although it is usually symmetrized to obtain the Hausdorff metric! Given a classical metric space $M$ we can take the set of compact, non-empty subsets, $S_M \coloneqq\{A \subseteq M \mid A\ \text{compact},A\ne \emptyset\}$. We can then define a generalized metric on this by $$S_M(A, B) \coloneqq \sup_{a\in A}\inf_{b\in B} M(a, b).$$

  

  One way of thinking about $S_M(A, B)$ is that it is measuring the furthest you would have to go if you were dropped at a random point in $A$ and wanted to take the shortest route to $B$. Because we are considering compact sets we have the following simple characterization of zero-distance, which is telling us that the generalized metric is encoding the usual order on subsets: $$S_M(A, B) = 0 \quad \Longleftrightarrow \quad A \subseteq B.$$ (If we hadn’t used compact subsets then we wouldn’t have had such a nice characterization.) This leads us to see that this generalized metric is naturally asymmetric even though we started with a symmetric metric on $M$. The usual Hausdorff metric $\operatorname{d}_{\mathrm{H}}$ is obtained by symmetrizing this generalized metric: $$\operatorname{d}_{\mathrm{H}}(A, B) = \max\big(S_M(A, B), S_M(B, A)\big).$$ Clearly, however, the Hausdorff metric is losing information, such as the partial order, that was contained in the generalized metric.

This concludes the quick introduction to metric spaces as enriched categories. However, this perspective does allow other categorical techniques to be used with metric spaces, leading to some fruitful mathematics.

Afterword

It’s probably worth mentioning my experiences a little. As a Part III student at Cambridge, like many students before and after me, I attended Peter Johnstone’s Category Theory course, so I had familiarity with category theory, but, as I didn’t take the exam, I didn’t have a deep understanding. I then went on to do my PhD and later work on things related to algebraic topology and topological quantum field theory which both have strong categorical underpinnings. Coming back to the basics of category theory when I was involved in the Catsters with Eugenia Cheng I realised that various things didn’t chime with my experience of using categories and certain things seemed almost arcane to me, for example the importance put on things like monoids and functors into the category of sets.

It wasn’t until many, many years later that I started to understand that this cognitive dissonance was because in areas like algebraic topology, topological field theory, representation theory and algebraic geometry it is much more useful to consider that you’re not working over the category of sets but over some other category like vector space or abelian groups or chain complexes. As a ‘working mathematician’, rather than monoids and functors into the category of sets, I would have been much more familiar with algebras or rings and with functors into the category of vector spaces or abelian groups.

Although I was aware of the notion of enriched category theory, I didn’t get a thorough appreciation of quite what it meant until I’d spent a lot of time thinking about metric spaces in this context in particular via Lawvere’s rather wonderful paper Metric spaces, generalized logic and closed categories. I think one reason that enriched category is not more appreciated as a perspective is that the main text, Kelly’s Basic Concepts of Enriched Category Theory, is not particularly welcoming and only entered into by the brave! I could probably try to make the area more welcoming by polishing up my first attempt at a draft book on enriched categories, but with all of the other things I want to do, unfortunately I don’t see that happening too soon.

Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood, Eric Redekopp and I have been creating software for modeling the spread of disease… with the help of category theory!

Lots of epidemiologists use “stock-flow diagrams” to describe ordinary differential equation (ODE) models of disease dynamics. We’ve created two tools to help them.

The first, called StockFlow.jl, is based on category theory and written in AlgebraicJulia, a framework for programming with categories that many people at or associated with Topos have been developing. The second, called ModelCollab, runs on web browsers and serves as a graphical user interface for StockFlow.jl.

Using ModelCollab requires no knowledge of Julia or category theory! This feature should be useful in “participatory modeling”, an approach where models are built with the help of diverse stakeholders. However, as we keep introducing new features in StockFlow.jl, it takes time to implement them in ModelCollab.

But what’s a stock-flow diagram, and what does our software let you do with them?

The picture here shows an example: a simple disease model where Susceptible people become Infective, then Recovered, then Susceptible again.

The boxes are “stocks” and the double-edged arrows are “flows”. There are also blue “links” from stocks to “variables”, and from stocks and variables to flows. This picture doesn’t show the formulas that say exactly how the variables depend on stocks, and how the flows depend on stocks and variables. So, this picture doesn’t show the whole thing. It’s really just what they call a “system structure diagram”: a stock-flow diagram missing the quantitative information that you need to get a system of ODEs from it. A stock-flow diagram, on the other hand, uniquely specifies a system of first-order ODEs.

Modelers often regard diagrams as an informal step toward a mathematically rigorous formulation of a model in terms of ODEs. However, we’ve shown that stock-flow diagrams have a precise mathematical syntax! They are objects in a category while “open” stock-flow diagrams, where things can flow in and out of the whole system, are horizontal 1-cells in a double category If you know category theory you can read a paper we wrote with Evan Patterson where we explain this:

• John C. Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams. To appear in Proceedings of Applied Category Theory 2022.

If you don’t, we have a gentler paper for you:

• John C. Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Evan Redekopp, A categorical framework for modeling with stock and flow diagrams, to appear in Mathematics for Public Health, Springer, Berlin.

Why does it help to formalize the syntax of stock-flow diagrams using category theory? There are many reasons, but here are three:

1. Functorial Semantics

Our software lets modelers separate the syntax of stock and flow diagrams from their semantics: that is, the various uses to which these diagrams are put. Different choices of semantics are described via different functors. This idea, called “functorial semantics”, goes back to Lawvere and is popular in certain realms of theoretical computer science.

Besides the ODE semantics, we have implemented functors that turn stock-flow diagrams into other widely used diagrams: “system structure diagrams”, which I already explained, and “causal loop diagrams”. It doesn’t really matter much here, but a causal loop diagram ignores the distinction between stocks, flows and variables, lumps them all together, and has arrows saying what affects what:

These other forms of semantics capture purely qualitative features of stock and flow models. In the future, people can implement still more forms of semantics, like stochastic differential equation models!

So, instead of a single monolithic model, we have something much more flexible.

2. Composition

ModelCollab provides a structured way to build complex stock-flow diagrams from small reusable pieces. These pieces are open stock-flow diagrams, and sticking together amounts to composing them.

ModelCollab lets users save these diagrams and retrieve them for reuse as parts of various larger models. Since ModelCollab can run on multiple web browsers, it lets members of a modeling team compose models collaboratively. This is a big advance on current systems, which are not optimized for collaborative work.

This picture shows two small stock-flow diagrams being composed in ModelCollab:

Some of the underlying math here was developed in earlier work using categories and epidemiological modeling, which was also done by people at Topos and their collaborators:

• Sophie Libkind, Andrew Baas, Micah Halter, Evan Patterson and James P. Fairbanks, An algebraic framework for structured epidemic modelling, Philosophical Transactions of the Royal Society A 380 (2022), 20210309.

3. Stratification

Our software also allows users to “stratify” models: that is, refine them by subdividing a single population (stock) into several smaller populations with distinct features. For example, you might take a disease model and break each stock into different age groups.

In contrast to the global changes commonly required to stratify stock-flow diagrams, our software lets users build a stratified diagram as a “pullback” of simpler diagrams, which can be saved for reuse. Pullbacks are a concept from category theory, and here we are using pullbacks in the category whose objects are system structure diagrams. Remember, these are like stock and flow diagrams, but lacking the quantitative information describing the rates of flows. After a system structure diagram has been constructed, this information can be added to obtain a stock and flow diagram.

This picture shows two different models stratified in two different ways, creating four larger models. I won’t try to really explain this here. But at least you can get a tiny glimpse of how complicated these models get. They get a lot bigger! That’s why we need software based on good math to deal with them efficiently.

References

[AJ] AlgebraicJulia: Bringing compositionality to technical computing.

[B1] John C. Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams. To appear in Proceedings of Applied Category Theory 2022.

[B2] John C. Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling with stock and flow diagrams, to appear in Mathematics for Public Health, Springer, Berlin.

[H] P. S. Hovmand, Community Based System Dynamics, Springer, Berlin, 2014.

[L] Sophie Libkind, Andrew Baas, Micah Halter, Evan Patterson and James P. Fairbanks, An algebraic framework for structured epidemic modelling, Philosophical Transactions of the Royal Society A 380 (2022), 20210309.

[MC] ModelCollab: A web-based application for collaborating on simulation models in real-time using Firebase.

[SF] Stockflow.jl.

The Appalachians are an old, worn-down mountain chain that runs down the eastern side of North America. The ecology of the Appalachians is fascinating. For example:

Ecologists have tested many species of Appalachian trees to see how much cold they can survive. As you’d expect, for many trees the killing temperature is just a bit colder than the lowest temperatures at the northern end of their range. That makes sense: presumably they’ve spread as far north—and as far up the mountains—as they can.

But some other trees can survive temperatures much lower than that! For example white and black spruce, aspen and balsam poplar can survive temperatures of -60° C, which is -80° F. Why is that?

One guess is that this extra hardiness is left over from the last glacial cycle, which peaked 20,000 years ago—or even previous glacial cycles. It got a lot colder then!

So, maybe these trees are native to the northern Appalachians—while others, even those occupying the same regions, have only spread there since it warmed up around 10,000 years ago. Ancient pollen shows that trees have been moving north and south with every glacial cycle.

I learned about this issue here:

• Scott Weidensaul, Mountains of the Heart: a Natural History of the Appalachians, Fulcrum Publishing, 2016.

I bought this book before a drive through the Appalachians.

To add some extra complexity to the story, David C. writes:

I’d love to understand more and reconcile that with the fact that none of these trees do well above around 4500 ft in the northern Appalachians (New Hampshire).

and Brian Hawthorne writes:

Don’t forget that all the tree species had to move back into the areas that were under the last glacier.

Scientists want to know everything, and we’ve been trying to get there since the dawn of science. So why aren’t we there yet? Why are there things we still don’t know?

Sometimes, the reason is obvious: we can’t do the experiments yet. Victorian London had neither the technology nor the wealth to build a machine like Fermilab, so they couldn’t discover the top quark. Even if Newton had the idea for General Relativity, the telescopes of the era wouldn’t have let astronomers see its effect on the motion of Mercury. As we grow (in technology, in resources, in knowledge, in raw number of human beings), we can test more things and learn more about the world.

But I’m a theoretical physicist, not an experimental physicist. I still want to understand the world, but what I contribute aren’t new experiments, but new ideas and new calculations. This brings back the question in a new form: why are there calculations we haven’t done yet? Why are there ideas we haven’t had yet?

Sometimes, we can track the reason down to bottlenecks. A bottleneck is a step in a calculation that, for some reason, is harder than the rest. As you try to push a calculation to new heights, the bottleneck is the first thing that slows you down, like the way liquid bubbles through the neck of a literal bottle. If you can clear the bottleneck, you can speed up your calculation and accomplish more.

In the clearest cases, we can see how these bottlenecks could be solved with more technology. As computers get faster and more powerful, calculations become possible that weren’t possible before, in the same way new experiments become possible with new equipment. This is essentially what has happened recently with machine learning, where relatively old ideas are finally feasible to apply on a massive scale.

In physics, a subtlety is that we rarely have access to the most powerful computers available. Some types of physics are done on genuine supercomputers, but for more speculative or lower-priority research we have to use small computer clusters, or even our laptops. Something can be a bottleneck not because it can’t be done on any computer, but because it can’t be done on the computers we can afford.

Most of the time, bottlenecks aren’t quite so obvious. That’s because in theoretical physics, often, we don’t know what we want to calculate. If we want to know why something happens, and not merely that it happens, then we need a calculation that we can interpret, that “makes sense” and that thus, hopefully, we can generalize. We might have some ideas for how that calculation could work: some property a mathematical theory might have that we already know how to understand. Some of those ideas are easy to check, so we check, and make progress. Others are harder, and we have to decide: is the calculation worth it, if we don’t know if it will give us the explanation we need?

Those decisions provide new bottlenecks, often hidden ones. As we get better at calculation, the threshold for an “easy” check gets easier and easier to meet. We put aside fewer possibilities, so we notice more things, which inspire yet more ideas. We make more progress, not because the old calculations were impossible, but because they weren’t easy enough, and now they are. Progress fuels progress, a virtuous cycle that gets us closer and closer to understanding everything we want to understand (which is everything).

The major developments in human history are always steeped in dark ironies. Yes, that’s my Law of Dark Irony, the whole thing.

I don’t know why it’s true, but it certainly seems to be. Taking WWII as the archetypal example, let’s enumerate just the more obvious ones:

• After the carnage of WWI, the world’s most sensitive and thoughtful people (many of them) learned the lesson that they should oppose war at any cost. This attitude let Germany rearm and set the stage for WWII.
• Hitler, who was neither tall nor blond, wished to establish the worldwide domination of tall, blond Aryans … and do so via an alliance with the Japanese.
• The Nazis touted the dream of eugenically perfecting the human race, then perpetrated a genocide against a tiny group that had produced Einstein, von Neumann, Wigner, Ulam, and Tarski.
• The Jews were murdered using a chemical—Zyklon B—developed in part by the Jewish chemist Fritz Haber.
• The Allied force that made the greatest sacrifice in lives to defeat Hitler was Stalin’s USSR, another of history’s most murderous and horrifying regimes.
• The man who rallied the free world to defeat Nazism, Winston Churchill, was himself a racist colonialist, whose views would be (and regularly are) denounced as “Nazi” on modern college campuses.
• The WWII legacy that would go on to threaten humanity’s existence—the Bomb—was created in what the scientists believed was a desperate race to save humanity. Then Hitler was defeated before the Bomb was ready, and it turned out the Nazis were never even close to building their own Bomb, and the Bomb was used instead against Japan.

When I think about the scenarios where superintelligent AI destroys the world, they rarely seem to do enough justice to the Law of Dark Irony. It’s like: OK, AI is created to serve humanity, and instead it turns on humanity and destroys it. Great, that’s one dark irony. One. What other dark ironies could there be? How about:

• For decades, the Yudkowskyans warned about the dangers of superintelligence. So far, by all accounts, the great practical effect of these warnings has been to inspire the founding of both DeepMind and OpenAI, the entities that Yudkowskyans believe are locked into a race to realize those dangers.
• Maybe AIs will displace humans … and they’ll deserve to, since they won’t be quite as wretched and cruel as we are. (This is basically the plot of Westworld, or at least of its first couple seasons, which Dana and I are now belatedly watching.)
• Maybe the world will get destroyed by what Yudkowsky calls a “pivotal act”: an act meant to safeguard the world from takeover from an unaligned AGI, for example by taking it over with an aligned AGI first. (I seriously worry about this; it’s a pretty obvious one.)
• Maybe AI will get the idea to take over the world, but only because it’s been trained on generations of science fiction and decades of Internet discussion worrying about the possibility of AI taking over the world. (I’m far from the first to notice this possibility.)
• Maybe AI will indeed destroy the world, but it will do so “by mistake,” while trying to save the world, or by taking a calculated gamble to save the world that fails. (A commenter on my last post brought this one up.)
• Maybe humanity will successfully coordinate to pause AGI development, and then promptly be destroyed by something else—runaway climate change, an accidental nuclear exchange—that the AGI, had it been created, would’ve prevented. (This, of course, would be directly analogous to one of the great dark ironies of all time: the one where decades of antinuclear activism, intended to save the planet, has instead doomed us to destroy the earth by oil and coal.)

Readers: which other possible dark ironies have I missed?

During the past seven months, I’ve steamed across the Atlantic, sailed in a flying castle, teleported across the globe, and shuttled forward and backward in time. Literarily, not literally—the Quantum-Steampunk Short-Story Contest began welcoming submissions in October 2022. We challenged everybody aged 13 and over to write a steampunk narrative that involves a real or imagined quantum technology. One hundred sixty-seven entries arrived from 29 countries. Professional writers submitted stories, as did 13-year-olds. Tenured physics professors, librarians, English and math teachers, undergraduates, physicians, graduate students, and a United States Senate staffer entered. Thanks to their creativity, I now have a folder full of other worlds.

I’m over the moon (in a steam-powered ship) to announce the winners. David Wakeham received the $1,500 grand prize for the story The Creature of Ashen House. First runner-up Gerard McCaul won $1,000 for Doctor Up and Mister Down, and second runner-up Paulo Barreto won $500 for Eikonal. The People’s Choice Award ($500) went to Cristina Legarda for Pursuit, also nominated by two judges for a “Please Turn This into a Novel” award. Thanks to the 261 of you who voted in the People’s Choice competition!

In addition to traditional awards, we created four idiosyncratic ones, each entailing $250. We recognized Jeff Provine’s Stealing Buttons for its badass steampunk heroine; Matt King’s Three Imperiled Scientists for its wit and (relatedly) its portrayal of academia; Rick Searle’s The Recurrence Machine for its steampunk atmosphere; and Claudia Clarke’s Looking Forward, Looking Back, for its heart-capturing automaton. You can read all the finalist stories here.

Sending our judges the finalists in March, I felt not only exhilaration (and relief, as whittling down 167 entries entails no little hand wringing), but also anxiety. Would the stories measure up? So I must have glowed when the first judge submitted his evaluations: Speculative-fiction author Ken Liu enthused, “The entries were so fun to read.” Similar reactions followed from across the panel, which featured experts in mathematics, philosophy, creative writing, experimental quantum physics, and history: “I had a very good time reading these stories,” another panelist wrote. “This was fun and some excellent spring break airplane (no dirigibles, I’m afraid) reading,” said another. Many thanks to our judges and short-listing committee for their input. University of Maryland undergraduates Hannah Cho and Jade Leschack led the team of students who narrowed down the candidates. I couldn’t resist treating the committee to a Victorian-inspired thank-you upon announcing the winners.

Although this year’s contest has ended, quantum-steampunk literature has just shipped out from its berth. Two contest entrants have posted their stories on their own online domains: You can read the mystery by Duke physics professor Ken Brown here and the adventure by quantum-algorithm designer Brian Siegelwax here. All other entrants, please feel free to post your stories and to submit them to other literary contests. Drop me a line, and leave a link in the chat below, when your story is published. I’d love to hear how your journey continues.

Also, stay tuned for v2.0 of the Quantum-Steampunk Short-Story Contest. An organization has expressed interest in a reboot during the 2024–2025 academic year. AI-collaboration category, anyone? Bonus points if you use a quantum neural network. Please email me if you’d like to support the effort!

The opportunity to helm this contest has been a privilege and a dream. Many thanks to our writers, readers, funder (the John Templeton Foundation), staff (especially webmaster Anıl Zenginoğlu), judges, and shortlisting committee. Keep writing, and keep experimenting.

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 Here is a little puzzle I just came up with when in today's hep-th serving I found 

  arXiv:2304.14351 [pdf, other]

Operator growth and black hole formation

Felix M. Haehl, Ying Zhao

Comments: 20+9 pages, 10 figures. arXiv admin note: text overlap with arXiv:2104.02736

Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Quantum Physics (quant-ph)

When two particles collide in an asymptotically AdS spacetime with high enough energy and small enough impact parameter, they can form a black hole. 

But to explain it, I should probably say one or two things about thermal states in the algebraic QFT language: There (as we teach for example in our "Mathematical Statistical Physics" class) you take take to distinguish (quasi-local) observables which form a C*-algebra and representations of these on a Hilbert space. In particular, like for example for Lie algebras, there can be inequivalent representations that is different Hilbert spaces where the observables act as operators but there are no (quasi-local) operators that you can use to act on a vector state in one Hilbert space that brings you to the other Hilbert space. The different Hilbert space representations are different super-selection sectors of the theory.

A typical example are states of different density in infinite volume: The difference in particle number is infinite but any finite product of creation and annihilation operators cannot change the particle number by an infinite amount. Or said differently: In Fock space, there are only states with arbitrary but finite particle number, trying to change that you run into IR divergent operators.

Similarly, assuming that the (weak closure) of the representation on one Hilbert space if a type III factor as it should be for a good QFT, states of different temperatures (KMS states in that language) are disjoint, meaning they live in different Hilbert spaces and you cannot go from one to the other by acting with a quasi-local operator. This is proven as Theorem 5.3.35 in volume 2 of the Bratelli/Robinson textbook.

Now to the AdS black holes: Start with empty AdS space also encoded by the vacuum in the holographic boundary theory. Now, at t=0 you act with two boundary operators (obviously quasi-local) to create two strong gravitational wave packets heading towards each other with very small impact parameter. Assuming the hoop conjecture, they will create a black hole when they collide (probably plus some outgoing gravitational radiation). 

Then we wait long enough for things to settle (but not so long as the black hole starts to evaporate in a significant amount). We should be left with some AdS-Kerr black hole. From the boundary perspective, this should now be a thermal state (of the black hole temperature) according to the usual dictionary.

So, from the point of the boundary, we started from the vacuum, acted with local operators and ended up in a thermal state. But this is exactly what the abstract reasoning above says is impossible.

How can this be? Comments are open!

The concept of chemical potential is one that seems almost deliberately obscure to many.  I’ve written about this here, and referenced this article.  What you may not realize is that the chemical potential, of water in particular, plays a crucial role in why my banana waffle recipe works so well.  

My waffle recipe starts with an old, peel-getting-brown banana, which I peel and put in a medium bowl with a couple of teaspoons of salt and a tablespoon of brown sugar.  With just a little mashing with a fork to mix with the salt and sugar, the banana basically liquefies in a couple of minutes.  That’s where the chemical potential comes in.  

Chemical potential, \(\mu\), describes how particles tend to diffuse, from regions of high chemical potential (more accurately, high \(\mu/T\)) to regions of low chemical potential \((\mu/T\)). The water molecules in the cells of the banana is already at a higher chemical potential than, e.g., the water vapor in the air around the banana.  That’s why if you let the banana sit around it would eventually dry out, and there is an “osmotic” pressure that pushes out against the cell membranes and cell walls.  Adding salt and sugar to the exterior of the cells lowers the chemical potential for water outside the cells even more (because there is an energetic benefit to the water molecules to form a solution with the salt and sugar - the polar water molecules have an attractive interaction with the ions from the salt, and an attractive interaction via hydrogen bonding with the sugar).  This increases the osmotic pressure, so that water leaks out of the cells (maybe even rupturing the cell membrane, though when people want to encourage that they throw in a little soap, not conducive to good waffles).  Wait a couple of minutes, stir, and then I have yummy banana goo that forms the beginning of my Sunday morning waffle batter.

This is a goofy example of the power of thermodynamics and statistical mechanics.  At room temperature, there are many more microscopic arrangements of the water molecules (in the presence of sugar and salt) with the banana forming liquefied goo than with the water sitting in the cells, and so the liquefaction happens spontaneously once the ingredients are put together.  (Osmosis can even be funny - I highly recommend reading this story of you can find a copy.)

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I completely disagree with the open letter signed by Elon Musk and numerous other luminaries in which they advocate a moratorium on advancing AI so that time can be taken to consider the implications and risks associated with this technology.

• While the intention is well-meaning and the risks are real, the analysis is superficial and unlikely to play out as suggested.

• Although there are undeniably major risks presented by advanced AI, a moratorium is unlikely to further progress in dealing with them and more likely hinder it. Political responses to disruptive forces tend to be reactionary rather than preemptive and it is not foreseeable that during such a moratorium political and regulatory solutions will be implemented. It is naive to think that if presented with a six month window of opportunity to consider the implications of AI that politicians and regulators are going to make use of it to formulate a master plan. Societal consensus and political responses to complex emerging problems do not take place over such short timescales, and attempts to do so are likely to be poorly considered.

• Such a moratorium is necessarily voluntary as there are no mechanisms for global enforcement, meaning that only good actors will participate, tilting the balance of power in the AI sphere in favour of bad actors.

• Technological advancement is inherently disruptive and there are many instances through modern history where technology has made types of human labour redundant. However, it is very clear that embracing technology has in general driven humanity forward not backward.

• Attempting to inhibit technological advancement is largely futile and unenforceable. Adapting to embrace it is by far the best approach. Adaptation is an evolutionary process, not something that can be decided in advance. We are not in a position to make advance determinations as there are too many unknowns and the spectrum of implications is unclear.

• Obstructing technological advancement that competes against us is a form of protectionism. Recently Italy placed a ban on ChatGPT, and some other EU nations are reportedly considering the same. Doing so, rather than encouraging home-grown development of AI industries represents a major economic setback, enforces competitive disadvantage, and missed opportunity that risks future economic irrelevance. This is not to say that Italy’s privacy-related concerns have no merit. However, placing an outright ban on emerging technologies, rather than adapting to them in tandem with their development is backward thinking. The same line of reasoning could equally be used to justify banning any of the cloud services we all rely on or the internet as a whole.

• Yes, advanced AI will be highly disruptive, but also transformative, with the potential to act as a huge multiplier on productivity, which drives economic progress and human development. Wilfully delaying or missing this opportunity is economically and strategically destructive, handing power to competitors and adversaries.

• We definitely should be acting quickly in considering the ethical and broader implications of AI upon society, but placing a halt on technological progress isn’t going to expedite this process. That will happen as the implications becomes tangible, and in the meantime we’ll have only delayed progress for no reason.

• Openness and transparency are the most powerful forces against malevolent misuse. Driving things underground inhibits this, imposing opaqueness on the sector.

• Turning AI into a black market is completely foolish.

The post Response to “Pause Giant AI Experiments: An Open Letter” appeared first on Peter Rohde.

The Elias M. Stein Prize for New Perspectives in Analysis is awarded for the development of groundbreaking methods in analysis which demonstrate promise to revitalize established areas or create new opportunities for mathematical discovery. The current prize amount is US$5,000 and the prize is awarded every three years for work published in the preceding six years.

This prize was endowed in 2022 by students, colleagues, and friends of Elias M. Stein (my former advisor) to honor his remarkable legacy in the area of mathematical analysis. Stein, who passed away in 2018, is remembered for identifying many deep principles and methods which transcend their original context, and for opening entirely new areas of research which captivated the attention and imagination of generations of analysts. This prize seeks to recognize mathematicians at any career stage who, like Stein, have found exciting new avenues for mathematical exploration in subjects old or new or made deep insights which demonstrate promise to reshape thinking across areas.

This will be the inaugural year for the prize, and I have agreed to serve on the prize committee. We welcome nominations for the prize, which will be accepted until June 30, 2023, and are seeking a strong and diverse pool of nominees. Nominations (submitted at this link) should include a letter of nomination and a brief citation to be used in the event that the nomination is successful. Alternatively, if you are aware of a strong potential candidate but are not able to provide the nomination yourself, we welcome your suggestion (by private email) along with — if possible — your suggestions of possible nominators. 

For questions about this award, please contact the AMS Secretary at secretary@ams.org.

pon receiving my speaking assignments for the Tucson Festival of Books, I mentally raised my eyebrows. I’d be participating in a panel discussion with Mike Evans, the founder of Grubhub? But I hadn’t created an app that’s a household name. I hadn’t transformed 30 million people’s eating habits. I’m a theoretical physicist; I build universes in my head for a living. I could spend all day trying to prove a theorem and failing, and no stocks would tumble as a result.

Once the wave of incredulity had crested, I noticed that the panel was entitled “The Future of Tech.” Grubhub has transformed technology, I reasoned, and quantum computing is in the process of doing so. Fair enough. 

Besides, my husband pointed out, the food industry requires fridges. Physicists building quantum computers from superconductors need fridges. The latter fridges require temperatures ten million times lower than restaurateurs do, but we still share an interest.

Very well, I thought. Game on.

Tucson hosts the third-largest book festival in the United States. And why shouldn’t it, as the festival takes place in early March, when much of the country is shivering and eyeing Arizona’s T-shirt temperatures with envy? If I had to visit any institution in the winter, I couldn’t object to the festival’s home, the University of Arizona.

The day before the festival, I presented a colloquium at the university, for the Arizona Quantum Alliance. The talk took place in the Wyant College of Optical Sciences, the home of an optical-instruments museum. Many of the instruments date to the 1800s and, built from brass and wood, smack of steampunk. I approved. Outside the optics building, workers were setting up tents to house the festival’s science activities.

The next day—a Saturday—dawned clear and bright. Late in the morning, I met Mike and our panel’s moderator, Bob Griffin, another startup veteran. We sat down at a table in the back of a broad tent, the tent filled up with listeners, and the conversation began.

I relished the conversation as I’d relished an early-morning ramble along the trails by my hotel at the base of the Santa Catalina Mountains. I joined theoretical physics for the love of ideas, and this exchange of ideas offered an intellectual workout. One of Mike’s points resonated with me most: Grubhub didn’t advance technology much. He shifted consumers from ordering pizza via phone call to ordering pizza via computer, then to ordering pizza via apps on phones. Yet these small changes, accumulated across a population and encouraged by a pandemic, changed society. Food-delivery services exploded and helped establish the gig economy (despite Mike’s concerns about worker security). One small step for technology, adopted by tens of millions, can constitute one giant leap for commerce.

To me, Grubhub offered a foil for quantum computing, which offers a giant leap in technology: The physical laws best-suited to describing today’s computers can’t describe quantum computers. Some sources portray this advance as bound to transform all our lives in countless ways. This portrayal strikes some quantum scientists as hype that can endanger quality work. 

Quantum computers will transform cybersecurity, being able to break the safeguards that secure our credit-card information when we order food via Grubhub. Yet most consumers don’t know what safeguards are protecting us. We simply trust that safeguards exist. How they look under the hood will change by the time large-scale quantum computers exist—will metamorphose perhaps as dramatically as did Gregor Samsa before he woke up as an insect. But consumers’ lives might not metamorphose.

Quantum scientists hope and anticipate that quantum computers will enable discoveries in chemistry, materials science, and pharmacology. Molecules are quantum, and many materials exhibit quantum properties. Simulating quantum systems takes classical (everyday) computers copious amounts of time and memory—in some cases, so much that a classical computer the size of the universe would take ages. Quantum computers will be able to simulate quantum subjects naturally. But how these simulations will impact everyday life remains a question.

For example, consider my favorite potential application of quantum computers: fertilizer production, as envisioned by Microsoft’s quantum team. Humanity spends about 3% of the world’s energy on producing fertilizer, using a technique developed in 1909. Bacteria accomplish the same goal far more efficiently. But those bacteria use a molecule—nitrogenase—too complicated for us to understand using classical computers. Being quantum, the molecule invites quantum computation. Quantum computers may crack the molecule’s secrets and transform fertilizer production and energy use. The planet and humanity would benefit. We might reduce famines or avert human-driven natural disasters. But would the quantum computation change my neighbor’s behavior as Grubhub has? I can’t say.

Finally, evidence suggests that quantum computers can assist with optimization problems. Imagine a company that needs to transport supplies to various places at various times. How can the company optimize this process—implement it most efficiently? Quantum computers seem likely to be able to help. The evidence isn’t watertight, however, and quantum computers might not solve optimization problems exactly. If the evidence winds up correct, industries will benefit. But would this advance change Jane Doe’s everyday habits? Or will she only receive pizza deliveries a few minutes more quickly?

Don’t get me wrong; quantum technology has transformed our lives. It’s enabled the most accurate, most precise clocks in the world, which form the infrastructure behind GPS. Quantum physics has awed us, enabling the detection of gravitational waves—ripples, predicted by Einstein, in spacetime. But large-scale quantum computers—the holy grail of quantum technology—don’t suit all problems, such as totting up the miles I traveled en route to Tucson; and consumers might not notice quantum computers’ transformation of cybersecurity. I expect quantum computing to change the world, but let’s think twice about whether quantum computing will change everyone’s life like a blockbuster app.

I’ve no idea how many people have made this pun about Mike’s work, but the panel discussion left me with food for thought. He earned his undergraduate degree at MIT, by the way; so scientifically inclined Quantum Frontiers readers might enjoy his memoir, Hangry. It conveys a strong voice and dishes on data and diligence through stories. (For the best predictor of whether you’ll enjoy a burrito, ignore the starred reviews. Check how many people have reordered the burrito.)

The festival made my week. After the panel, I signed books; participated in a discussion about why “The Future Is Quantum!” with law professor Jane Bambauer; and narrowly missed a talk by Lois Lowry, a Newbury Award winner who wrote novels that I read as a child. (The auditorium filled up before I reached the door, but I’m glad that it did; Lois Lowry deserves a packed house and then some.) I learned—as I’d wondered—that yes, there’s something magical to being an author at a book festival. And I learned about how the future of tech depends on more than tech.

I was doing guest lecture / Q&A in a science communication course at Cornell, where I was a professor at large last week. A friend of mine was in the back of the room, and later told me: “A bunch of students were on their laptops, but only about half of them were looking at Instagram and the other half were looking at information about you and your books!”

He meant this to be a compliment but my initial reaction was — what? Students were looking at their laptops while we were talking? I had no idea. In my mental construction of the event, both while it was happening and afterwards, every eye in the room was on me.

And this made me reflect — when I am giving a talk, or even a classroom lecture, I’m inclined to always think I’m being listened to. But of course that’s not true! It couldn’t be true!

There are limits, of course. If I’m lecturing and I’ve lost the whole room, I see their eyes die and I notice it. I stop and regroup and change course. But if half the kids are tuned out? I’m just gonna be honest, I probably don’t notice that.

Now you can read this as saying I’m a huge egotist who relies on unrealistic assessments of how interesting I’m being, and thanks to this reliance am failing to engage the class. Or you could say it’s very, very hard to teach class in such a way that there’s not some notable proportion of students tuned out at any given moment, and that it would be even harder to teach class well if you were constantly aware of which students those were. And as a counterpoint to that sympathetic assessment, you could say it’s not a random and constantly shifting sample of students who are tuned out; there might be a notable proportion who are almost tuned out and who I’m allowing myself to fail, or rather to not even try, to reach.

I don’t really know!

The “made-up language” task that GPT-3 couldn’t do is handled much better by GPT-4, or at least the version of GPT-4 built into Bing Chat. The prompt:

Henderson’s native language is Gravinic, and when he speaks English, it sounds like this:

“I to the store did go and some cabbage did buy. Now I some dinner am making and I some cake will later eat. I you am not very much liking. I for you to go away am politely asking.”

Write more dialogue where Henderson tells you about his day.

The output:

Henderson continued to speak in his native language, Gravinic, which was very difficult for me to understand. He seemed to have a lot of things to say about his day, but I could only catch some words here and there.

“I today at work did have a meeting with my boss. He me did give a new project that I by next week must finish. It very hard and complicated is. I not sure if I it can do. He me also did say that I more hours must work and less pay will get. He very mean and unfair is. I him do not like at all.”

He paused for a moment and looked at me expectantly. I nodded politely, pretending to follow his story.

“Then I at lunch did go with my friend Bob. He me did tell that he a new girlfriend has. She very beautiful and smart is. He her met online on a dating site. He me showed her picture on his phone. I it did see and very jealous was. I also a girlfriend want but no one me likes.”

He sighed and shook his head sadly.

“Then I after work did go to the gym. I some exercise did do and some weights did lift. I very strong and fit am. But no one me notices or compliments gives. They only at me stare and laugh do behind my back. They very rude and mean are. I them do not like either.”

He frowned and clenched his fists.

“Then I home did come and you here did find waiting for me. You me always questions ask and annoy do with your presence. You very annoying and boring are. I you do not like too.”

He glared at me angrily.

“I you am now telling that you must leave do right now or else I you will hit do with this frying pan.”

He pointed at the frying pan on the stove menacingly.

I decided it was time to end this conversation before he got violent.

I quickly got up from my seat and headed towards the door.

Not perfect, but it really has picked up that in this version of English, verbs go last and past tense is indicated with “did.” It hasn’t really picked up how to do present tense (i.e. it should have said “They only are at me staring and are behind my back laughing”) and it is inconsistent in what it does with adverb phrases and there are some weird uses of “do” towards the end, but altogether a big improvement.

“If I had a nickel for every unsolicited and very personal health question I’ve gotten at parties, I’d have paid off my medical school loans by now,” my doctor friend complained. As a physicist, I can somewhat relate. I occasionally find myself nodding along politely to people’s eccentric theories about the universe. A gentleman once explained to me how twin telepathy (the phenomenon where, for example, one twin feels the other’s pain despite being in separate countries) comes from twins’ brains being entangled in the womb. Entanglement is a nonclassical correlation that can exist between spatially separated systems. If two objects are entangled, it’s possible to know everything about both of them together but nothing about either one. Entangling two particles (let alone full brains) over tens of kilometres (let alone full countries) is incredibly challenging. “Using twins to study entanglement, that’ll be the day,” I thought. Well, my last paper did something like that. 

In theory, a twin study consists of two people that are as identical as possible in every way except for one. What that allows you to do is isolate the effect of that one thing on something else. Aleksander Lasek (postdoc at QuICS), David Huse (professor of physics at Princeton), Nicole Yunger Halpern (NIST physicist and Quantum Frontiers blogger), and I were interested in isolating the effects of quantities’ noncommutation (explained below) on entanglement. To do so, we first built a pair of twins and then compared them. 

Consider a well-insulated thermos filled with soup. The heat and the number of “soup particles” inside the thermos are conserved. So the energy and the number of “soup particles” are conserved quantities. In classical physics, conserved quantities commute. This means that we can simultaneously measure the amount of each conserved quantity in our system, like the energy and number of soup particles. However, in quantum mechanics, this needn’t be true. Measuring one property of a quantum system can change another measurement’s outcome.

Conserved quantities’ noncommutation in thermodynamics has led to some interesting results. For example, it’s been shown that conserved quantities’ noncommutation can decrease the rate of entropy production. For the purposes of this post, entropy production is something that limits engine efficiency—how well engines can convert fuel to useful work. For example, if your car engine had zero entropy production (which is impossible), it would convert 100% of the energy in your car’s fuel into work that moved your car along the road. Current car engines can convert about 30% of this energy, so it’s no wonder that people are excited about the prospective application of decreasing entropy production. Other results (like this one and that one) have connected noncommutation to potentially hindering thermalization—the phenomenon where systems interact until they have similar properties, like when a cup of coffee cools. Thermalization limits memory storage and battery lifetimes. Thus, learning how to resist thermalization could also potentially lead to better technologies, such as longer-lasting batteries. 

One can measure the amount of entanglement within a system, and as quantum particles thermalize, they entangle. Given the above results about thermalization, we might expect that noncommutation would decrease entanglement. Testing this expectation is where the twins come in.

Say we built a pair of twins that were identical in every way except for one. Nancy, the noncommuting twin, has some features that don’t commute, say, her hair colour and height. This means that if we measure her height, we’ll have no idea what her hair colour is. For Connor, the commuting twin, his hair colour and height commute, so we can determine them both simultaneously. Which twin has more entanglement? It turns out it’s Nancy.

Disclaimer: This paragraph is written for an expert audience. Our actual models consist of 1D chains of pairs of qubits. Each model has three conserved quantities (“charges”), which are sums over local charges on the sites. In the noncommuting model, the three local charges are tensor products of Pauli matrices with the identity (XI, YI, ZI). In the commuting model, the three local charges are tensor products of the Pauli matrices with themselves (XX, YY, ZZ). The paper explains in what sense these models are similar. We compared these models numerically and analytically in different settings suggested by conventional and quantum thermodynamics. In every comparison, the noncommuting model had more entanglement on average.

Our result thus suggests that noncommutation increases entanglement. So does charges’ noncommutation promote or hinder thermalization? Frankly, I’m not sure. But I’d bet the answer won’t be in the next eccentric theory I hear at a party.

Asgar Jamneshan, Or Shalom, and myself have just uploaded to the arXiv our preprints “A Host–Kra -system of order 5 that is not Abramov of order 5, and non-measurability of the inverse theorem for the norm” and “The structure of totally disconnected Host–Kra–Ziegler factors, and the inverse theorem for the Gowers uniformity norms on finite abelian groups of bounded torsion“. These two papers are both concerned with advancing the inverse theory for the Gowers norms and Gowers-Host-Kra seminorms; the first paper provides a counterexample in this theory (in particular disproving a conjecture of Bergelson, Ziegler and myself), and the second paper gives new positive results in the case when the underlying group is bounded torsion, or the ergodic system is totally disconnected. I discuss the two papers more below the fold.

— 1. System of order which is not Abramov of order —

I gave a talk on this paper recently at the IAS; the slides for that talk are available here.

This project can be motivated by the inverse conjecture for the Gowers norm in finite fields, which is now a theorem:

Theorem 1 (Inverse conjecture for the Gowers norm in finite fields) Let be a prime and . Suppose that is a one-bounded function with a lower bound on the Gowers uniformity norm. Then there exists a (non-classical) polynomial of degree at most such that .

This is now known for all (see this paper of Ziegler and myself for the first proof of the general case, and this paper of Milicevic for the most recent developments concerning quantitative bounds), although initial results focused on either small values of , or the “high characteristic” case when is large compared to . One approach to this theorem proceeds via ergodic theory. Indeed it was observed in this previous paper of Ziegler and myself that for a given choice of and , the above theorem follows from the following ergodic analogue:

Conjecture 2 (Inverse conjecture for the Gowers-Host-Kra semi-norm in finite fields) Let be a prime and . Suppose that with an ergodic -system with positive Gowers-Host-Kra seminorm (see for instance this previous post for a definition). Then there exists a measurable polynomial of degree at most such that has a non-zero inner product with . (In the language of ergodic theory: every -system of order is an Abramov system of order .)

The implication proceeds by a correspondence principle analogous to the Furstenberg correspondence principle developed in that paper (see also this paper of Towsner for a closely related principle, and this paper of Jamneshan and I for a refinement). In a paper with Bergelson and Ziegler, we were able to establish Conjecture 2 in the “high characteristic” case , thus also proving Theorem 1 in this regime, and conjectured that Conjecture 2 was in fact true for all . This was recently verified in the slightly larger range by Candela, Gonzalez-Sanchez, and Szegedy.

Even though Theorem 1 is now known in full generality by other methods, there are still combinatorial reasons for investigating Conjecture 2. One of these is that the implication of Theorem 1 from Corollary 2 in fact gives additional control on the polynomial produced by Theorem 1, namely that it is some sense “measurable in the sigma-algebra generated by ” (basically because the ergodic theory polynomial produced by Conjecture 2 is also measurable in , as opposed to merely being measurable in an extension of ). What this means in the finitary setting of is a bit tricky to write down precisely (since the naive sigma-algebra generated by the translates of will mostly likely be the discrete sigma-algebra), but roughly speaking it means that can be approximated to arbitrary accuracy by functions of boundedly many (random) translates of . This can be interpreted in a complexity theory sense by stating that Theorem 1 can be made “algorithmic” in a “probabilistic bounded time oracle” or “local list decoding” sense which we will not make precise here.

The main result of this paper is

Theorem 3 Conjecture 2 fails for . In fact the “measurable inverse theorem” alluded to above also fails in this case.

Informally, this means that for large , we can find -bounded “pseudo-quintic” functions with large norm, which then must necessarily correlate with at least one quintic by Theorem 1, but such that none of these quintics can be approximated to high accuracy by functions of (random) shifts of . Roughly speaking, this means that the inverse theorem cannot be made locally algorithmic (though it is still possible that a Goldreich-Levin type result of polynomial time algorithmic inverse theory is still possible, as is already known for for ; see this recent paper of Kim, Li and Tidor for further discussion).

The way we arrived at this theorem was by (morally) reducing matters to understanding a certain “finite nilspace cohomology problem”. In the end it boiled down to locating a certain function from a -element set to a two-element set which was a “strongly -homogeneous cocycle” but not a “coboundary” (these terms are defined precisely in the paper). This strongly -homogeneous cocycle can be expressed in terms of a simpler function that takes values on a -element space . The task of locating turned out to be one that was within the range of our (somewhat rudimentary) SAGE computation abilities (mostly involving computing the Smith normal form of some reasonably large integer matrices), but the counterexample functions this produced were initially somewhat opaque to us. After cleaning up these functions by hand (by subtracting off various “coboundaries”), we eventually found versions of these functions which were nice enough that we could verify all the claims needed in a purely human-readable fashion, without any further computer assistance. As a consequence, we can now describe the pseudo-quintic explicitly, though it is safe to say we would not have been able to come up with this example without the initial computer search, and we don’t currently have a broader conceptual understanding of which could potentially generate such counterexamples. The function takes the form

— 2. Structure of totally disconnected systems —

Despite the above negative result, in our other paper we are able to get a weak version of Conjecture 2, that also extends to actions of bounded-torsion abelian groups:

Theorem 4 (Weak inverse conjecture for the Gowers-Host-Kra semi-norm in bounded torsion groups) Let be a bounded-torsion abelian group and . Suppose that with an ergodic -system with positive Gowers-Host-Kra seminorm . Then, after lifting to a torsion-free group , there exists a measurable polynomial of degree at most defined on an extension of which has a non-zero inner product with .

Combining this with the correspondence principle and some additional tools, we obtain a weak version of Theorem 1 that also extends to bounded-torsion groups:

Theorem 5 (Inverse conjecture for the Gowers norm in bounded torsion groups) Let be a finite abelian -torsion group for some and . Suppose that is a one-bounded function with . Then there exists a (non-classical) polynomial of degree at most such that .

The way Theorem 4 (and hence Theorem 5) is proven is as follows. The now-standard machinery of Host and Kra (as discussed for instance in their book) allows us to reduce to a system of order , which is a certain tower of extensions of compact abelian structure groups by various cocycles . In the -torsion case, standard theory allows us to show that these structure groups are also -torsion, hence totally disconnected. So it would now suffice to understand the action of torsion-free groups on totally disconnected systems . For the purposes of proving Theorem 4 we have the freedom to extend as we please, and we take advantage of this freedom by “extending by radicals”, in the sense that whenever we locate a polynomial in the system, we adjoin to it roots of that polynomial (i.e., solutions to ) that are polynomials of the same degree as ; this is usually not possible to do in the original system , but can always be done in a suitable extension, analogously to how roots do not always exist in a given field, but can always be located in some extension of that field. After applying this process countably many times it turns out that we can arrive at a system which is -divisible in the sense that polynomials of any degree have roots of any order that are of the same degree. In other words, the group of polynomials of any fixed degree is a divisible abelian group, and thus injective in the category of such groups. This makes a lot of short exact sequences that show up in the theory split automatically, and greatly simplifies the cohomological issues one encounters in the theory, to the point where all the cocycles mentioned previously can now be “straightened” into polynomials of the expected degree (or, in the language of ergodic theory, this extension is a Weyl system of order , and hence also Abramov of order ). This is sufficient to establish Theorem 4. To get Theorem 5, we ran into a technical obstacle arising from the fact that the remainder map is not a polynomial mod if is not itself a prime power. To resolve this, we established ergodic theory analogues of the Sylow decomposition of abelian -torsion groups into -groups , as well as the Schur-Zassenhaus theorem. Roughly speaking, the upshot of these theorems is that any ergodic -system , with -torsion, can be split as the “direct sum” of ergodic -systems for primes dividing , where is the subgroup of consisting of those elements whose order is a power of . This allows us to reduce to the case when is a prime power without too much difficulty.

In fact, the above analysis gives stronger structural classifications of totally disconnected systems (in which the acting group is torsion-free). Weyl systems can also be interpreted as translational systems , where is a nilpotent Polish group and is a closed cocompact subgroup, with the action being given by left-translation by various elements of . Perhaps the most famous examples of such translational systems are nilmanifolds, but in this setting where the acting group is not finitely generated, it turns out to be necessary to consider more general translational systems, in which need not be a Lie group (or even locally compact), and not discrete. Our previous results then describe totally disconnected systems as factors of such translational systems. One natural candidate for such factors are the double coset systems formed by quotienting out by the action of another closed group that is normalized by the action of . We were able to show that all totally disconnected systems with torsion-free acting group had this double coset structure. This turned out to be surprisingly subtle at a technical level, for at least two reasons. Firstly, after locating the closed group (which in general is Polish, but not compact or even locally compact), it was not immediately obvious that was itself a Polish space (this amounts to the orbits of a closed set still being closed), and also not obvious that this double coset space had a good nilspace structure (in particular that the factor map from to is a nilspace fibration). This latter issue we were able to resolve with a tool kindly shared to us in a forthcoming work by Candela, Gonzales-Sanchez, and Szegedy, who observed that the nilspace fibration property was available if the quotient groups obeyed an algebraic “groupable” axiom which we were able to verify in this case (they also have counterexamples showing that the nilspace structure can break down without this axiom). There was however one further rather annoying complication. In order to fully obtain the identification of our system with a double coset system, we needed the equivalence

The International Center for Mathematical Sciences in Edinburgh recently launched its “Mathematics for Humanity” initiative with a call for research activity proposals (ranging from small collaborations to courses, workshops and conferences) aimed at using mathematics to contributing to the betterment of humanity. (I have agreed to serve on the scientific committee to evaluate these proposals.) We launched this initiative in January and initially set the deadline for April 15, but several people who had expressed interest felt that this was insufficient time to prepare a quality proposal, so we have now extended the deadline to June 1, and welcome further applications.

See also this Mathstodon post from fellow committee member John Baez last year where he solicited some preliminary suggestions for proposals, and my previous Mathstodon announcement of this programme.

I have a new paper up on the arXiv today with TriThang Tran and Craig Westerland, “Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle’s conjecture for function fields.”

There’s a bit of a story behind this, but before I tell it, let me say what the paper’s about. The main result is an upper bound for the number of extensions with bounded discriminant and fixed Galois group of a rational function field F_q(t). More precisely: if G is a subgroup of S_n, and K is a global field, we can ask how many degree-n extensions of K there are whose discriminant is at most X and whose Galois closure has Galois group G. A long-standing conjecture of Malle predicts that this count is asymptotic to c X^a (log X)^b for explicitly predicted exponents a and b. This is a pretty central problem in arithmetic statistics, and in general it still seems completely out of reach; for instance, Bhargava’s work allows us to count quintic extensions of Q, and this result was extended to global fields of any characteristic other than 2 by Bhargava, Shankar, and Wang. But an asymptotic for the number of degree 6 extensions would be a massive advance.

The point of the present paper is to prove upper bounds for counting field extensions in the case of arbitrary G and rational function fields K = F_q(t) with q prime to and large enough relative to |G|; upper bounds which agree with Malle’s conjecture up to the power of log X. I’m pretty excited about this! Malle’s conjecture by now has very robust and convincing heuristic justification, but there are very few cases where we actually know anything about G-extensions for any but very special classes of finite groups G. There are even a few very special cases where the method gives both upper and lower bounds (for instance, A_4-extensions over function fields containing a cube root of 3.)

The central idea, as you might guess from the authors, is to recast this question as a problem about counting F_q-rational points on moduli spaces of G-covers, called Hurwitz spaces; by the Grothendieck-Lefschetz trace formula, we can bound these point counts if we can bound the etale Betti numbers of these spaces, and by comparison between characteristic p and characteristic 0 we can turn this into a topological problem about bounding cohomology groups of the braid group with certain coefficients.

Actually, let me say what these coefficients are. Let c be a subset of a finite group G closed under conjugacy, k a field, and V the k-vectorspace spanned by c. Then is spanned by the set of n-tuples (g_1, … , g_n) in c^n, and this set carries a natural action of the braid group, where twining strand i past strand i+1 corresponds to the permutation

So for each n we have a representation of the braid group Br_n, and it turns out that everything we desire would be downstream from good bounds on

So far, this is the same strategy (expressed a little differently) than was used in our earlier paper with Akshay Venkatesh to get results towards the Cohen-Lenstra conjecture over F_q(t). That paper concerned itself with the case where G was a (modestly generalized) dihedral group; there was a technical barrier that prevented us from saying anything about more general groups, and the novelty of the present paper is to find a way past that restriction. I’m not going to say very much about it here! I’ll just say it turns out that there’s a really nice way to package the cohomology groups above — indeed, even more generally, whenever V is a braided vector space, you have these braid group actions on the tensor powers, and the cohomology groups can be packaged together as the Ext groups over the quantum shuffle algebra associated to V. And it is this quantum shuffle algebra (actually, mostly its more manageable subalgebra, the Nichols algebra) that the bulk of this bulky paper studies.

But now to the story. You might notice that the arXiv stamp on this paper starts with 17! So yes — we have claimed this result before. I even blogged about it! But… that proof was not correct. The overall approach was the same as it is now, but our approach to bounding the cohomology of the Nichols algebra just wasn’t right, and we are incredibly indebted to Oscar Randall-Williams for making us aware of this.

For the last six years, we’ve been working on and off on fixing this. We kept thinking we had the decisive fix and then having it fall apart. But last spring, we had a new idea, Craig came and visited me for a very intense week, and by the end I think we were confident that we had a route — though getting to the present version of the paper occupied months after that.

A couple of thoughts about making mistakes in mathematics.

• I don’t think we really handled this properly. Experts in the field certainly knew we weren’t standing by the original claim, and we certainly told lots of people this in talks and in conversations, and I think in general there is still an understanding that if a preprint is sitting up on the arXiv for years and hasn’t been published, maybe there’s a reason — we haven’t completely abandoned the idea that a paper becomes more “official” when it’s refereed and published. But the right thing to do in this situation is what we did with an earlier paper with an incorrect proof — replaced the paper on arXiv with a placeholder saying it was inaccurate, and issued a public announcement. So why didn’t we do that? Probably because we were constantly in a state of feeling like we had a line on fixing the paper, and we wanted to update it with a correct version. I don’t actually think that’s a great reason — but that was the reason.
• When you break a bone it never exactly sets back the same way. And I think, having gotten this wrong before, I find it hard to be as self-assured about it as I am about most things I write. It’s long and it’s grainy and it has a lot of moving parts. But we have checked it as much as it’s possible for us to check it, over a long period of time. We understand it and we think we haven’t missed anything and so we think it’s correct now. And there’s no real alternative to putting it out into the world and saying we think it’s correct now.

My best friend—who’s held the title of best friend since kindergarten—calls me the keeper of her childhood memories. I recall which toys we played with, the first time I visited her house,¹ and which beverages our classmates drank during snack time in kindergarten.² She wouldn’t be surprised to learn that the first workshop I’ve co-organized centered on memory.

Memory—and the loss of memory—stars in thermodynamics. As an example, take what my husband will probably do this evening: bake tomorrow’s breakfast. I don’t know whether he’ll bake fruit-and-oat cookies, banana muffins, pear muffins, or pumpkin muffins. Whichever he chooses, his baking will create a scent. That scent will waft across the apartment, seep into air vents, and escape into the corridor—will disperse into the environment. By tomorrow evening, nobody will be able to tell by sniffing what my husband will have baked. 

That is, the kitchen’s environment lacks a memory. This lack contributes to our experience of time’s arrow: We sense that time passes partially by smelling less and less of breakfast. Physicists call memoryless systems and processes Markovian.

Our kitchen’s environment is Markovian because it’s large and particles churn through it randomly. But not all environments share these characteristics. Metaphorically speaking, a dispersed memory of breakfast may recollect, return to a kitchen, and influence the following week’s baking. For instance, imagine an atom in a quantum computer, rather than a kitchen in an apartment. A few other atoms may form our atom’s environment. Quantum information may leak from our atom into that environment, swish around in the environment for a time, and then return to haunt our atom. We’d call the atom’s evolution and environment non-Markovian.

I had the good fortune to co-organize a workshop about non-Markovianity—about memory—this February. The workshop took place at the Banff International Research Station, abbreviated BIRS, which you pronounce like the plural of what you say when shivering outdoors in Canada. BIRS operates in the Banff Centre for Arts and Creativity, high in the Rocky Mountains. The Banff Centre could accompany a dictionary entry for pristine, to my mind. The air feels crisp, the trees on nearby peaks stand out against the snow like evergreen fringes on white velvet, and the buildings balance a rustic-mountain-lodge style with the avant-garde. 

The workshop balanced styles, too, but skewed toward the theoretical and abstract. We learned about why the world behaves classically in our everyday experiences; about information-theoretic measures of the distances between quantum states; and how to simulate, on quantum computers, chemical systems that interact with environments. One talk, though, brought our theory back down to (the snow-dusted) Earth.

Gabriela Schlau-Cohen runs a chemistry lab at MIT. She wants to understand how plants transport energy. Energy arrives at a plant from the sun in the form of light. The light hits a pigment-and-protein complex. If the plant is lucky, the light transforms into a particle-like packet of energy called an exciton. The exciton traverses the receptor complex, then other complexes. Eventually, the exciton finds a spot where it can enable processes such as leaf growth. 

A high fraction of the impinging photons—85%—transform into excitons. How do plants convert and transport energy as efficiently as they do?

Gabriela’s group aims to find out—not by testing natural light-harvesting complexes, but by building complexes themselves. The experimentalists mimic the complex’s protein using DNA. You can fold DNA into almost any shape you want, by choosing the DNA’s base pairs (basic units) adroitly and by using “staples” formed from more DNA scraps. The sculpted molecules are called DNA origami.

Gabriela’s group engineers different DNA structures, analogous to complexes’ proteins, to have different properties. For instance, the experimentalists engineer rigid structures and flexible structures. Then, the group assesses how energy moves through each structure. Each structure forms an environment that influences excitons’ behaviors, similarly to how a memory-containing environment influences an atom.

The Banff environment influenced me, stirring up memories like powder displaced by a skier on the slopes above us. I first participated in a BIRS workshop as a PhD student, and then I returned as a postdoc. Now, I was co-organizing a workshop to which I brought a PhD student of my own. Time flows, as we’re reminded while walking down the mountain from the Banff Centre into town: A cemetery borders part of the path. Time flows, but we belong to that thermodynamically remarkable class of systems that retain memories…memories and a few other treasures that resist change, such as friendships held since kindergarten.

¹Plushy versions of Simba and Nala from The Lion King. I remain grateful to her for letting me play at being Nala.

²I’d request milk, another kid would request apple juice, and everyone else would request orange juice.

If , a Poisson random variable with mean is a random variable taking values in the natural numbers with probability distribution

Proposition 1 (Bennett’s inequality) One has

for and

for , where

From the Taylor expansion for we conclude Gaussian type tail bounds in the regime (and in particular when (in the spirit of the Chernoff, Bernstein, and Hoeffding inequalities). but in the regime where is large and positive one obtains a slight gain over these other classical bounds (of type, rather than ).

Proof: We use the exponential moment method. For any , we have from Markov’s inequality that

Remark 2 Bennett’s inequality also applies for (suitably normalized) sums of bounded independent random variables. In some cases there are direct comparison inequalities available to relate those variables to the Poisson case. For instance, suppose is the sum of independent Boolean variables of total mean and with for some . Then for any natural number , we have

As such, for small, one can efficiently control the tail probabilities of in terms of the tail probability of a Poisson random variable of mean close to ; this is of course very closely related to the well known fact that the Poisson distribution emerges as the limit of sums of many independent boolean variables, each of which is non-zero with small probability. See this paper of Bentkus and this paper of Pinelis for some further useful (and less obvious) comparison inequalities of this type.

In this note I wanted to record the observation that one can improve the Bennett bound by a small polynomial factor once one leaves the Gaussian regime , in particular gaining a factor of when . This observation is not difficult and is implicitly in the literature (one can extract it for instance from the much more general results of this paper of Talagrand, and the basic idea already appears in this paper of Glynn), but I was not able to find a clean version of this statement in the literature, so I am placing it here on my blog. (But if a reader knows of a reference that basically contains the bound below, I would be happy to know of it.)

Proposition 3 (Improved Bennett’s inequality) One has

for and

for .

Proof: We begin with the first inequality. We may assume that , since otherwise the claim follows from the usual Bennett inequality. We expand out the left-hand side as

Now we turn to the second inequality. As before we may assume that . We first dispose of a degenerate case in which . Here the left-hand side is just

It remains to consider the regime where and . The left-hand side expands as

The same analysis can be reversed to show that the bounds given above are basically sharp up to constants, at least when (and ) are large.

Thanks to the hard work of Frédéric Wang and the folks at Igalia, the Blink engine in Chrome 109 now supports MathML Core.

It took a little bit of work to get it working correctly in Instiki and on this blog.

• The columnalign attribute is not supported, so a shim is needed to get the individual <mtd> to align correctly.
• This commit enabled the display of SVG embedded in equations and got rid of the vertical scroll bars in equations.
• Since Chrome does not support hyperlinks (either href or xlink:href attributes) on MathML elements, this slightly hacky workaround enabled hyperlinks in equations, as created by \href{url}{expression}.

There are a number of remaining issues.

• Math accents don’t stretch, when they’re supposed to. Here are a few examples of things that (currently) render incorrectly in Chrome (some of them, admittedly, are incorrect in Safari too):

  $$\mathbf{V}_{1} \times \mathbf{V}_{2} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\\\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\ \end{vmatrix}$$

  $$\left\vert\frac{f(z)-f(a)}{1-\overline{f(a)}f(z)}\right\vert\le \left\vert\frac{z-a}{1-\overline{a}z}\right\vert$$

  $$\widetilde{PGL}(N)$$

  $$\begin{matrix} P_1(Y) &\to& P_1(X) \\ \downarrow &\Downarrow\mathrlap{\sim}& \downarrow \\ T' &\to& T \end{matrix}$$

  $$p_3 (x) = \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}} + \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}} + \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}} + \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}}$$

• This equation $$\boxed{(i\slash{D}+m)\psi = 0}$$ doesn’t display remotely correctly, because Chrome doesn’t implement the <menclose> element. Fixed now.

• …

But, hey, this is amazing for a first release.

Update:

I added support for \boxed{} and \slash{}, both of which use <menclose>, which is not supported by Chrome. So now the above equation should render correctly in Chrome. Thanks to Monica Kang, for help with the CSS.

After doing a night bottle feed of our youngest in the wee hours of the morning some nights earlier this week, in order to help me get back to sleep I decided to turn on BBC Sounds to find a programme to listen to... and lo and behold, look what had just aired live! The programme that I'd recorded at Broadcasting House a few weeks ago in London.

So it is out now. It is an episode of Jim Al-Khalili's excellent BBC Radio 4 programme "The Life Scientific". The show is very much in the spirit of what (as you know) I strive to do in my work in the public sphere (including this blog): discuss the science an individual does right alongside aspects of the broader life of that individual. I recommend listening to [...] Click to continue reading this post →

The post The Life Scientific Interview appeared first on Asymptotia.

I’m sitting, for the second night in a row, in a rather pleasant restaurant in Oxford, somewhere on the walk between the physics department and my hotel. They pour a pretty good Malbec, and tonight I’ve had the wood-fired Guinea Fowl. I can hear snippets of conversation in the distance, telling me that many people who come here are regulars, and that correlates well with the fact that I liked the place immediately last night and decided I’d come back. The friendly staff remembered me and greeted me like a regular upon my return, which I liked. Gee’s is spacious with a high ceiling, and so I can sit away from everyone in a time where I’d still rather not be too cavalier with regards covid. On another occasion I might have sought out a famous pub with some good pub food and be elbow-to-elbow with students and tourists, but the phrase “too soon” came to mind when I walked by such establishments and glanced into the windows.

However, I am not here to do a restaurant review, although you might have thought that from the previous paragraph (the guinea fowl was excellent though, and the risotto last night was tasty, if a tiny bit over-salted for my tastes). Instead I find myself reflecting on […] Click to continue reading this post →

The post What a Week! appeared first on Asymptotia.

As I mentioned in the previous post, I had business at BBC Broadcasting House this week. I was recording an interview that I’ll fill you in on later on, closer to release of the finished programme. Recall that in the post I mentioned how amusing it would be for me … Click to continue reading this post →

The post BBC Fun! appeared first on Asymptotia.

Happy 2023 everyone!  You’ve noticed, no doubt, that the blog has been quiet recently.  That’s because I’ve got a book contract, with a deadline of March 31, 2023.  [The book itself won’t be published til spring 2024.]  I’ll tell you more about this in future posts. But over the next couple of months I’ll be a bit slow to answer questions and even slower to write content.  Fortunately, much of the content on this website is still current — the universe seems to be much the same in 2023 as it was in 2011 when the site was born. So poke around; I’m sure you’ll find something that interests you!

Last week, Science Twitter was roiled by claims that “disruptive science” was on the wane and that this might be reversed by “reading widely”, taking “year long sabbaticals” and “focussing less on quantity … and more on …quality”. It blew up, which is probably not surprising given that it first pandered to our collective angst and then suggested some highly congenial remedies.

The Nature paper that kicked off this storm in our social media teacup is profusely illustrated with graphs and charts. The data is not uninteresting and does suggest that something about the practice of science has changed over the course of the last eight or nine decades. The problem is that it could also be Exhibit A in a demonstration of how data science can generate buzz while remaining largely disconnected from reality.

“Disruption” is a useful framework for discussing technological innovation (digital cameras render film obsolete; Netflix kills your neighbourhood video store, streaming music replaces CDs) but it is less clear to me that it can be applied directly to high-value science. “What is good?” is perhaps the oldest question in the book but the paper seems to skate past it.

The problem is (at least as I see it) many if not most scientific breakthroughs [1] extend the frontiers of knowledge rather than demolishing their forebears [2]. Even the biggest “paradigm shifts” often left their predecessors largely intact. Einstein arguably “disrupted” Newton but while film cameras and vinyl records are now the preserve of hipsters and purists, Newtonian physics is still at the heart of the field – as anyone who has taken first year physics or built a bridge that stood up can attest.

Similarly, quantum mechanics shattered the then-prevailing clockwork conception of the cosmos. However, its technical content was effectively a greenfield development since at a detailed level there was nothing for quantum mechanics to replace. By the end of the 1920s, however, quantum mechanics had given us the tools to explain almost everything that happens inside of an atom.

Consequently, as I see it, neither relativity or quantum mechanics really fits a conventional understanding of “disruption” even though they combine to create one the biggest revolutions ever seen in science. So that should be a problem if you are using “disruption” as a template for identifying interesting and important science.

Rather than making a qualitative assessment, the authors deploy a metric to measure disruption based on citation counts [3] – a widely cited paper whose own bibliographic antecedents then become less prominent is judged to be “disruptive” [4]. This leads to plots like the one below which focuses on Nobel winning papers and three “prestige” journals (Figure 5 from the paper).

If we take this study at its word, “disruption” has largely flatlined for the last fifty years. But one of the specific papers they identify – Riess et al.’s co-discovery of “dark energy” (or, more properly, observations suggesting that the rate at which the universe expands is picking up speed) is not rated as “disruptive” despite being the biggest upheaval in our understanding of the cosmos in a couple of generations.

Conversely, the discovery of the DNA double helix is measured to be “disruptive” — and it is certainly a watershed in our understanding of the the chemistry of life. The authors explain that it displaced an earlier “triple helix” model proposed by Linus Pauling – but Pauling’s scenario was less than a year old at this point so it was hardly an established incumbent knocked off its perch by a unexpected upstart. In fact, Watson and Crick’s 1953 discovery paper has only six references, and only one of those was published prior to 1952. Dirac’s 1928 paper scores well and it likewise has a handful of references and most those were similarly only a year or so old at the time of publication. However, the “disruption metric” looks for changes in citation patterns five years either side of publication. Consequently, even though there is no way their metric can produce meaningful data for these papers (given its reliance on a five year before-and-after comparison of citation counts) they single them out for special attention rather than filtering them and papers like them from their dataset.

What this suggests to me is that there has not been a sufficiently rigorous sniff-testing of the output of this algorithm. So on top of adopting a model of progress without really asking whether or not it captures the essence of “breakthrough” science the output of the metric used to assess it was often reverse-engineered to justify the numerical values it yields.

The concern that science is increasingly driven by “bean counting” and a publish or perish mentality that is at odds with genuine progress is widespread, and my own view (like most scientists, I would guess) is that there is truth to it. There is certainly a lot of frog-boiling in academia: it is indeed a challenge for working scientists to get long periods to reflect and explore and junior scientists are locked into a furiously competitive job market that offers little security to its participants.

Ironically, though, one key contributor to this pressure-cooker in which we find ourselves is Nature itself, the journal that published this paper. And Nature not only published it but hyped it in a news article – an incestuous coupling between peer reviewed content and “news” that can make the careers of those fortunate enough to participate in it. However, it is widely argued that this practice makes Nature itself a contributor to any decline of scientific quality that may be taking place by nudging authors to hype their work in ways not fully justified by their actual results. But “turning off the hype machine” is not one of the proposed solutions to our problems — and a cynic might suggest that this could be because it would also disable the money spigot that generates many millions of dollars a year for Nature’s very-definitely for-profit owners.

To some extent this is just me being cranky, since I spent part of last week at a slow simmer every time I saw this work flash by on a screen. But it matters, because this sort of analysis can find its way into debates about how to “fix” the supposed problems of science. And there certainly are many ways in which we could make science better. But before we prescribe we would be wise to accurately determine the symptoms of its illness. Coming up with numerical metrics to measure quality and impact in science is enormously tempting since it converts an otherwise laborious and qualitative process into something that it is both quantitative and automated [5] — but it is also very difficult, and it hasn’t happened here.

Ironically, the authors of this work are a professor in a management school, his PhD student and a sociologist who claim all expertise in “innovation” and “entrepreneurship”. Physicists are often seen as more willing than most to have opinions on matters outside of our professional domain and we are increasingly likely to be rebuked for failures to “stay in our lane”. But that advice cuts both ways; if you want to have opinions on science maybe you should work with people who have real expertise in the fields you hope to assess?

[1] I am going to focus on physics, since that is what I know best – but the pattern is claimed to be largely field-independent.

[2] There are exceptions. The heliocentric solar system supplanted the geocentric view and “caloric fluid” is no longer seen as a useful description of heat, but the norm for physics (and much of 20th century chemistry and biology, so far as I can see) is to “amend and extend”. There are often competing explanations for a phenomenon – e.g. Big Bang cosmology v. Steady State – only one of which can “win”, but these more closely resemble rivalries like the contest between BetaMax and VHS than “disruption”.

[3] They also make an argument that the language we use to talk about scientific results has changed over time, but most of the story has been based on their “disruption” metric.

[4] It had been used previously on patent applications (which must list “prior art”) by one of the authors, where it may actually make more sense.

[5] See also my views on the h-index.

Banner image: https://memory-alpha.fandom.com/wiki/Disruptor

[caption id="attachment_20038" align="aligncenter" width="499"] Brompton bicycle rental lockers.[/caption]
I’ll be visiting Broadcasting House during my time here in London this week, for reasons I’ll mention later. Needless to say (almost), as a Brompton rider, and fan of the wonderful show W1A, I feel a sense of regret that I don’t have my bike here so that I can ride up to the front of the building on it. you won’t know what I’m talking about if you don’t know the show. Well, last night I was a-wandering and saw the rental option shown in the photo. It is very tempting…

-cvj
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[I was originally going to use the title “Back Home”, but then somehow this choice had a resonance to it that I liked. (Also reminds me of a lovely Joshua Redman album…)] So I am back in London, my home town. And since I’ve got 8 hour jet lag, I’m … Click to continue reading this post →

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