Planet Musings

An European Union commission has voted to ban the use of lead ammunition near wetlands and waterways! The proposal now needs to be approved by the European Parliament and Council. They are expected to approve the ban. If so, it will go into effect in 2022. The same commission, called REACH, may debate a complete ban on lead ammunition and fishing weights later this year.

Why does this matter? The European Chemicals Agency has estimated that as many as 1.5 million aquatic birds die annually from lead poisoning because they swallow some of the 5000 tonnes of lead shot that land in European wetlands each year. Water birds are more likely to be poisoned by lead because they mistake small lead shot pellets for stones they deliberately ingest to help grind their food.

In fact, about 20,000 tonnes of lead shot is fired each year in the EU, and 60,000 in the US. Eating game shot with lead is not good for you—but also, even low levels of lead in the environment can cause health damage and negative changes in behavior.

How much lead is too much? This is a tricky question, so I’ll just give some data. In the U.S., the geometric mean of the blood lead level among adults was 1.2 micrograms per deciliter (μg/dL) in 2009–2010. Blood lead concentrations in poisoning victims ranges from 30-80 µg/dL in children exposed to lead paint in older houses, 80–100 µg/dL in people working with pottery glazes, 90–140 µg/dL in individuals consuming contaminated herbal medicines, 110–140 µg/dL in indoor shooting range instructors and as high as 330 µg/dL in those drinking fruit juices from glazed earthenware containers!

The amount of lead that US children are exposed to has been dropping, thanks to improved regulations:

However, what seem like low levels now may be high in the grand scheme of things. The amount of lead in the environment has increased by a factor of about 300 in the Greenland ice sheet during the past 3000 years. Most of this is due to industrial emissions:

• Amy Ng and Clair Patterson, Natural concentrations of lead in ancient Arctic and Antarctic ice, Geochimica et Cosmochimica Acta 45 (1981), 2109–2121.

I spent my research time this morning typing math into a document co-authored by Ana Bonaca (Harvard) about a likelihood function for asteroseismology. It is tedious and repetitive to type the math out in LaTeX, and I feel like I have done the nearly-same typing over and over again. And, adding to the feeling of futility: The linear-algebra objects we define in the paper are more conceptual than operational: In many cases we don't actually construct these objects; we avoid constructing things that will make the execution either slow or imprecise. So in addition to typing these, I find myself also typing lots of non-trivial implementation advice.

If you ever think metaphysics is easy, learn a little quantum field theory.

Someone asked me recently about virtual particles. When talking to the public, physicists sometimes explain the behavior of quantum fields with what they call “virtual particles”. They’ll describe forces coming from virtual particles going back and forth, or a bubbling sea of virtual particles and anti-particles popping out of empty space.

The thing is, this is a metaphor. What’s more, it’s a metaphor for an approximation. As physicists, when we draw diagrams with more and more virtual particles, we’re trying to use something we know how to calculate with (particles) to understand something tougher to handle (interacting quantum fields). Virtual particles, at least as you’re probably picturing them, don’t really exist.

I don’t really blame physicists for talking like that, though. Virtual particles are a metaphor, sure, a way to talk about a particular calculation. But so is basically anything we can say about quantum field theory. In quantum field theory, it’s pretty tough to say which things “really exist”.

I’ll start with an example, neutrino oscillation.

You might have heard that there are three types of neutrinos, corresponding to the three “generations” of the Standard Model: electron-neutrinos, muon-neutrinos, and tau-neutrinos. Each is produced in particular kinds of reactions: electron-neutrinos, for example, get produced by beta-plus decay, when a proton turns into a neutron, an anti-electron, and an electron-neutrino.

Leave these neutrinos alone though, and something strange happens. Detect what you expect to be an electron-neutrino, and it might have changed into a muon-neutrino or a tau-neutrino. The neutrino oscillated.

Why does this happen?

One way to explain it is to say that electron-neutrinos, muon-neutrinos, and tau-neutrinos don’t “really exist”. Instead, what really exists are neutrinos with specific masses. These don’t have catchy names, so let’s just call them neutrino-one, neutrino-two, and neutrino-three. What we think of as electron-neutrinos, muon-neutrinos, and tau-neutrinos are each some mix (a quantum superposition) of these “really existing” neutrinos, specifically the mixes that interact nicely with electrons, muons, and tau leptons respectively. When you let them travel, it’s these neutrinos that do the traveling, and due to quantum effects that I’m not explaining here you end up with a different mix than you started with.

This probably seems like a perfectly reasonable explanation. But it shouldn’t. Because if you take one of these mass-neutrinos, and interact with an electron, or a muon, or a tau, then suddenly it behaves like a mix of the old electron-neutrinos, muon-neutrinos, and tau-neutrinos.

That’s because both explanations are trying to chop the world up in a way that can’t be done consistently. There aren’t electron-neutrinos, muon-neutrinos, and tau-neutrinos, and there aren’t neutrino-ones, neutrino-twos, and neutrino-threes. There’s a mathematical object (a vector space) that can look like either.

Whether you’re comfortable with that depends on whether you think of mathematical objects as “things that exist”. If you aren’t, you’re going to have trouble thinking about the quantum world. Maybe you want to take a step back, and say that at least “fields” should exist. But that still won’t do: we can redefine fields, add them together or even use more complicated functions, and still get the same physics. The kinds of things that exist can’t be like this. Instead you end up invoking another kind of mathematical object, equivalence classes.

If you want to be totally rigorous, you have to go a step further. You end up thinking of physics in a very bare-bones way, as the set of all observations you could perform. Instead of describing the world in terms of “these things” or “those things”, the world is a black box, and all you’re doing is finding patterns in that black box.

Is there a way around this? Maybe. But it requires thought, and serious philosophy. It’s not intuitive, it’s not easy, and it doesn’t lend itself well to 3d animations in documentaries. So in practice, whenever anyone tells you about something in physics, you can be pretty sure it’s a metaphor. Nice describable, non-mathematical things typically don’t exist.

 I've written before about the Einstein-deHaas effect - supposedly Einstein's only experimental result (see here, too) - a fantastic proof that spin really is angular momentum.  In that experiment, a magnetic field is flipped, causing the magnetization of a ferromagnet to reorient itself to align with the new field direction.  While Einstein and deHaas thought about amperean current loops (the idea that magnetization came from microscopic circulating currents that we would now call orbital magnetism), we now know that magnetization in many materials comes from the spin of the electrons.  When those spins reorient, angular momentum has to be conserved somehow, so it is transferred to/from the lattice, resulting in a mechanical torque that can be measured.

Less well-known is the complement, the Barnett effect.  Take a ferromagnetic material and rotate it. The mechanical rotational angular momentum gets transferred (via rather complicated physics, it turns out) at some rate to the spins of the electrons, causing the material to develop a magnetization along the rotational axis.  This seems amazing to me now, knowing about spin.  It must've really seemed nearly miraculous back in 1915 when it was measured by Barnett.

So, how did Barnett actually measure this, with the technology available in 1915?  Here's the basic diagram of the scheme from the original paper:

There are two rods that can each be rotated about its long axis.  The rods pass through counterwound coils, so that if there is a differential change in magnetic flux through the two coils, that generates a current that flows through the fluxmeter.  The Grassot fluxmeter is a fancy galvanometer - basically a coil suspended on a torsion fiber between poles of a magnet.  Current through that coil leads to a torque on the fiber, which is detected in this case by deflection of a beam of light bounced off a mirror mounted on the fiber.  The paper describes the setup in great detail, and getting this to work clearly involved meticulous experimental technique and care.  It's impressive how people were able to do this kind of work without all the modern electronics that we take for granted.  Respect.

Michael Weiss and I have been carrying on a dialog on nonstandard models of arithmetic, and after a long break we’re continuing, here:

• Michael Weiss and John Baez, Non-standard models of arithmetic (Part 18).

In this part we reach a goal we’ve been heading toward for a long time! We’ve been reading this paper:

• Ali Enayat, Standard models of arithmetic.

and we’ve finally gone through the main theorems and explained what they say. We’ll talk about the proofs later.

The simplest one is this:

• Every ZF-standard model of PA that is not V-standard is recursively saturated.

What does this theorem mean, roughly? Let me be very sketchy here, to keep things simple and give just a flavor of what’s going on.

Peano arithmetic is a well-known axiomatic theory of the natural numbers. People study different models of Peano arithmetic in some universe of sets, say U. If we fix our universe U there is typically one ‘standard’ model of Peano arithmetic, built using the set

or in other words

All models of Peano arithmetic not isomorphic to this one are called ‘nonstandard’. You can show that any model of Peano arithmetic contains an isomorphic copy of standard model as an initial segment. This uniquely characterizes the standard model.

But different axioms for set theory give different concepts of U, the universe of sets. So the uniqueness of the standard model of Peano arithmetic is relative to that choice!

Let’s fix a choice of axioms for set theory: the Zermelo–Fraenkel or ‘ZF’ axioms are a popular choice. For the sake of discussion I’ll assume these axioms are consistent. (If they turn out not to be, I’ll correct this post.)

We can say the universe U is just what ZF is talking about, and only theorems of ZF count as things we know about U. Or, we can take a different attitude. After all, there are a couple of senses in which the ZF axioms don’t completely pin down the universe of sets.

First, there are statements in set theory that are neither provable nor disprovable from the ZF axioms. For any of these statements we’re free to assume it holds, or it doesn’t hold. We can add either it or its negation to the ZF axioms, and still get a consistent theory.

Second, a closely related sense in which the ZF axioms don’t uniquely pin down U is this: there are many different models of the ZF axioms.

Here I’m talking about models in some universe of sets, say V. This may seem circular! But it’s not really: first we choose some way to deal with set theory, and then we study models of the ZF axioms in this context. It’s a useful thing to do.

So fix this picture in mind. We start with a universe of sets V. Then we look at different models of ZF in V, each of which gives a universe U. U is sitting inside V, but from inside it looks like ‘the universe of all sets’.

Now, for each of these universes U we can study models of Peano arithmetic in U. And as I already explained, inside each U there will be a standard model of Peano arithmetic. But of course this depends on U.

So, we get lots of standard models of Peano arithmetic, one for each choice of U. Enayat calls these ZF-standard models of Peano arithmetic.

But there is one very special model of ZF in V, namely V itself. In other words, one choice of U is to take U = V. There’s a standard model of Peano arithmetic in V itself. This is an example of a ZF-standard model, but this is a very special one. Let’s call any model of Peano arithmetic isomorphic to this one V-standard.

Enayat’s theorem is about ZF-standard models of Peano arithmetic that aren’t V-standard. He shows that any ZF-standard model that’s not V-standard is ‘recursively saturated’.

What does it mean for a model M of Peano arithmetic to be ‘recursively saturated’? The idea is very roughly that ‘anything that can happen in any model, happens in M’.

Let me be a bit more precise. It means that if you write any computer program that prints out an infinite list of properties of an n-tuple of natural numbers, and there’s some model of Peano arithmetic that has an n-tuple with all these properties, then there’s an n-tuple of natural numbers in the model M with all these properties.

For example, there are models of Peano arithmetic that have a number x such that

0 < x
1 < x
2 < x
3 < x

and so on, ad infinitum. These are the nonstandard models. So a recursively saturated model must have such a number x. So it must be nonstandard.

In short, Enayat has found that ZF-standard models of Peano arithmetic in the universe V come in two drastically different kinds. They are either ‘as standard as possible’, namely V-standard. Or, they are ‘extremely rich’, containing n-tuples with all possible lists of consistent properties that you can print out with a computer program: they are recursively saturated.

I am probably almost as confused about this as you are. But Michael and I will dig into this more in our series of posts.

In fact we’ve been at this a while already. Here is a description of the whole series of posts so far:

Posts 1–10 are available as pdf files, formatted for small and medium screens.

Non-standard Models of Arithmetic 1: John kicks off the series by asking about recursively saturated models, and Michael says a bit about n-types and the overspill lemma. He also mentions the arithmetical hierarchy.

Non-standard Models of Arithmetic 2: John mention some references, and sets a goal: to understand this paper:

• Ali Enayat, Standard models of arithmetic.

John describes his dream: to show that “the” standard model is a much more nebulous notion than many seem to believe. He says a bit about the overspill lemma, and Joel David Hamkins gives a quick overview of saturation.

Non-standard Models of Arithmetic 3: A few remarks on the ultrafinitists Alexander Yessenin-Volpin and Edward Nelson; also Michael’s grad-school friend who used to argue that 7 might be nonstandard.

Non-standard Models of Arithmetic 4: Some back-and-forth on Enayat’s term “standard model” (or “ZF-standard model”) for the omega of a model of ZF. Philosophy starts to rear its head.

Non-standard Models of Arithmetic 5: Hamlet and Polonius talk math, and Michael holds forth on his philosophies of mathematics.

Non-standard Models of Arithmetic 6: John weighs in with why he finds “the standard model of Peano arithmetic” a problematic phrase. The Busy Beaver function is mentioned.

Non-standard Models of Arithmetic 7: We start on Enayat’s paper in earnest. Some throat-clearing about Axiom SM, standard models of ZF, inaccessible cardinals, and absoluteness. “As above, so below”: how ZF makes its “gravitational field” felt in PA.

Non-standard Models of Arithmetic 8: A bit about the Paris-Harrington and Goodstein theorems. In preparation, the equivalence (of sorts) between PA and ZF¬∞. The universe V_ω of hereditarily finite sets and its correspondence with . A bit about Ramsey’s theorem (needed for Paris-Harrington). Finally, we touch on the different ways theories can be “equivalent”, thanks to a comment by Jeffrey Ketland.

Non-standard Models of Arithmetic 9: Michael sketches the proof of the Paris-Harrington theorem.

Non-Standard Models of Arithmetic 10: Ordinal analysis, the function growth hierarchies, and some fragments of PA. Some questions that neither of us knows how to answer.

Non-standard Models of Arithmetic 11: Back to Enayat’s paper: his definition of PA^T for a recursive extension T of ZF. This uses the translation of formulas of PA into formulas of ZF, . Craig’s trick and Rosser’s trick.

Non-standard Models of Arithmetic 12: The strength of PA^T for various T‘s. PA^ZF is equivalent to PA^ZFC+GCH, but PA^ZFI is strictly stronger than PA^ZF. (ZFI = ZF + “there exists an inaccessible cardinal”.)

Non-standard Models of Arithmetic 13: Enayat’s “natural” axiomatization of PA^T, and his proof that this works. A digression into Tarski’s theorem on the undefinability of truth, and how to work around it. For example, while truth is not definable, we can define truth for statements with at most a fixed number of quantifiers.

Non-standard Models of Arithmetic 14: The previous post showed that PA^T implies Φ_T, where Φ_T is Enayat’s “natural” axiomatization of PA^T. Here we show the converse. We also interpret Φ_T as saying, “Trust T”.

Non-standard Models of Arithmetic 15: We start to look at Truth (aka Satisfaction). Tarski gave a definition of Truth, and showed that Truth is undefinable. Less enigmatically put, there is no formula True(x) in the language of Peano arithmetic (L(PA)) that holds exactly for the Gödel numbers of true sentences of Peano arithmetic. On the other hand, Truth for Peano arithmetic can be formalized in the language of set theory (L(ZF)), and there are other work-arounds. We give an analogy with the Cantor diagonal argument.

Non-standard Models of Arithmetic 16: We look at the nitty-gritty of True_d(x), the formula in L(PA) that expresses truth in PA for formulas with parse-tree depth at most d. We see how the quantifiers “bleed through”, and why this prevents us from combining the whole series of True_d(x)’s into a single formula True(x). We also look at the variant Sat_Σn(x,y).

Non-standard Models of Arithmetic 17: More about how True_d evades Tarski’s undefinability theorem. The difference between True_d and Sat_Σn, and how it doesn’t matter for us. How True_d captures Truth for models of arithmetic: PA ⊢ True_d(⌜φ⌝) φ, for any φ of parse-tree depth at most d. Sketch of why this holds.

• Non-standard Models of Arithmetic 18: The heart of Enayat’s paper: characterizing countable nonstandard T-standard models of PA (Prop. 6, Thm. 7, Cor. 8). Refresher on types. Meaning of ‘recursively saturated’. Standard meaning of ‘nonstandard’; standard and nonstandard meanings of ‘standard’.

• Non-standard Models of Arithmetic 19: We marvel a bit over Enayat’s Prop. 6, and especially Cor. 8. The triple-decker sandwich, aka three-layer cake: ω^U⊂U⊂V. More about why the omegas of standard models of ZF are standard. Refresher on Φ_T. The smug confidence of a ZF-standard model.

Because of a certain movie from earlier this Summer (which I have not yet got around to mentioning here on the blog), I’ve been doing a lot of interviews recently, so sorry in advance for my face showing up in all your media. And I know many will sneer because … Click to continue reading this post →

The post Early Career Musings appeared first on Asymptotia.

Hurwitz’s theorem says that there are only 4 normed division algebras over the real numbers, up to isomorphism: the real numbers, the complex numbers, the quaternions, and the octonions. The proof was published in 1923. It’s a famous result, and several other proofs are known. I’ve spent a lot of time studying them.

Thus you can imagine my surprise today when I learned Hurwitz’s theorem was false!

• Joy Christian, Eight-dimensional octonion-like but associative normed division algebra, Communications in Algebra (2020), 1-10.

Abstract. We present an eight-dimensional even sub-algebra of the $2^4=16$-dimensional associative Clifford algebra $\mathrm{Cl}_{4,0}$ and show that its eight-dimensional elements denoted as $\mathbf{X}$ and $\mathbf{Y}$ respect the norm relation $\| \mathbf{X} \mathbf{Y}\| = \| \mathbf{X} \| \| \mathbf{Y} \|$, thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.

Even more wonderful is that the author has discovered that the unit vectors in his normed division algebra form a 7-sphere that is not homeomorphic to the standard 7-sphere. Exotic 7-spheres are a dime a dozen, but those merely fail to be diffeomorphic to the standard 7-sphere.

Well, no: this article must be wrong.

Doesn’t Communications in Algebra have some mechanism where a few editors look at every paper — just to be sure they make sense?

The category of cocommutative comonoid objects in a symmetric monoidal category is cartesian, with their tensor product serving as their product. This result seems to date back to here:

• Thomas Fox, Coalgebras and Cartesian categories, Comm. Alg. 4 (1976), 665–667.

Dually, the category of commutative monoid objects in a symmetric monoidal category is cocartesian. This was proved in Fox’s suspiciously similar paper in Cocomm. Coalg.

I’m working on a paper with Todd Trimble and Joe Moeller, and right now we need something similar one level up — that is, for symmetric pseudomonoids. (For example, a symmetric pseudomonoid in Cat is a symmetric monoidal category.)

The 2-category of symmetric pseudomonoids in a symmetric monoidal 2-category should be cocartesian, with their tensor product serving as their coproduct. I imagine the coproduct universal property will hold only up to 2-iso.

Has someone proved this, so we don’t need to?

Hmm, I just noticed that this paper:

• Brendan Fong and David I, Spivak, Supplying bells and whistles in symmetric monoidal categories.

proves the result I want in the special case where the symmetric monoidal 2-category is Cat. Namely:

Theorem 2.3. The 2-category SMC of symmetric monoidal categories, strong monoidal functors, and monoidal natural transformations has 2-categorical biproducts.

Unfortunately their proof is not purely ‘formal’, so it doesn’t instantly generalize to other symmetric monoidal 2-categories. And surely the fact that the coproducts in SMC are biproducts must rely on the fact that Cat is a cartesian 2-category; this must fail for symmetric pseudomonoids in a general symmetric monoidal 2-category.

They do more:

Theorem 2.2. The 2-category SMC has all small products and coproducts, and products are strict.

read more

Why is biodiversity ‘good’? To what extent is this sort of goodness even relevant to ecosystems—as opposed to us humans? I’d like to study this mathematically.

To do this, we’d need to extract some answerable questions out of the morass of subtlety and complexity. For example: what role does biodiversity play in the ability of ecosystems to be robust under sudden changes of external conditions? This is already plenty hard to study mathematically, since it requires understanding ‘biodiversity’ and ‘robustness’.

Luckily there has already been a lot of work on the mathematics of biodiversity and its connection to entropy. For example:

• Tom Leinster, Measuring biodiversity, Azimuth, 7 November 2011.

But how does biodiversity help robustness?

There’s been a lot of work on this. This paper has some inspiring passages:

• Robert E. Ulanowicz,, Sally J. Goerner, Bernard Lietaer and Rocio Gomez, Quantifying sustainability: Resilience, efficiency and the return of information theory, Ecological Complexity 6 (2009), 27–36.

I’m not sure the math lives up to their claims, but I like these lines:

In other words, (14) says that the capacity for a system to undergo evolutionary change or self-organization consists of two aspects: It must be capable of exercising sufficient directed power (ascendancy) to maintain its integrity over time. Simultaneously, it must possess a reserve of flexible actions that can be used to meet the exigencies of novel disturbances. According to (14) these two aspects are literally complementary.

The two aspects are ‘ascendancy’, which is something like efficiency or being optimized, and ‘reserve capacity’, which is something like random junk that might come in handy if something unexpected comes up.

You know those gadgets you kept in the back of your kitchen drawer and never needed… until you did? If you’re aiming for ‘ascendancy’ you’d clear out those drawers. But if you keep that stuff, you’ve got more ‘reserve capacity’. They both have their good points. Ideally you want to strike a wise balance. You’ve probably sensed this every time you clean out your house: should I keep this thing because I might need it, or should I get rid of it?

I think it would be great to make these concepts precise. The paper at hand attempts this by taking a matrix of nonnegative numbers to describe flows in an ecological network. They define a kind of entropy for this matrix, very similar in look to Shannon entropy. Then they write this as a sum of two parts: a part closely analogous to mutual information, and a part closely analogous to conditional entropy. This decomposition is standard in information theory. This is their equation (14).

If you want to learn more about the underlying math, click on this picture:

The new idea of these authors is that in the context of an ecological network, the mutual information can be understood as ‘ascendancy’, while the conditional entropy can be understood as ‘reserve capacity’.

I don’t know if I believe this! But I like the general idea of a balance between ascendancy and reserve capacity.

They write:

While the dynamics of this dialectic interaction can be quite subtle and highly complex, one thing is boldly clear—systems with either vanishingly small ascendancy or insignificant reserves are destined to perish before long. A system lacking ascendancy has neither the extent of activity nor the internal organization needed to survive. By contrast, systems that are so tightly constrained and honed to a particular environment appear ‘‘brittle’’ in the sense of Holling (1986) or ‘‘senescent’’ in the sense of Salthe (1993) and are prone to collapse in the face of even minor novel disturbances. Systems that endure—that is, are sustainable—lie somewhere between these extremes. But, where?

I spent part of the weekend looking at issues with chemical abundances as a function of position and velocity in the Solar neighborhood. In the data, it looks like somehow the main-sequence abundances in the APOGEE and GALAH data sets are different for stars in different positions along the same orbit. That's bad for my Chemical Torus Imaging (tm) project with Adrian Price-Whelan (Flatiron)! But slowly we realized that the issue is that the abundances depend on stellar effective temperatures, and different temperatures are differently represented in different parts of the orbit. Phew. But there is a problem with the data. (Okay actually this can be a real, physical effect or a problem with the data; either way, we have to deal.) Time to call Christina Eilers (MIT), who is thinking about exactly this kind of abundance-calibration problem.

Some good news! According to this article, we’re rapidly approaching the tipping point when, even without subsidies, it will be as cheaper to own an electric car than one that burns fossil fuels.

• Jack Ewing, The age of electric cars is dawning ahead of schedule, New York Times, September 20, 2020.

FRANKFURT — An electric Volkswagen ID.3 for the same price as a Golf. A Tesla Model 3 that costs as much as a BMW 3 Series. A Renault Zoe electric subcompact whose monthly lease payment might equal a nice dinner for two in Paris.

As car sales collapsed in Europe because of the pandemic, one category grew rapidly: electric vehicles. One reason is that purchase prices in Europe are coming tantalizingly close to the prices for cars with gasoline or diesel engines.

At the moment this near parity is possible only with government subsidies that, depending on the country, can cut more than $10,000 from the final price. Carmakers are offering deals on electric cars to meet stricter European Union regulations on carbon dioxide emissions. In Germany, an electric Renault Zoe can be leased for 139 euros a month, or $164.

Electric vehicles are not yet as popular in the United States, largely because government incentives are less generous. Battery-powered cars account for about 2 percent of new car sales in America, while in Europe the market share is approaching 5 percent. Including hybrids, the share rises to nearly 9 percent in Europe, according to Matthias Schmidt, an independent analyst in Berlin.

As electric cars become more mainstream, the automobile industry is rapidly approaching the tipping point when, even without subsidies, it will be as cheap, and maybe cheaper, to own a plug-in vehicle than one that burns fossil fuels. The carmaker that reaches price parity first may be positioned to dominate the segment.

A few years ago, industry experts expected 2025 would be the turning point. But technology is advancing faster than expected, and could be poised for a quantum leap. Elon Musk is expected to announce a breakthrough at Tesla’s “Battery Day” event on Tuesday that would allow electric cars to travel significantly farther without adding weight.

The balance of power in the auto industry may depend on which carmaker, electronics company or start-up succeeds in squeezing the most power per pound into a battery, what’s known as energy density. A battery with high energy density is inherently cheaper because it requires fewer raw materials and less weight to deliver the same range.

“We’re seeing energy density increase faster than ever before,” said Milan Thakore, a senior research analyst at Wood Mackenzie, an energy consultant which recently pushed its prediction of the tipping point ahead by a year, to 2024.

However, the article also points out that this tipping point is of the overall lifetime cost of the vehicle! The sticker price of electric cars will still be higher for a while. And there aren’t nearly enough charging stations!

My next car will be electric. But first I’m installing solar power for my house. I’m working on it now.

Nina Holden won the 2021 Maryam Mirzakhani New Frontiers Prize for her work on random surfaces and the mathematics of quantum gravity. I’d like to tell you what she did… but I’m so far behind I’ll just explain a bit of the background.

Suppose you randomly choose a triangulation of the sphere with n triangles. This is a purely combinatorial thing, but you can think of it as a metric space if each of the triangles is equilateral with all sides of length 1.

This is a distorted picture of what you might get, drawn by Jérémie Bettinelli:

The triangles are not drawn as equilateral, so we can fit this shape into 3d space. Visit Bettinelli’s page for images that you can rotate:

• Jérémie Bettinelli, Computer simulations of random maps.

I’ve described how to build a random space out of triangles. In the limit if you rescale the resulting space by a factor of so it doesn’t get bigger and bigger, it converges to a ‘random metric space’ with fascinating properties. It’s called the Brownian map.

This random metric space is on average so wrinkly and crinkly that ‘almost surely’—that is, with probability 1—its Hausdorff dimension is not 2 but 4. And yet it is almost surely homeomorphic to a sphere!

Rigorously proving this is hard: a mix of combinatorics, probability theory and geometry.

Ideas from physics are also important here. There’s a theory called
‘Liouville quantum gravity’ that describes these random 2-dimensional surfaces. So, physicists have ways of—nonrigorously—figuring out answers to some questions faster than the mathematicians!

A key step in understanding the Brownian map was this paper from 2013:

• Jean-François Le Gall, Uniqueness and universality of the Brownian map, Annals of Probability 41 (2013), 2880–2960.

The Brownian map is to surfaces what Brownian motion is to curves. For example, the Hausdorff dimension of Brownian motion is almost surely 2: twice the dimension of a smooth curve. For the Brownian map it’s almost surely 4, twice the dimension of a smooth surface.

Let me just say one more technical thing. There’s a ‘space of all compact metric spaces’, and the Brownian map is actually a probability measure on this space! It’s called the Gromov-Hausdorff space, and it itself is a metric space… but not compact. (So no, we don’t have a set containing itself as an element.)

There’s a lot more to say about this… but I haven’t gotten very close to understanding Nina Holden’s work yet. She wrote a 7-paper series leading up to this one:

• Nina Holden and Xin Sun, Convergence of uniform triangulations under the Cardy embedding.

They show that random triangulations of a disk can be chosen to a random metric on the disk which can also be obtained from Liouville quantum gravity.

This is a much easier place to start learning this general subject:

• Ewain Gwynne, Random surfaces and Liouville quantum gravity.

One reason I find all this interesting is that when I worked on ‘spin foam models’ of quantum gravity, we were trying to develop combinatorial theories of spacetime that had nice limits as the number of discrete building blocks approached infinity. We were studying theories much more complicated than random 2-dimensional triangulations, and it quickly became clear to me how much work it would be to carefully analyze these. So it’s interesting to see how mathematicians have entered this subject—starting with a much more tractable class of theories, which are already quite hard.

While the theory I just described gives random metric spaces whose Hausdorff dimension is twice their topological dimension, Liouville quantum gravity actually contains an adjustable parameter that lets you force these metric spaces to become less wrinkled, with lower Hausdorff dimension. Taming the behavior of random triangulations gets harder in higher dimensions. Renate Loll, Jan Ambjørn and others have argued that we need to work with Lorentzian rather than Riemannian geometries to get physically reasonable behavior. This approach to quantum gravity is called causal dynamical triangulations.

The Department of Electrical and Computer Engineering at Rice University invites applications for a tenure track Assistant Professor Position in the area of experimental quantum engineering, broadly defined. Under exceptional circumstances, more experienced senior candidates may be considered. Specific areas of interest include, but are not limited to: quantum computation, quantum sensing, quantum simulation, and quantum networks.

The department has a vibrant research program in novel, leading-edge research areas, has a strong culture of interdisciplinary and multidisciplinary research with great national and international visibility, and is ranked #1 nationally in faculty productivity.* With multiple faculty involved in quantum materials, quantum devices, optics and photonics, and condensed matter physics, Rice ECE considers these areas as focal points of quantum engineering research in the coming decade. The successful applicant will be required to teach undergraduate courses and build a successful research program.

The successful candidate will have a strong commitment to teaching, advising, and mentoring undergraduate and graduate students from diverse backgrounds. Consistent with the National Research Council’s report, Convergence: Facilitating Transdisciplinary Integration of Life Sciences, Physical Sciences, Engineering, and Beyond, we are seeking candidates who have demonstrated ability to lead and work in research groups that “… [integrate] the knowledge, tools, and ways of thinking…” from engineering, mathematics, and computational, natural, social and behavioral sciences to solve societal problems using a convergent approach.

Applicants should submit a cover letter, curriculum vitae, statements of research and teaching interests, and at least three references through the Rice faculty application website: http://jobs.rice.edu/postings/24582. The deadline for applications is January 15, 2021; review of applications will commence November 15, 2020. The position is expected to be available July 1, 2021. Additional information can be found on our website: http://www.ece.rice.edu.

Rice University is a private university with a strong reputation for academic excellence in both undergraduate and graduate education and research. Located in the economically dynamic, internationally diverse city of Houston, Texas, 4th largest city in the U.S., Rice attracts outstanding undergraduate and graduate students from across the nation and around the world. Rice provides a stimulating environment for research, teaching, and joint projects with industry.

The George R. Brown School of Engineering ranks among the top 20 of undergraduate engineering programs (US News & World Report) and is strongly committed to nurturing the aspirations of faculty, staff, and students in an inclusive environment. Rice University is an Equal Opportunity Employer with commitment to diversity at all levels and considers for employment qualified applicants without regard to race, color, religion, age, sex, sexual orientation, gender identity, national or ethnic origin, genetic information, disability, or protected veteran status. We seek greater representation of women, minorities, people with disabilities, and veterans in disciplines in which they have historically been underrepresented; to attract international students from a wider range of countries and backgrounds; to accelerate progress in building a faculty and staff who are diverse in background and thought; and we support an inclusive environment that fosters interaction and understanding within our diverse community.

*http://news.rice.edu/2007/11/30/rices-electrical-engineering-and-computer-science-programs-rank-no-1/ Rice University is an Equal Opportunity Employer with commitment to diversity at all levels, and considers for employment qualified applicants without regard to race, color, religion, age, sex, sexual orientation, gender identity, national or ethnic origin, genetic information, disability or protected veteran status.

The Department of Physics and Astronomy at Rice University invites applications for a tenure-track faculty position in astronomy in the general field of galactic star formation, including the formation and evolution of planetary systems. We seek an outstanding theoretical, observational, or computational astronomer whose research will complement and extend existing activities in these areas within the Department. In addition to developing an independent and vigorous research program, the successful applicant will be expected to teach, on average, one undergraduate or graduate course each semester, and contribute to the service missions of the Department and University. The Department anticipates making the appointment at the assistant professor level. A Ph.D. in astronomy/astrophysics or related field is required. 

Applications for this position must be submitted electronically at http://jobs.rice.edu/postings/24588. Applicants will be required to submit the following: (1) cover letter; (2) curriculum vitae; (3) statement of research; (4) statement on teaching, mentoring, and outreach; (5) PDF copies of up to three publications; and (6) the names, affiliations, and email addresses of three professional references. Rice University is committed to a culturally diverse intellectual community. In this spirit, we particularly welcome applications from all genders and members of historically underrepresented groups who exemplify diverse cultural experiences and who are especially qualified to mentor and advise all members of our diverse student population. We will begin reviewing applications December 1, 2020. To receive full consideration, all application materials must be received by January 1, 2021. The appointment is expected to begin in July, 2021. 

Rice University is an Equal Opportunity Employer with a commitment to diversity at all levels, and considers for employment qualified applicants without regard to race, color, religion, age, sex, sexual orientation, gender identity, national or ethnic origin, genetic information, disability, or protected veteran status. We encourage applicants from diverse backgrounds to apply.

In his day, Agent 3203.7 had stopped people from trying to kill Adolf Hitler, Richard Nixon, and even, in the case of one unusually thoughtful assassin, Henry David Thoreau. But this was a new one on him.

“So…” drawled the seventh version of Agent 3203. His prosthetic hand crushed the simple 21st-century gun into fused metal and dropped it. “You traveled to the past in order to kill… of all people… Donald Trump. Care to explain why?”

The time-traveller’s eyes looked wild. Crazed. Nothing unusual. “How can you ask me that? You’re a time-traveler too! You know what he does!”

That was a surprising level of ignorance even for a 21st-century jumper. “Different timelines, kid. Some are pretty obscure. What the heck did Trump do in yours that’s worth taking your one shot at time travel to assassinate him of all people?”

“He’s destroying my world!”

Agent 3203.7 took a good look at where Donald Trump was pridefully addressing the unveiling of the Trump Taj Mahal in New Jersey, then took another good look at the errant time-traveler. “Destroying it how, exactly? Did Trump turn mad scientist in your timeline?”

“He’s President of the United States!”

Agent 3203.7 took another long stare at his new prisoner. He was apparently serious. “How did Trump become President in your timeline? Strangely advanced technology, subliminal messaging?”

“He was elected in the usual way,” the prisoner said bitterly.

Agent 3203.7 shook his head in amazement. Talk about shooting the messenger. “Kid, I doubt Trump was your timeline’s main problem.”

(thanks to Eliezer for giving me permission to reprint here)

Today Bonaca (Harvard) and I settled on a full scope for a first paper on asteroseismology of giant stars in the NASA TESS survey. We are going to find that our marginalized-likelihood formalism confirms beautifully many of the classical asteroseismology results; we are going to find that some get adjusted by us; we are going to find that some are totally invisible to us. And we will have reasons or discussion for all three kinds of cases. And then we will run on “everything” (subject to some cuts). That's a good paper! If we can do it. This weekend I need to do some writing to get our likelihood properly recorded in math.

An 81-year-old medical doctor has fallen off a ladder in his house. His pet bird hopped out of his reach, from branch to branch of a tree on the patio. The doctor followed via ladder and slipped. His servants cluster around him, the clamor grows, and he longs for his wife to join him before he dies. She arrives at last. He gazes at her face; utters, “Only God knows how much I loved you”; and expires.

I set the book down on my lap and looked up. I was nestled in a wicker chair outside the Huntington Art Gallery in San Marino, California. Busts of long-dead Romans kept me company. The lawn in front of me unfurled below a sky that—unusually for San Marino—was partially obscured by clouds. My final summer at Caltech was unfurling. I’d walked to the Huntington, one weekend afternoon, with a novel from Caltech’s English library.¹

What a novel.

You may have encountered the phrase “love in the time of corona.” Several times. Per week. Throughout the past six months. Love in the Time of Cholera predates the meme by 35 years. Nobel laureate Gabriel García Márquez captured the inhabitants, beliefs, architecture, mores, and spirit of a Colombian city around the turn of the 20th century. His work transcends its setting, spanning love, death, life, obsession, integrity, redemption, and eternity. A thermodynamicist couldn’t ask for more-fitting reading.

Love in the Time of Cholera centers on a love triangle. Fermina Daza, the only child of a wealthy man, excels in her studies. She holds herself with poise and self-assurance, and she spits fire whenever others try to control her. The girl dazzles Florentino Ariza, a poet, who restructures his life around his desire for her. Fermina Daza’s pride impresses Dr. Juvenal Urbino, a doctor renowned for exterminating a cholera epidemic. After rejecting both men, Fermina Daza marries Dr. Juvenal Urbino. The two personalities clash, and one betrays the other, but they cling together across the decades. Florentino Ariza retains his obsession with Fermina Daza, despite having countless affairs. Dr. Juvenal Urbino dies by ladder, whereupon Florentino Ariza swoops in to win Fermina Daza over. Throughout the book, characters mistake symptoms of love for symptoms of cholera; and lovers block out the world by claiming to have cholera and self-quarantining.

As a thermodynamicist, I see the second law of thermodynamics in every chapter. The second law implies that time marches only forward, order decays, and randomness scatters information to the wind. García Márquez depicts his characters aging, aging more, and aging more. Many characters die. Florentino Ariza’s mother loses her memory to dementia or Alzheimer’s disease. A pawnbroker, she buys jewels from the elite whose fortunes have eroded. Forgetting the jewels’ value one day, she mistakes them for candies and distributes them to children.

The second law bites most, to me, in the doctor’s final words, “Only God knows how much I loved you.” Later, the widow Fermina Daza sighs, “It is incredible how one can be happy for so many years in the midst of so many squabbles, so many problems, damn it, and not really know if it was love or not.” She doesn’t know how much her husband loved her, especially in light of the betrayal that rocked the couple and a rumor of another betrayal. Her husband could have affirmed his love with his dying breath, but he refused: He might have loved her with all his heart, and he might not have loved her; he kept the truth a secret to all but God. No one can retrieve the information after he dies.² 

Love in the Time of Cholera—and thermodynamics—must sound like a mouthful of horseradish. But each offers nourishment, an appetizer and an entrée. According to the first law of thermodynamics, the amount of energy in every closed, isolated system remains constant: Physics preserves something. Florentino Ariza preserves his love for decades, despite Fermina Daza’s marrying another man, despite her aging.

The latter preservation can last only so long in the story: Florentino Ariza, being mortal, will die. He claims that his love will last “forever,” but he won’t last forever. At the end of the novel, he sails between two harbors—back and forth, back and forth—refusing to finish crossing a River Styx. I see this sailing as prethermalization: A few quantum systems resist thermalizing, or flowing to the physics analogue of death, for a while. But they succumb later. Florentino Ariza can’t evade the far bank forever, just as the second law of thermodynamics forbids his boat from functioning as a perpetuum mobile.

Though mortal within his story, Florentino Ariza survives as a book character. The book survives. García Márquez wrote about a country I’d never visited, and an era decades before my birth, 33 years before I checked his book out of the library. But the book dazzled me. It pulsed with the vibrancy, color, emotion, and intellect—with the fullness—of life. The book gained another life when the coronavius hit. Thermodynamics dictates that people age and die, but the laws of thermodynamics remain.³ I hope and trust—with the caveat about humanity’s not destroying itself—that Love in the Time of Cholera will pulse in 350 years. 

What’s not to love?

¹Yes, Caltech has an English library. I found gems in it, and the librarians ordered more when I inquired about books they didn’t have. I commend it to everyone who has access.

²I googled “Only God knows how much I loved you” and was startled to see the line depicted as a hallmark of romance. Please tell your romantic partners how much you love them; don’t make them guess till the ends of their lives.

³Lee Smolin has proposed that the laws of physics change. If they do, the change seems to have to obey metalaws that remain constant.

Back to Kenny Courser’s thesis:

• Kenny Courser, Open Systems: A Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020.

One thing Kenny does here is explain the flaws in a well-known framework for studying open systems: decorated cospans. Decorated cospans were developed by my student Brendan Fong. Since I was Brendan’s advisor at the time, a hefty helping of blame for not noticing the problems belongs to me! But luckily, Kenny doesn’t just point out the problems: he shows how to fix them. As a result, everything we’ve done with decorated cospans can be saved.

The main theorem on decorated cospans is correct; it’s just less useful than we’d hoped! The idea is to cook up categories where the morphisms are open systems. The objects of such a category could be something simple like finite sets, but morphisms from $X$ to $Y$ could be something more interesting, like ‘open graphs’:

Here $X$ and $Y$ are mapped into a third set in the middle, but this set in the middle is the set of nodes of a graph. We say the set in the middle has been ‘decorated’ with a graph.

Here’s how the original theory of decorated cospans seeks to make this precise.

Fong’s Theorem. Suppose $C$ is a category with finite colimits, and make $C$ into a symmetric monoidal category with its coproduct as the tensor product. Suppose $F\colon (C,+) \to (\mathrm{Set},\times)$ is a lax symmetric monoidal functor. Define an F-decorated cospan to be a cospan

in $C$ together with an element of $F(N)$. Then there is a symmetric monoidal category with

• objects of $C$ as objects,
• isomorphism classes of $F$-decorated cospans as morphisms.

I won’t go into many details, but let me say how to compose two decorated spans, and also how this ‘isomorphism class’ business works.

Given two decorated cospans we compose their underlying cospans in the usual way, via pushout:

We get a cospan from $X$ to $Z$. To decorate this we need an element of $F(M +_Y N)$. So, we take the decorations we have on the cospans being composed, which together give an element of $F(N) \times F(M)$, and apply this composite map:

$$F(N) \times F(M) \longrightarrow F(N+M) \longrightarrow F(N+_Y M)$$

Here the first map, called the laxator, comes from the fact that $F$ is a lax monoidal functor, while the second comes from applying $F$ to the canonical map $N+M \to N+_Y M$.

Since composing cospans involves a pushout, which is defined via a universal property, the composite is only well-defined up to isomorphism. So, to get an actual category, we take isomorphism classes of decorated cospans as our morphisms.

Here an isomorphism of cospans is a commuting diagram like this:

where $h$ is an isomorphism. If the first cospan here has a decoration $d \in F(N)$ and the second has a decoration $d' \in F(N')$, then we have an isomorphism of decorated cospans if $F(h)(d) = d'$.

So, that’s the idea. The theorem is true, and it works fine for some applications — but not so well for others, like the example of open graphs!

Why not? Well, let’s look at that example in detail. Given a finite set $N$, let’s define a graph on $N$ to be a finite set $E$ together with two functions $s, t \colon E \to N$. We call $N$ the set of nodes, $E$ the set of edges, and the functions $s$ and $t$ map each edge to its source and target, respectively. So, a graph on $N$ is a way of choosing a graph whose set of nodes is $N$.

We can try to apply the above theorem taking $C = \mathrm{FinSet}$ and letting $F \colon \mathrm{FinSet} \to \mathrm{Set}$be the functor sending each finite set $N$ to the set of all graphs on $N$.

The first problem, which Brendan and I noticed right away, is that there’s not really a set of graphs on $N$. There’s a proper class! $E$ ranges over all possible finite sets, and there’s not a set of all finite sets.

This should have set alarm bells ringing right away. But we used a standard dodge. In fact there are two. One is to replace $\mathrm{FinSet}$ with an equivalent small category, and define a graph just as before but taking $N$ and $E$ to be objects in this equivalent category. Another is to invoke the axiom of universes. Either way, we get a set of graphs on each $N$.

Then Fong’s theorem applies, and we get a decorated cospan category with:

• ‘finite sets’ as objects,
• isomorphism classes of open graphs as morphisms.

Here I’m putting ‘finite sets’ in quotes because of the trickery I just mentioned, but it’s really not a big deal so I’ll stop now. An open graph has a finite set $N$ of nodes, a finite set $E$ of edges, maps $s,t \colon E \to N$ saying the source and target of each edge, and two maps $f \colon X \to N$ and $g \colon Y \to N$.

These last two maps are what make it an open graph going from $X$ to $Y$:

Isomorphism classes of open graphs from $X$ to $Y$ are the morphisms from $X$ to $Y$ in our decorated cospan category.

But later, when Kenny was devising a bicategory of decorated cospans, we noticed a second problem. The concept of ‘isomorphic decorated cospan’ doesn’t behave well in this example: the concept of isomorphism is too narrowly defined!

Suppose you and I have isomorphic decorated cospans:

In the example at hand, this means you have a graph on the finite set $N$ and I have a graph on the finite set $N'$. Call yours $d \in F(N)$ and mine $d' \in F(N')$.

We also have a bijection $h \colon N \to N'$ such that

$$F(h)(d) = d'$$

What does this mean? I haven’t specified the functor $F$ in detail so you’ll have to take my word for it, but it should be believable. It means that my graph is exactly like yours except that we replace the nodes of your graph, which are elements of $N$, by the elements of $N'$ that they correspond to. But this means the edges of my graph must be exactly the same as the edges of yours. It’s not good enough for our graphs to have isomorphic sets of edges: they need to be equal!.

For a more precise account of this, with pictures, read the introduction to Chapter 3 of Kenny’s thesis.

So, our decorated cospan category has ‘too many morphisms’. Two open graphs will necessarily define different morphisms if they have different sets of edges.

This set Kenny and I to work on a new formalism, structured cospans, that avoids this problem. Later Kenny and Christina Vasilakopoulou also figured out a way to fix the decorated cospan formalism. Kenny’s thesis explains all this, and also how structured cospans are related to the ‘new, improved’ decorated cospans.

But then something else happened! Christina Vasilakopoulou was a postdoc at U.C. Riverside while all this was going on. She and my grad student Joe Moeller wrote a paper on something called the monoidal Grothendieck construction, which plays a key role in relating structured cospans to the new decorated cospans. But the anonymous referee of their paper pointed out another problem with the old decorated cospans!

Briefly, the problem is that the functor $F \colon (\mathrm{FinSet},+) \to (\mathrm{Set},\times)$ that sends each $N$to the set of graphs having $N$ as their set of vertices cannot be made lax monoidal in the desired way. To make $F$ lax monoidal, we must pick a natural transformation called the laxator:

$$\phi_{N,M} \colon F(N) \times F(M) \to F(N+M)$$

I used this map earlier when explaining how to compose decorated cospans.

The idea seems straightforward enough: given a graph on $N$ and a graph on $M$ we get a graph on their disjoint union $N+M$. This is true, and there is a natural transformation $\phi_{N,M}$ that does this.

But the definition of lax monoidal functor demands that the laxator make a certain hexagon commute! And it does not!

I won’t draw this hexagon here; you can see it at the link or in the intro to Chapter 3 of Kenny’s thesis, where he explains this problem. The problem arises because when we have three sets of edges, say $E,E',E''$, we typically have

$$(E + E') + E'' \ne E + (E' + E'')$$

There is a sneaky way to partially get around this problem, which he also explains: define graphs using a category equivalent to $\mathrm{FinSet}$ where $+$ is strictly associative, not just up to isomorphism!

This is good enough to make $F$ lax monoidal. But Kenny noticed yet another problem: $F$ is still not lax symmetric monoidal. If you try to show it is, you wind up needing two graphs to be equal: one with $E+E'$ as its set of edges, and another with $E'+E$ as its set of edges. These aren’t equal! And at this point it seems we hit a dead end. While there’s a category equivalent to $\mathrm{FinSet}$ where $+$ is strictly associative, there’s no such category where $+$ is also strictly commutative.

In the end, by going through all these contortions, we can use a watered-down version of Fong’s theorem to get a category with open graphs as morphisms, and we can make it monoidal — but not symmetric monoidal.

It’s clear that something bad is going on here. We are not following the tao of mathematics. We keep wanting things to be equal, that should only be isomorphic. The problem is that we’re treating a graph on a set as extra structure on that set, when it’s actually extra stuff.

Structured cospans, and the new decorated cospans, are the solution! For example, the new decorated cospans let us use a category of graphs with a given set of nodes, instead of a mere set. Now $F(N)$ is a category rather than a set. This category lets us talk about isomorphic graphs with the same set of vertices, and all our problems evaporate.

There are a few theorems in abstract category theory in which specific numbers play an important role. For example:

Theorem. Let $\mathsf{S}$ be the free symmetric monoidal category on an object $x$. Regard $\mathsf{S}$ as a mere category. Then there exists an equivalence $F \colon \mathsf{S} \to \mathsf{S}$ such that:

• $F$ is not naturally isomorphic to the identity,
• $F$ acts as the identity on all objects,
• $F$ acts as the identity on all endomorphisms $f \colon x^{\otimes n} \to x^{\otimes n}$ except when $n = 6$.

This theorem would become false if we replaced $6$ by any other number.

The proof is lurking here. The point is that $\mathsf{S}$ is the groupoid of finite sets and bijections, so $hom(x^{\otimes n} , x^{\otimes n})$ is the symmetric group $S_n$ — and of all the symmetric groups, only $S_6$ has an outer automorphism.

If we replaced the free symmetric monoidal category on one object by some higher-dimensional analogues we could create theorems with all sorts of crazy numbers showing up, like 24 or 240, since we could get homotopy groups of spheres.

Still, it’s a surprise when a theorem with purely category-theoretic assumptions has a specific number other than 0, 1, or 2 in its conclusion. We were talking about these on Category Theory Community Server. Here’s one pointed out by Peter Arndt:

Theorem. The only category for which the Yoneda embedding is the rightmost of a string of 5 adjoints is the category $\mathsf{Set}$.

The proof is here:

• Robert Rosebrugh and R. J. Wood, An adjoint characterization of the category of sets, Proc. Amer. Math. Soc. 122 (1994), 409–413.

Here’s another one that Arndt pointed out:

Theorem. There are just 3 possible lengths of maximal chains of adjoint functors between compactly generated tensor-triangulated categories: 3, 5 and $\infty$.

The proof is here:

• Paul Balmer, Ivo Dell’Ambrogio and Beren Sander, Grothendieck-Neeman duality and the Wirthmüller isomorphism, Compositio Mathematica 152 (2016), 1740–1776.

Reid Barton pointed out another:

Theorem. There are just 9 model category structures on $Set$.

This was mentioned without proof by Tom Goodwillie on MathOverflow and explained here:

• Omar Antolín Camarena, The nine model category structures on the category of sets.

Do you know other nice theorems like this: hypotheses that sound like ‘general abstract nonsense’, with a surprising conclusion that involves a specific natural number other than 0, 1, and 2?

Today, Tyler Pritchard (NYU) and I assembled a group of time-domain-interested astrophysicists from around NYC (and a few who are part of the NYC community but more far-flung). In a two-hour meeting, all we did was introduce ourselves and our interests in time domain, multi-messenger, and Vera Rubin Observatory LSST, and then discuss what we might do, collectively, as a group. Time-domain expertise spanned an amazing range of scales, from asteroid search to exoplanet characterization to stellar rotation to classical novae to white-dwarf mergers with neutron stars, supernovae, light echoes, AGN variability, tidal-disruption events, and black-hole mergers. As we had predicted in advance, the group recognized a clear opportunity to create some kind of externally funded “Gotham” (the terminology we often use for NYC-area efforts these days) center for time-domain astrophysics.

Also, as we predicted, there was more confusion about whether we should be thinking about a real-time event broker for LSST. But we identified some themes in the group that might make for a good project: We have very good theorists working, who could help on physics-driven multi-messenger triggers. We have very good machine-learners working, who could help on data-driven triggers. And we have lots of non-supernovae (and weird-supernova) science cases among us. Could we make something that serves our collective science interests but is also extremely useful to global astrophysics? I think we could.

In Fourier transforms, or periodograms, or signal processing in general, when you look at a time stream that is generated by a single frequency f (plus noise, say) and has a total time length T, you expect the power-spectrum ppeak, or the likelihood function for f to have a width that is no narrower than 1/T in the frequency direction. This is for extremely deep reasons, that relate to—among other things—the uncertainty principle. You can't localize a signal in both frequency and time simultaneously.

Ana Bonaca (Harvard) and I are fitting combs of frequencies to light curves. That is, we are fitting a model with K frequencies, equally spaced. We are finding that the likelihood function has a peak that is a factor of K narrower than 1/T, in both the central-frequency direction, and the frequency-spacing direction. Is this interesting or surprising? I have ways to justify this point, heuristically. But is there a fundamental result here, and where is it in the literature?

I have uploaded to the arXiv my paper “Exploring the toolkit of Jean Bourgain“. This is one of a collection of papers to be published in the Bulletin of the American Mathematical Society describing aspects of the work of Jean Bourgain; other contributors to this collection include Keith Ball, Ciprian Demeter, and Carlos Kenig. Because the other contributors will be covering specific areas of Jean’s work in some detail, I decided to take a non-overlapping tack, and focus instead on some basic tools of Jean that he frequently used across many of the fields he contributed to. Jean had a surprising number of these “basic tools” that he wielded with great dexterity, and in this paper I focus on just a few of them:

• Reducing qualitative analysis results (e.g., convergence theorems or dimension bounds) to quantitative analysis estimates (e.g., variational inequalities or maximal function estimates).
• Using dyadic pigeonholing to locate good scales to work in or to apply truncations.
• Using random translations to amplify small sets (low density) into large sets (positive density).
• Combining large deviation inequalities with metric entropy bounds to control suprema of various random processes.

Each of these techniques is individually not too difficult to explain, and were certainly employed on occasion by various mathematicians prior to Bourgain’s work; but Jean had internalized them to the point where he would instinctively use them as soon as they became relevant to a given problem at hand. I illustrate this at the end of the paper with an exposition of one particular result of Jean, on the Erdős similarity problem, in which his main result (that any sum of three infinite sets of reals has the property that there exists a positive measure set that does not contain any homothetic copy of ) is basically proven by a sequential application of these tools (except for dyadic pigeonholing, which turns out not to be needed here).

I had initially intended to also cover some other basic tools in Jean’s toolkit, such as the uncertainty principle and the use of probabilistic decoupling, but was having trouble keeping the paper coherent with such a broad focus (certainly I could not identify a single paper of Jean’s that employed all of these tools at once). I hope though that the examples given in the paper gives some reasonable impression of Jean’s research style.

You’ve probably heard of the myth of the solitary scientist. While Newton might have figured out calculus isolated on his farm, most scientists work better when they communicate. If we reach out to other scientists, we can make progress a lot faster.

Even if you understand that, you might not know what that reaching out actually looks like. I’ve seen far too many crackpots who approach scientific communication like a spammer: sending out emails to everyone in a department, commenting in every vaguely related comment section they can find. While commercial spammers hope for a few gullible people among the thousands they contact, that kind of thing doesn’t benefit crackpots. As far as I can tell, they communicate that way because they genuinely don’t know any better.

So in this post, I want to give a road map for how we scientists reach out to other scientists. Keep these steps in mind, and if you ever need to reach out to a scientist you’ll know what to do.

First, decide what you want to know. This may sound obvious, but sometimes people skip this step. We aren’t communicating just to communicate, but because we want to learn something from the other person. Maybe it’s a new method or idea, maybe we just want confirmation we’re on the right track. We don’t reach out just to “show our theory”, but because we hope to learn something from the response.

Then, figure out who might know it. To do this, we first need to decide how specialized our question is. We often have questions about specific papers: a statement we don’t understand, a formula that seems wrong, or a method that isn’t working. For those, we contact an author from that paper. Other times, the question hasn’t been addressed in a paper, but does fall under a specific well-defined topic: a particular type of calculation, for example. For those we seek out a specialist on that specific topic. Finally, sometimes the question is more general, something anyone in our field might in principle know but we happen not to. For that kind of question, we look for someone we trust, someone we have a prior friendship with and feel comfortable asking “dumb questions”. These days, we can supplement that with platforms like PhysicsOverflow that let us post technical questions and invite anyone to respond.

Note that, for all of these, there’s some work to do first. We need to read the relevant papers, bone up on a topic, even check Wikipedia sometimes. We need to put in enough work to at least try to answer our question, so that we know exactly what we need the other person for.

Finally, contact them appropriately. Papers will usually give contact information for one, or all, of the authors. University websites will give university emails. We’d reach out with something like that first, and switch to personal email (or something even more casual, like Skype or social media) only for people we already have a track record of communicating with in that way.

By posing and directing our questions well, scientists can reach out and get help when we struggle. Science is a team effort, we’re stronger when we work together.

Update (Sep. 17): Several people, here and elsewhere, wrote to tell me that while they completely agreed with my strategic and moral stance in this post, they think that it’s the ads of Republican Voters Against Trump, rather than the Lincoln Project, that have been most effective in changing Trump supporters’ minds. So please consider donating to RVAT instead or in addition! In fact, what the hell, I’ll match donations to RVAT up to $1000.

For the past few months, I’ve alternated between periods of debilitating depression and (thankfully) longer stretches when I’m more-or-less able to work. Triggers for my depressive episodes include reading social media, watching my 7-year daughter struggle with prolonged isolation, and (especially) contemplating the ongoing apocalypse in the American West, the hundreds of thousands of pointless covid deaths, and an election in 48 days that if I didn’t know such things were impossible in America would seem likely to produce a terrifying standoff as a despot and millions of his armed loyalists refuse to cede control. Meanwhile, catalysts for my relatively functional periods have included teaching my undergrad quantum information class, Zoom calls with my students, life on Venus?!? (my guess is no, but almost entirely due to priors), learning new math (fulfilling a decades-old goal, I’m finally working my way through Paul Cohen’s celebrated proof of the independence of the Continuum Hypothesis—more about that later!).

Of course, when you feel crushed by the weight of the world’s horribleness, it improves your mood to be able even just to prick the horribleness with a pin. So I was gratified that, in response to a previous post, Shtetl-Optimized readers contributed at least $3,000, the first $2,000 of which I matched, mostly to the Biden-Harris campaign but a little to the Lincoln Project.

Alas, a commenter was unhappy with the latter:

Lincoln Project? Really? … Pushing the Overton window rightward during a worldwide fascist dawn isn’t good. I have trouble understanding why even extremely smart people have trouble with this sort of thing.

Since this is actually important, I’d like to spend the rest of this post responding to it.

For me it’s simple.

What’s the goal right now? To defeat Trump. In the US right now, that’s the prerequisite to every other sane political goal.

What will it take to achieve that goal? Turnout, energizing the base, defending the election process … but also, if possible, persuading a sliver of Trump supporters in swing states to switch sides, or at least vote third party or abstain.

Who is actually effective at that goal? Well, no one knows for sure. But while I thought the Biden campaign had some semi-decent ads, the Lincoln Project’s best stuff seems better to me, just savagely good.

Why are they effective? The answer seems obvious: for the same reason why a jilted ex is a more dangerous foe than a stranger. If anyone understood how to deprogram a Republican from the Trump cult, who would it be: Alexandria Ocasio-Cortez, or a fellow Republican who successfully broke from the cult?

Do I agree with the Lincoln Republicans about most of the “normal” issues that Americans once argued about? Not at all. Do I hold them, in part, morally responsible for creating the preconditions to the current nightmare? Certainly.

And should any of that cause me to boycott them? Not in my moral universe. If Churchill and FDR could team up with Stalin, then surely we in the Resistance can temporarily ally ourselves with the rare Republicans who chose their stated principles over power when tested—their very rarity attesting to the nontriviality of their choice.

To my mind, turning one’s back on would-be allies, in a conflict whose stakes obviously overshadow what’s bad about those allies, is simultaneously one of the dumbest and the ugliest things that human beings can do. It abandons reason for moral purity and ends up achieving neither.

I mentioned in a previous post Sundholm’s rendition in dependent type theory of the Donkey sentence:

Every farmer who owns a donkey beats it.

For those who find this unnatural, I offered

Anyone who owns a gun should register it.

The idea then is that sentences of the form

Every $A$ that $R$s a $B$, $S$s it,

are rendered in type theory as

$$\prod_{z: \sum_{x: A} \sum_{y: B} R(x, y)} S(p(z), p(q(z)),$$ where $p$ and $q$ are the projections to first and second components. We see ‘it’ corresponds to $p(q(z))$.

This got me wondering if we could make life even harder for the advocate of first-order logic – let’s give them the typed version to be generous – by constructing a natural language sentence which it would be even more awkward to formalise.

One thing to note is that the type above has been formed from a dependency structure of the kind:

$$x: A, y: B, w: R(x, y) \vdash s: S(x, y).$$

Now I’ve been writing up on the nLab some of what Mike told us about 3-theories back here. Note in particular his comments on the limited dependencies which may be expressed in the ‘first-order 3-theory’:

one layer of dependency: there is one layer of types which cannot depend on anything, and then there is a second layer of types (“propositions” in the logical reading) which can depend on types in the first layer, but not on each other.

So to stretch this 3-theory we should look for a maximally dependent structure. Take something of the form

$$x: A, y: B(x), z: C(x, y) \vdash d:D(x, y, z).$$

Ideally the types will all be sets.

By the time quantification has been applied, the left hand side swept up into an iterated dependent sum and a dependent product formed, the proposition should be bristling with awkward anaphoric pronouns.

The best I could come up with was:

Whenever someone’s child speaks politely to them, they should praise them for it.

A gendered version with, say, men and daughters would help with the they/them clashes.

Whenever a man’s daughter speaks politely to him, he should praise her for it.

Any better candidates?

Starting on Oct 2, I will be teaching Math 246A, the first course in the three-quarter graduate complex analysis sequence at the math department here at UCLA.  This first course covers much of the same ground as an honours undergraduate complex analysis course, in particular focusing on the basic properties of holomorphic functions such as the Cauchy and residue theorems, the classification of singularities, and the maximum principle, but there will be more of an emphasis on rigour, generalisation and abstraction, and connections with other parts of mathematics.  The main text I will be using for this course is Stein-Shakarchi (with Ahlfors as a secondary text), but I will also be using the blog lecture notes I wrote the last time I taught this course in 2016. At this time I do not expect to significantly deviate from my past lecture notes, though I do not know at present how different the pace will be this quarter when the course is taught remotely. As with my 247B course last spring, the lectures will be open to the public, though other coursework components will be restricted to enrolled students.

Another Update (Sep. 15): Sorry for the long delay; new post coming soon! To tide you over—or just to distract you from the darkness figuratively and literally engulfing our civilization—here’s a Fortune article about today’s announcement by IBM of its plans for the next few years in superconducting quantum computing, with some remarks from yours truly.

Another Update (Sep. 8): A reader wrote to let me know about a fundraiser for Denys Smirnov, a 2015 IMO gold medalist from Ukraine who needs an expensive bone marrow transplant to survive Hodgkin’s lymphoma. I just donated and I hope you’ll consider it too!

Update (Sep. 5): Here’s another quantum computing podcast I did, “Dunc Tank” with Duncan Gammie. Enjoy!

Thanks so much to Shtetl-Optimized readers, so far we’ve raised $1,371 for the Biden-Harris campaign and $225 for the Lincoln Project, which I intend to match for $3,192 total. If you’d like to donate by tonight (Thursday night), there’s still $404 to go!

Meanwhile, a mere three days after declaring my “new motto,” I’ve come up with a new new motto for this blog, hopefully a more cheerful one:

When civilization seems on the brink of collapse, sometimes there’s nothing left to talk about but maximal separations between randomized and quantum query complexity.

On that note, please enjoy my new one-hour podcast on Spotify (if that link doesn’t work, try this one) with Matthew Putman of Utility+. Alas, my umming and ahhing were more frequent than I now aim for, but that’s partly compensated for by Matthew’s excellent decision to speed up the audio. This was an unusually wide-ranging interview, covering everything from SlateStarCodex to quantum gravity to interdisciplinary conferences to the challenges of teaching quantum computing to 7-year-olds. I hope you like it!

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A small proportion of UW-Madison courses were being given in person, until last week, that is, but not mine. I’m teaching two graduate courses, introduction to algebra (which I’ve taught several times before) and introduction to algebraic number theory, which I’ve taught before but not for quite a few years. And I’m teaching them sitting in my chair at home. So I thought I’d write down a bit about what that’s like, since depending on who you ask, we’ll never do it again (in which case it’s good to record the memory) or this is the way we’ll all teach in the future (in which case it’s good to record my first impression.)

First of all, it’s tiring. Just as tiring as teaching in the classroom, even though I don’t have to leave my chair. This surprised me! But, introspecting, I think I actually draw energy from the state of being in a room with people, talking at the board, walking around, interacting. I usually leave class feeling less tired than when I walked in.

On the screen, no. I teach lectures at 10 and 11 and at noon when both are done I’m wiped out.

My rig, settled on after other setups kept glitching out: Notability open on iPad, I write notes as if on blackboard with the Apple Pencil, iPad connected by physical cable to laptop, screensharing to a window on the laptop which window I am sharing in Microsoft Teams to the class while the laptop camera and mic capture my face and voice.

What I have not done:

• Gotten a pro-quality microphone
• Set up a curated “lecture space” from which to broadcast
• Recorded lecture videos in advance so I can use the lecture hour for discussion
• Used breakout rooms in Teams to let the students discuss among themselves

All of these seem like good ideas.

So far (but I am still in the part of both courses where the material isn’t too hard) the students and I seem to find this… OK. My handwriting is somewhat worse on the tablet than it is on the blackboard and it’s not great on the blackboard. The only student who has told me they prefer online is one who reports being too shy to speak in class, sometimes too shy even to attend, and who feels more able to participate by typing in the chat window with the camera turned off. That makes sense!

I have it easy — these courses have only thirty students each, so the logistical work of handling student questions, homework, etc. isn’t overwhelming. Teaching big undergraduate courses presents its own problems. What happens with calculus quizzes? In the spring it was reported that cheating was universal (there are lots of websites that will compute integrals for you in another window!) So we now have a system called Honorlock which inhabits the student’s browser, watches IP traffic for visits to cheating sites, and commandeers the student’s webcam (!) to check whether their eye motions indicate cheating (!!) This sounds awful and frankly kind of creepy and not worth it. And the students, unsurprisingly, hate it. But then how does assessment work? The obvious answer is to give exams which are open book and which measure something more contentful about the material than can be tested by a usual quiz. I can think of two problems:

• Fluency with the basic manipulations (of both algebra and calculus) is actually one of the skills the class is meant to impart: yes, there are things a computer can do it’s good to be able to do mentally. (I don’t think I place a complicated trig substitution in this category, but knowing that the integral of x^n is on order x^{n+1}, yes.
• Tests that measured understanding would be different from and a lot harder than what students are used to! And this is a crappy time to be an undergraduate. I don’t think it’s a great idea for their calculus course to become, without warning, much more difficult than the one they signed up for.

I thought it was gonna work.

Really! I thought we could sort-of-open college again and not cause a big outbreak. Most of our students live here year-round. By all accounts, there have been fraternity parties all summer. We had a spike of cases in the campus area when bars opened back up at the end of June, which subsided when the county put back those restrictions (though never back down to the levels we’d seen in March, April, and May.) At the end of July I wrote “statewide, cases are growing and growing, and the situation is much worse in the South. I would fight back if you said this was a predictable consequence; nothing about this disease is predictable with any confidence. It could have worked.”

And maybe it could have; but it didn’t. As soon as school started last Wednesday, the percentage of student tests coming back positive, started growing, about 20% higher every day. On Saturday, nine Greek houses were quarantined. A week into school, with about 8% of tests positive, the University halted in-person classes and completely quarantined two first-year dormitories with two hours notice. Food is being brought in three times a day. Hope you like your roommate.

A lot of people, unlike me, saw this coming.

Maybe we can beat this back. Who knows? We did in July. But this outbreak is bigger.

Public schools in Madison are fully online right now. With a summer to prepare it’s working better than it did last spring. But it’s not great, and I would guess that for poor kids it’s a lot worse than “not great.” Private schools are allowed to be open in grades K-2, and a court decision that came down today has, at least for now, allowed them to open to all grades. More outbreaks? To be a broken record, who knows? The argument for opening K-2 sounds pretty good to me; while it’s not definite, most people seem to think younger children are less likely to spread and contract the disease, and that age range is where having kids at home limits parents most. Schools in Georgia have been open, and there have been lots of school outbreaks, and those schools get closed for a while and then reopen, but it doesn’t seem to have created big wave of cases statewide.

This article is good. Beating COVID isn’t all-or-nothing, but people seem to see it that way. If the bar’s open, that means it’s safe, and you can drink with whoever you want, as close as you want. No! Nothing is safe, if you mean safe safe. But also nothing is a guarantee of disaster. If everybody would do 50% of what they felt like doing, we could beat it. Or maybe 75%, who knows. But it feels like if we don’t insist on 0%, people will understand us to mean that 100% is OK. I don’t have any good ideas about how to fix this.

I’ve been busy running a conference this week, Elliptics and Beyond.

After Amplitudes was held online this year, a few of us at the Niels Bohr Institute were inspired. We thought this would be the perfect time to hold a small online conference, focused on the Calabi-Yaus that have been popping up lately in Feynman diagrams. Then we heard from the organizers of Elliptics 2020. They had been planning to hold a conference in Mainz about elliptic integrals in Feynman diagrams, but had to postpone it due to the pandemic. We decided to team up and hold a joint conference on both topics: the elliptic integrals that are just starting to be understood, and the mysterious integrals that lie beyond. Hence, Elliptics and Beyond.

The conference has been fun thus far. There’s been a mix of review material bringing people up to speed on elliptic integrals and exciting new developments. Some are taking methods that have been successful in other areas and generalizing them to elliptic integrals, others have been honing techniques for elliptics to make them “production-ready”. A few are looking ahead even further, to higher-genus amplitudes in string theory and Calabi-Yaus in Feynman diagrams.

We organized the conference along similar lines to Zoomplitudes, but with a few experiments of our own. Like Zoomplitudes, we made a Slack space for the conference, so people could chat physics outside the talks. Ours was less active, though. I suspect that kind of space needs a critical mass of people, and with a smaller conference we may just not have gotten there. Having fewer people did allow us a more relaxed schedule, which in turn meant we could mostly keep things on-time. We had discussion sessions in the morning (European time), with talks in the afternoon, so almost everyone could make the talks at least. We also had a “conference dinner”, which went much better than I would have expected. We put people randomly into Zoom Breakout Rooms of five or six, to emulate the tables of an in-person conference, and folks chatted while eating their (self-brought of course) dinner. People seemed to really enjoy the chance to just chat casually with the other folks at the conference. If you’re organizing an online conference soon, I’d recommend trying it!

Holding a conference online means that a lot of people can attend who otherwise couldn’t. We had over a hundred people register, and while not all of them showed up there were typically fifty or sixty people on the Zoom session. Some of these were specialists in elliptics or Calabi-Yaus who wouldn’t ordinarily make it to a conference like this. Others were people from the rest of the amplitudes field who joined for parts of the conference that caught their eye. But surprisingly many weren’t even amplitudeologists, but students and young researchers in a variety of topics from all over the world. Some seemed curious and eager to learn, others I suspect just needed to say they had been to a conference. Both are responding to a situation where suddenly conference after conference is available online, free to join. It will be interesting to see if, and how, the world adapts.

Sometimes it takes a while to answer a scientific question, and sometimes that answer ends up being a bit unexpected.  Three years ago, I wrote about a paper from our group, where we had found, much to our surprise, that the thermoelectric response of polycrystalline gold wires varied a lot as a function of position within the wire, even though the metal was, by every reasonable definition, a good, electrically homogeneous material.  (We were able to observe this by using a focused laser as a scannable heat source, and measuring the open-circuit photovoltage of the device as a function of the laser position.)  At the time, I wrote "Annealing the wires does change the voltage pattern as well as smoothing it out.  This is a pretty good indicator that the grain boundaries really are important here."

What would be the best way to test the idea that somehow the grain boundaries within the wire were responsible for this effect?  Well, the natural thought experiment would be to do the same measurement in a single crystal gold wire, and then ideally do a measurement in a wire with, say, a single grain boundary in a known location.  

Fig. 4 from this paper

We embarked on a rewarding collaboration that turned out to be a long, complicated path of measuring many many device structures of various shapes, sizes, and dimensions.  My student Charlotte Evans, measuring the photothermoelectric (PTE) response of these, worked closely with members of Prof. Fan's group - Rui Yang grew and prepared devices, and Lucia Gan did many hours of back-scatter electron diffraction measurements and analysis, for comparison with the photovoltage maps.  My student Mahdiyeh Abbasi learned the intricacies of finite element modeling to see what kind of spatial variation of Seebeck coefficient \(S\) would be needed to reproduce the photovoltage maps.  

From Fig. 1 of our new paper.  Panel g upper shows the local crystal misorientation as found from electron back-scatter diffraction, while  panel g lower shows a spatial map of the PTE response.  The two  patterns definitely resemble each other (panel h), and this is seen consistently across many devices.

Vaughan Jones, who made fundamental contributions in operator algebras and knot theory (in particular developing a surprising connection between the two), died this week, aged 67.

Vaughan and I grew up in extremely culturally similar countries, worked in adjacent areas of mathematics, shared (as of this week) a coauthor in Dima Shylakhtenko, started out our career with the same postdoc position (as UCLA Hedrick Assistant Professors, sixteen years apart) and even ended up in sister campuses of the University of California, but surprisingly we only interacted occasionally, via chance meetings at conferences or emails on some committee business. I found him extremely easy to get along with when we did meet, though, perhaps because of our similar cultural upbringing.

I have not had much occasion to directly use much of Vaughan’s mathematical contributions, but I did very much enjoy reading his influential 1999 preprint on planar algebras (which, for some odd reason has never been formally published). Traditional algebra notation is one-dimensional in nature, with algebraic expressions being described by strings of mathematical symbols; a linear operator , for instance, might appear in the middle of such a string, taking in an input on the right and returning an output on its left that might then be fed into some other operation. There are a few mathematical notations which are two-dimensional, such as the commutative diagrams in homological algebra, the tree expansions of solutions to nonlinear PDE (particularly stochastic nonlinear PDE), or the Feynman diagrams and Penrose graphical notations from physics, but these are the exception rather than the rule, and the notation is often still concentrated on a one-dimensional complex of vertices and edges (or arrows) in the plane. Planar algebras, by contrast, fully exploit the topological nature of the plane; a planar “operator” (or “operad”) inhabits some punctured region of the plane, such as an annulus, with “inputs” entering from the inner boundaries of the region and “outputs” emerging from the outer boundary. These algebras arose for Vaughan in both operator theory and knot theory, and have since been used in some other areas of mathematics such as representation theory and homology. I myself have not found a direct use for this type of algebra in my own work, but nevertheless I found the mere possibility of higher dimensional notation being the natural choice for a given mathematical problem to be conceptually liberating.

I’m pleased to announce my new comic series, “The Peacock”.

Abdul Basit, Artem Chernikov, Sergei Starchenko, Chiu-Minh Tran and I have uploaded to the arXiv our paper Zarankiewicz’s problem for semilinear hypergraphs. This paper is in the spirit of a number of results in extremal graph theory in which the bounds for various graph-theoretic problems or results can be greatly improved if one makes some additional hypotheses regarding the structure of the graph, for instance by requiring that the graph be “definable” with respect to some theory with good model-theoretic properties.

A basic motivating example is the question of counting the number of incidences between points and lines (or between points and other geometric objects). Suppose one has points and lines in a space. How many incidences can there be between these points and lines? The utterly trivial bound is , but by using the basic fact that two points determine a line (or two lines intersect in at most one point), a simple application of Cauchy-Schwarz improves this bound to . In graph theoretic terms, the point is that the bipartite incidence graph between points and lines does not contain a copy of (there does not exist two points and two lines that are all incident to each other). Without any other further hypotheses, this bound is basically sharp: consider for instance the collection of points and lines in a finite plane , that has incidences (one can make the situation more symmetric by working with a projective plane rather than an affine plane). If however one considers lines in the real plane , the famous Szemerédi-Trotter theorem improves the incidence bound further from to . Thus the incidence graph between real points and lines contains more structure than merely the absence of .

More generally, bounding on the size of bipartite graphs (or multipartite hypergraphs) not containing a copy of some complete bipartite subgraph (or in the hypergraph case) is known as Zarankiewicz’s problem. We have results for all and all orders of hypergraph, but for sake of this post I will focus on the bipartite case.

In our paper we improve the bound to a near-linear bound in the case that the incidence graph is “semilinear”. A model case occurs when one considers incidences between points and axis-parallel rectangles in the plane. Now the condition is not automatic (it is of course possible for two distinct points to both lie in two distinct rectangles), so we impose this condition by fiat:

Theorem 1 Suppose one has points and axis-parallel rectangles in the plane, whose incidence graph contains no ‘s, for some large .

• (i) The total number of incidences is .
• (ii) If all the rectangles are dyadic, the bound can be improved to .
• (iii) The bound in (ii) is best possible (up to the choice of implied constant).

We don’t know whether the bound in (i) is similarly tight for non-dyadic boxes; the usual tricks for reducing the non-dyadic case to the dyadic case strangely fail to apply here. One can generalise to higher dimensions, replacing rectangles by polytopes with faces in some fixed finite set of orientations, at the cost of adding several more logarithmic factors; also, one can replace the reals by other ordered division rings, and replace polytopes by other sets of bounded “semilinear descriptive complexity”, e.g., unions of boundedly many polytopes, or which are cut out by boundedly many functions that enjoy coordinatewise monotonicity properties. For certain specific graphs we can remove the logarithmic factors entirely. We refer to the preprint for precise details.

The proof techniques are combinatorial. The proof of (i) relies primarily on the order structure of to implement a “divide and conquer” strategy in which one can efficiently control incidences between points and rectangles by incidences between approximately points and boxes. For (ii) there is additional order-theoretic structure one can work with: first there is an easy pruning device to reduce to the case when no rectangle is completely contained inside another, and then one can impose the “tile partial order” in which one dyadic rectangle is less than another if and . The point is that this order is “locally linear” in the sense that for any two dyadic rectangles , the set is linearly ordered, and this can be exploited by elementary double counting arguments to obtain a bound which eventually becomes after optimising certain parameters in the argument. The proof also suggests how to construct the counterexample in (iii), which is achieved by an elementary iterative construction.

This past week was a great one for my institution, as the Robert A. Welch Foundation and Rice University announced the creation of the Welch Institute for Advanced Materials.  Exactly how this is going to take shape and grow is still in the works, but the stated goals of materials-by-design and making Rice and Houston a global destination for advanced materials research are very exciting.  

Long-time readers of this blog know my view that the amazing physics of materials is routinely overlooked in part because materials are ubiquitous - for example, the fact that the Pauli principle in some real sense is what is keeping you from falling through the floor right now.  I'm working on refining a few key concepts/topics that I think are translatable to the general reading public.  Emergence, symmetry, phases of matter, the most important physical law most people have never heard about (the Pauli principle), quasiparticles, the quantum world (going full circle from the apparent onset of the classical to using collective systems to return to quantum degrees of freedom in qubits).   Any big topics I'm leaving out?

Dimitri Shlyakhtenko and I have uploaded to the arXiv our paper Fractional free convolution powers. For me, this project (which we started during the 2018 IPAM program on quantitative linear algebra) was motivated by a desire to understand the behavior of the minor process applied to a large random Hermitian matrix , in which one takes the successive upper left minors of and computes their eigenvalues in non-decreasing order. These eigenvalues are related to each other by the Cauchy interlacing inequalities

When is large and the matrix is a random matrix with empirical spectral distribution converging to some compactly supported probability measure on the real line, then under suitable hypotheses (e.g., unitary conjugation invariance of the random matrix ensemble ), a “concentration of measure” effect occurs, with the spectral distribution of the minors for for any fixed converging to a specific measure that depends only on and . The reason for this notation is that there is a surprising description of this measure when is a natural number, namely it is the free convolution of copies of , pushed forward by the dilation map . For instance, if is the Wigner semicircular measure , then . At the random matrix level, this reflects the fact that the minor of a GUE matrix is again a GUE matrix (up to a renormalizing constant).

As first observed by Bercovici and Voiculescu and developed further by Nica and Speicher, among other authors, the notion of a free convolution power of can be extended to non-integer , thus giving the notion of a “fractional free convolution power”. This notion can be defined in several different ways. One of them proceeds via the Cauchy transform

-transform

Nica and Speicher also gave a free probability interpretation of the fractional free convolution power: if is a noncommutative random variable in a noncommutative probability space with distribution , and is a real projection operator free of with trace , then the “minor” of (viewed as an element of a new noncommutative probability space whose elements are minors , with trace ) has the law of (we give a self-contained proof of this in an appendix to our paper). This suggests that the minor process (or fractional free convolution) can be studied within the framework of free probability theory.

One of the known facts about integer free convolution powers is monotonicity of the free entropy

Our first main result is to extend the monotonicity results of Shylakhtenko to fractional . We give two proofs of this fact, one using free probability machinery, and a more self contained (but less motivated) proof using integration by parts and contour integration. The free probability proof relies on the concept of the free score of a noncommutative random variable, which is the analogue of the classical score. The free score, also introduced by Voiculescu, can be defined by duality as measuring the perturbation with respect to semicircular noise, or more precisely

The free score interacts very well with the free minor process , in particular by standard calculations one can establish the identity

After an extensive amount of calculation of all the quantities that were implicit in the above free probability argument (in particular computing the various terms involved in the application of Pythagoras’ theorem), we were able to extract a self-contained proof of monotonicity that relied on differentiating the quantities in and using the differential equation (1). It turns out that if for sufficiently regular , then there is an identity

These monotonicity properties hint at the minor process being associated to some sort of “gradient flow” in the parameter. We were not able to formalize this intuition; indeed, it is not clear what a gradient flow on a varying noncommutative probability space even means. However, after substantial further calculation we were able to formally describe the minor process as the Euler-Lagrange equation for an intriguing Lagrangian functional that we conjecture to have a random matrix interpretation. We first work in “Lagrangian coordinates”, defining the quantity on the “Gelfand-Tsetlin pyramid”

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Listen to a certain flavor of crackpot, or a certain kind of science fiction, and you’ll hear about zero-point energy. Limitless free energy drawn from quantum space-time itself, zero-point energy probably sounds like bullshit. Often it is. But lurking behind the pseudoscience and the fiction is a real physics concept, albeit one that doesn’t really work like those people imagine.

In quantum mechanics, the zero-point energy is the lowest energy a particular system can have. That number doesn’t actually have to be zero, even for empty space. People sometimes describe this in terms of so-called virtual particles, popping up from nothing in particle-antiparticle pairs only to annihilate each other again, contributing energy in the absence of any “real particles”. There’s a real force, the Casimir effect, that gets attributed to this, a force that pulls two metal plates together even with no charge or extra electromagnetic field. The same bubbling of pairs of virtual particles also gets used to explain the Hawking radiation of black holes.

I’d like to try explaining all of these things in a different way, one that might clear up some common misconceptions. To start, let’s talk about, not zero-point energy, but zero-point diagrams.

Feynman diagrams are a tool we use to study particle physics. We start with a question: if some specific particles come together and interact, what’s the chance that some (perhaps different) particles emerge? We start by drawing lines representing the particles going in and out, then connect them in every way allowed by our theory. Finally we translate the diagrams to numbers, to get an estimate for the probability. In particle physics slang, the number of “points” is the total number of particles: particles in, plus particles out. For example, let’s say we want to know the chance that two electrons go in and two electrons come out. That gives us a “four-point” diagram: two in, plus two out. A zero-point diagram, then, means zero particles in, zero particles out.

(Note that this isn’t why zero-point energy is called zero-point energy, as far as I can tell. Zero-point energy is an older term from before Feynman diagrams.)

Remember, each Feynman diagram answers a specific question, about the chance of particles behaving in a certain way. You might wonder, what question does a zero-point diagram answer? The chance that nothing goes to nothing? Why would you want to know that?

To answer, I’d like to bring up some friends of mine, who do something that might sound equally strange: they calculate one-point diagrams, one particle goes to none. This isn’t strange for them because they study theories with defects.

Normally in particle physics, we think about our particles in an empty, featureless space. We don’t have to, though. One thing we can do is introduce features in this space, like walls and mirrors, and try to see what effect they have. We call these features “defects”.

If there’s a defect like that, then it makes sense to calculate a one-point diagram, because your one particle can interact with something that’s not a particle: it can interact with the defect.

You might see where this is going: let’s say you think there’s a force between two walls, that comes from quantum mechanics, and you want to calculate it. You could imagine it involves a diagram like this:

Roughly speaking, this is the kind of thing you could use to calculate the Casimir effect, that mysterious quantum force between metal plates. And indeed, it involves a zero-point diagram.

Here’s the thing, though: metal plates aren’t just “defects”. They’re real physical objects, made of real physical particles. So while you can think of the Casimir effect with a “zero-point diagram” like that, you can also think of it with a normal diagram, more like the four-point diagram I showed you earlier: one that computes, not a force between defects, but a force between the actual electrons and protons that make up the two plates.

A lot of the time when physicists talk about pairs of virtual particles popping up out of the vacuum, they have in mind a picture like this. And often, you can do the same trick, and think about it instead as interactions between physical particles. There’s a story of roughly this kind for Hawking radiation: you can think of a black hole event horizon as “cutting in half” a zero-point diagram, and see pairs of particles going out from the black hole…but you can also do a calculation that looks more like particles interacting with a gravitational field.

This also might help you understand why, contra the crackpots and science fiction writers, zero-point energy isn’t a source of unlimited free energy. Yes, a force like the Casimir effect comes “from the vacuum” in some sense. But really, it’s a force between two particles. And just like the gravitational force between two particles, this doesn’t give you unlimited free power. You have to do the work to move the particles back over and over again, using the same amount of power you gained from the force to begin with. And unlike the forces you’re used to, these are typically very small effects, as usual for something that depends on quantum mechanics. So it’s even less useful than more everyday forces for this.

Why do so many crackpots and authors expect zero-point energy to be a massive source of power? In part, this is due to mistakes physicists made early on.

Sometimes, when calculating a zero-point diagram (or any other diagram), we don’t get a sensible number. Instead, we get infinity. Physicists used to be baffled by this. Later, they understood the situation a bit better, and realized that those infinities were probably just due to our ignorance. We don’t know the ultimate high-energy theory, so it’s possible something happens at high energies to cancel those pesky infinities. Without knowing exactly what happened, physicists would estimate by using a “cutoff” energy where they expected things to change.

That kind of calculation led to an estimate you might have heard of, that the zero-point energy inside single light bulb could boil all the world’s oceans. That estimate gives a pretty impressive mental image…but it’s also wrong.

This kind of estimate led to “the worst theoretical prediction in the history of physics”, that the cosmological constant, the force that speeds up the expansion of the universe, is 120 orders of magnitude higher than its actual value (if it isn’t just zero). If there really were energy enough inside each light bulb to boil the world’s oceans, the expansion of the universe would be quite different than what we observe.

At this point, it’s pretty clear there is something wrong with these kinds of “cutoff” estimates. The only unclear part is whether that’s due to something subtle or something obvious. But either way, this particular estimate is just wrong, and you shouldn’t take it seriously. Zero-point energy exists, but it isn’t the magical untapped free energy you hear about in stories. It’s tiny quantum corrections to the forces between particles.

For the triumphant final video in the Biggest Ideas series, we look at a big idea indeed: Science. What is science, and why is it so great? And I also take the opportunity to dip a toe into the current state of fundamental physics — are predictions that unobservable universes exist really science? What if we never discover another particle? Is it worth building giant expensive experiments? Tune in to find out.

Thanks to everyone who has watched along the way. It’s been quite a ride.