Planet Musings

There is a very nice algorithm and set of methods called “Robust PCA,” originating in a paper by Candès. This method makes use of ideas from convex optimization to simultaneously learn a low-rank representation of the data plus a sparse representation for the outliers. This kind of situation comes up in astronomy all the time.

Way back, Tsalmantza and I made a replacement for PCA called HMF that deals with heteroskedastic data (variable error bars or variable data weights) and also missing data; we used it to build low-rank models of quasar spectra. This weekend I built a robust version of HMF (Robust HMF maybe?) that uses ideas from iteratively reweighted least squares to mimic the algorithm behind Robust PCA. It works! And it works well. Unfortunately, right now the investigator has to tune the rank of the low-rank part and also the soft outlier cutoff used in the IRLS. I would love to figure out principled ways to choose both. If you want to follow along, development is happening here for now.

The Kepler problem studies a particle moving in an inverse square force, like a planet orbiting the Sun. Last time I talked about an extra conserved quantity associated to this problem, which keeps elliptical orbits from precessing or changing shape. This extra conserved quantity is sometimes called the Laplace–Runge–Lenz vector, but since it was first discovered by none of these people, I prefer to call it the ‘eccentricity vector’

In 1847, Hamilton noticed a fascinating consequence of this extra conservation law. For a particle moving in an inverse square force, its momentum moves along a circle!

Greg Egan has given a beautiful geometrical argument for this fact:

• Greg Egan, The ellipse and the atom.

I will not try to outdo him; instead I’ll follow a more dry, calculational approach. One reason is that I’m trying to amass a little arsenal of formulas connected to the Kepler problem.

Let’s dive in. Remember from last time: we’re studying a particle whose position obeys

Its momentum is

Its momentum is not conserved. Its conserved quantities are energy:

the angular momentum vector:

and the eccentricity vector:

Now for the cool part: we can show that

Thus, the momentum stays on a circle of radius centered at the point And since and are conserved, this circle doesn’t change! Let’s call it Hamilton’s circle.

Now let’s actually do the calculations needed to show that the momentum stays on Hamilton’s circle. Since

we have

Taking the dot product of this vector with itself, which is 1, we get

Now, notice that and are orthogonal since Thus

I actually used this fact and explained it in more detail last time. Substituting this in, we get

Similarly, and are orthogonal! After all,

The first term is orthogonal to since it’s the cross product of and some other vector. And the second term is orthogonal to since is the cross product of and some other vector! So, we have

and thus

Substituting this in, we get

Using the cyclic property of the scalar triple product, we can rewrite this as

This is nicer because it involves in two places. If we divide both sides by we get

And now for the final flourish! The right hand is the dot product of a vector with itself:

This is the equation for Hamilton’s circle!

Now, beware: the momentum doesn’t usually move at a constant rate along Hamilton’s circle, since that would force the particle’s orbit to itself be circular.

But on the bright side, the momentum moves along Hamilton’s circle regardless of whether the particle’s orbit is elliptical, parabolic or hyperbolic. And we can easily distinguish the three cases using Hamilton’s circle!

After all, the center of Hamilton’s circle is the point and

so the distance of this center from the origin is

On the other hand, the radius of Hamilton’s circle is So his circle encloses the origin, goes through the origin or does not enclose the origin depending on whether or But we saw last time that these three cases correspond to elliptical, parabolic and hyperbolic orbits!

Summarizing:

• If the particle’s orbit is an ellipse and the origin lies inside Hamilton’s circle. The momentum goes round and round Hamilton’s circle as time passes.

• If the particle’s orbit is a parabola and the origin lies exactly on Hamilton’s circle. The particle’s momentum approaches zero as time approaches so its momentum goes around Hamilton’s circle exactly once as time passes.

• If the particle’s orbit is a hyperbola and the origin lies outside Hamilton’s circle. The particle’s momentum approaches distinct nonzero values as time approaches so its momentum goes around just a portion of Hamilton’s circle.

By the way, in general the curve traced out by the momentum vector of a particle is called a hodograph. So you can learn more about Hamilton’s circle with the help of that buzzword.

I’ve been working on a math project involving the periodic table of elements and the Kepler problem—that is, the problem of a particle moving in an inverse square force law. I started in 2021, the year I retired from U. C. Riverside. I had driven my wife across the country to DC, and spent the fall with her at the Center for Hellenic Studies, where she was working. They own a compound of dinky little houses right across the street from Hilary Clinton’s much grander home. The brilliant fall leaves inspired thoughts of chemistry: electrons and photons in resonance.

I felt very free, with nothing particular to do, so I did something just for fun: I explored a curious relation between the Kepler problem, the periodic table, and quantum field theory.

When I drove back to Riverside at the end of 2021 I got distracted by other things. But this year I’ve been in the mood for wrapping up half-done projects. I finished this project today after a week of hard work. It should appear on the arXiv soon.

I want to explain it! But I need some things to be fresh in your mind, so I’m going to republish some blog articles I wrote in 2021 to explain this work, starting with this one. (For Part 1 go here, but you won’t need that for what’s to come.)

Let’s start with the ‘eccentricity vector’. This is a conserved quantity for the Kepler problem. It always points in the direction of the orbit’s perihelion, and its magnitude equals the eccentricity of the orbit. There’s a simple formula for it in terms of the position, momentum and angular momentum of the particle:

But it never changes.

The eccentricity vector was named the ‘Runge–Lenz vector’ after Lenz used it in 1924 to study the hydrogen atom in the framework of the ‘old quantum mechanics’ of Bohr and Sommerfeld: Lenz cited Runge’s popular German textbook on vector analysis from 1919, which explains this vector. But Runge never claimed any originality: he attributed this vector to Gibbs, who wrote about it in his book on vector analysis in 1901!

Nowadays many people call it the ‘Laplace–Runge–Lenz vector’, honoring Laplace’s discussion of it in his famous treatise on celestial mechanics in 1799. But in fact this vector goes back at least to Jakob Hermann, who wrote about it in 1710, triggering further work on this topic by Johann Bernoulli in the same year.

Nobody has seen signs of this vector in work before Hermann. So, we might call it the Hermann–Bernoulli–Laplace–Gibbs–Runge–Lenz vector, or just the Hermann vector. But I prefer to call it the eccentricity vector, because for a particle in an inverse square law its magnitude is the eccentricity of that orbit!

Let’s suppose we have a particle whose position obeys this version of the inverse square force law:

where I remove the arrow from a vector when I want to talk about its magnitude. So, I’m setting the mass of this particle equal to 1, along with the constant saying the strength of the force. That’s because I want to keep the formulas clean! With these conventions, the momentum of the particle is

For this system it’s well-known that the following energy is conserved:

as well as the angular momentum vector:

But the interesting thing for me today is the eccentricity vector:

Let’s check that it’s conserved! Taking its time derivative,

But angular momentum is conserved so the second term vanishes, and

so we get

But the inverse square force law says

so

How can we see that this vanishes? Mind you, there are various geometrical ways to think about this, but today I’m in the mood for checking that my skills in vector algebra are sufficient for a brute-force proof—and I want to record this proof so I can see it later!

To get anywhere we need to deal with the cross product in the above formula:

There’s a nice identity for the vector triple product:

I could have fun talking about why this is true, but I won’t now! I’ll just use it:

and plug this into our formula

getting

But look—everything cancels! So

and the eccentricity vector is conserved!

So, it seems that the inverse square force law has 7 conserved quantities: the energy the 3 components of the angular momentum and the 3 components of the eccentricity vector . But they can’t all be independent, since the particle only has 6 degrees of freedom: 3 for position and 3 for momentum. There can be at most 5 independent conserved quantities, since something has to change. So there have to be at least two relations betwen the conserved quantities we’ve found.

The first of these relations is pretty obvious: and are at right angles, so

But wait, why are they at right angles? Because

The first term is orthogonal to because it’s a cross product of and the second is orthogonal to because is a cross product of and .

The second relation is a lot less obvious, but also more interesting. Let’s take the dot product of with itself:

or in other words,

But remember this nice cross product identity:

Since and are at right angles this gives

so

Then we can use the cyclic identity for the scalar triple product:

to rewrite this as

or simply

or even better,

But this means that

which is our second relation between conserved quantities for the Kepler problem!

This relation makes a lot of sense if you already know some facts about the eccentricity vector: it points in the direction of the orbit’s perihelion, and its magnitude equals the eccentricity of the orbit. From the relation above you can then see:

• if then and the orbit is a hyperbola.

• if then and the orbit is a parabola.

• if then and the orbit is an ellipse (or circle).

But why is the eccentricity? And why does the particle move in a hyperbola, parabola or ellipse in the first place? We can show both of these things by taking the dot product of and

Using the cyclic property of the scalar triple product we can rewrite this as

Now, we know that moves in the plane orthogonal to . In this plane, which contains the vector the equation defines a conic of eccentricity . I won’t show this from scratch, but it may seem more familiar if we rotate the whole situation so this plane is the plane and points in the direction. Then in polar coordinates this equation says

or

This is well-known, at least among students of physics who have solved the Kepler problem, to be the equation of a conic of eccentricity .

Another thing that’s good to do is define a rescaled eccentricity vector. In the case of elliptical orbits, where we define this by

Then we can take our relation

and rewrite it as

and then divide by getting

or

This suggests an interesting similarity between and which turns out to be very important in a deeper understanding of the Kepler problem. And with more work, you can use this idea to show that is the Hamiltonian for a free particle on the 3-sphere. But more about that some other time, I hope!

For now, you might try this:

• Wikipedia, Laplace–Runge–Lenz vector.

and of course this:

• The Kepler problem (part 1).

Jesper Grimstrup kindly sent me an electronic copy of his new book, The Ant Mill. He was also kind enough to give me some feedback on a first version of this review.

The story is an interweaving of the author's personal experiences in academia with his general observations. This personal story and his experiences are interesting, much like I expect those of everyone who has spent several years in academia would be. And he has clearly spent a lot of time thinking about thinking about research. I love meta-activities of this sort; the best example that I know of is You and Your Research by Hamming, which I stumbled on as a postdoc. Indeed, the existence of these sorts of things that are shared by young researchers is actually evidence against the central thesis of Grimstrup's book.

The market attacking High-Energy Physics seems to be burgeoning. On the one hand Hossenfelder believes that we have become 'lost in math,' and on the other Woit believes we are not mathematical enough; both attack string theory as a failed program. Grimstrup's book is in the mathematical camp, with the novelty that he piles scorn on all popular approaches to quantum gravity, in particular loop quantum gravity and noncommutative geometry, since he has come into closest contact with them. His observations about string theorists are mainly about the shoddy way that he was treated during his time at the NBI, with several egregious examples of bad behaviour. We are lead to conclude that it is not just string theorists who have formed a closed tribe, but that there are several such groups crowding out innovation.

Grimstrup refers to his own research program and gives examples of how it has just generally been ignored within academia. For example, he starts the book with a copy of a grant application by a 31-year-old Niels Bohr for an entire institute, and contrasts this with a grant application of his that was refused that effectively ended his career within academia (my understanding is that at the NBI in Copenhagen it is common to ask for and obtain grants to pay your own salary and prolong temporary contracts). He writes that he does not do this to compare himself to Niels Bohr, but inadvertently this is the impression I got from the book -- that he was doing this not in a self-aggrandising way, but in the sense that you can almost feel his frustration coming through the pages that his expectations did not meet reality. It seems like bait at times, inviting anyone who disagrees with the general thesis to attack him personally. Instead, I will have a look at his papers with an open mind, after writing this review, and keep my thoughts on them to myself.

The book made me think of how many of us enter academia. We grow up reading popular science accounts idolising physicists from a century ago. And it made me think more of the self-actualisation messages that were rammed down all our throats in the popular culture in the 80s and 90s: follow your dreams, stick to your principles, be true to yourself, this is the most important thing in life and you shouldn't worry about money, just be happy. And: working hard and getting good grades is the way to get to the top. The problem is that this is largely obsolete: it's based on the world that existed post world war two when there was a scarcity of labour and an economic boom. Then -- if you were from the right background and your face fit -- you could work hard, get a PhD and walk into a permanent academic job (yes this is a caricature). Science was respected and so were scientists; high-energy physics was at the top of the tree because of the connection with technological advancements and nuclear weapons. That world doesn't exist any more; while in many ways for the better, it is undeniable that we live in a world of much greater competition and public skepticism about science is increasing.

The scientific community has expanded, as has the population; and more importantly education throughout the world and global travel and communication has meant that the number of people around the world who are involved in research is much greater than it was. Grimstrup notes that increasing the size of the academic community has led to fundamental changes of behaviour: professionalisation of research and group think, and that this leads to an increasing incentive to work on mainstream topics. He has done bibliographical research to demonstrate this (in papers and presented in the book). It is clearly true that the Matthew effect exists in many branches of society, and therefore also in academia; governments wanting to exert some form of oversight in exchange for the funds that they provide has definitely led to changes in incentives for researchers. One aspect of this is that it is hard to judge the work of people from other fields, but we are required to do so; and then it is difficult to argue with quantitative measures such as number of papers, citations, h-indices. Then of course the measure becomes the target for certain people. 

Grimstrup rails against all these changes; he clearly believed that the correct thing to do for an aspiring researcher would be to work on their own ideas, stick to their principles and not compromise. They should work for a long time, in isolation, on a major paper, put it on arxiv.org and the next day their colleagues would read it and ask interesting questions about them. Fame and fortune would follow. The thing that shocked Grimstrup was that not only did people not even care about any papers he posted, a young competitor even once told him some ideas are simply not worth pursuing even though they may be interesting. For sure, this is horrible and shocking behaviour, and does not reflect well on the anonymous person who said it.

For my part I am still naive enough to think that if new ideas are good, someone will recognise them as such, and network effects will make them known. I know that many researchers already think more deeply about what they are doing than he gives us credit for: and we discuss it, during seminars, over a drink with colleagues, in the coffee-breaks of conferences, during our annual or five-year reviews, or in grant applications. When I discussed this review with a string-theorist colleague they remarked "of course we know the situation sucks!''  I think Grimstrup is therefore wrong to tar everyone with the same brush: the diversity in our community has increased greatly with time, and this means that there are indeed strong incentives to take a risk on a novel idea, because the rewards of opening a new research direction are immense! Being the originator of an idea, or the first to recognise the merit in even an old forgotten idea, can yield tremendous results and even greater recognition nowadays thanks to the same effects. Hence, starting a new field, or even a subfield, is something that most researchers aspire to; the rewards for doing so are even greater now than in times gone by, and the evidence that this is possible is even given in this book: the existence of several communities working on different approaches to quantum gravity. He argues that these are now old and stale, but my point is that the way that they were able to take root at all is an example of how this can happen. There are many subfields that have sprung up more recently, and in other branches of HEP there are of course many examples. Nowadays things can change very quickly: a new good idea will be very rapidly jumped on once it is recognised, and people are constantly on the lookout. 

Grimstrup also, like Lee Smolin, divides researchers into visionaries and technicians. He then complains that the technicians have taken over, with lots of disparaging comments about them digging endless holes. He then complains that there is an incentive to collaborate in modern research, only collaborators survive in the system: he has evidence that being a lone wolf is a poor survival strategy. He believes that we should work on our own; yet at the same time visionaries need to collaborate with technicians. I found this very jarring. Other than the facile placing of people into boxes, he is overlooking the benefits of collaboration -- his opinion is that it is just about inflating the number of papers one person can sign (and for sure there are people who cynically do this). But to me, discussing with other people, even just explaining something, is often the quickest way to generate genuinely new ideas or solutions to problems that we may never have come up with alone. At the same time, there are plenty of people who do write papers alone; to take a leaf from his book and share a personal story, I once had a comment on a postdoc application that I had no single-author papers and therefore did not demonstrate independence. Hence, there are incentives and a good reason for young researches to work alone sometimes. I then wrote a single-author paper, as I have occasionally done since (and got the fellowship next time I applied); I would agree that there is a pleasure and some advantages in doing this, but to do this all the time would mean I would risk missing out on lots of new ideas and other perspectives, as well as the pleasure of regular interactions with collaborators, and it would also limit the scope of my projects, where I benefit from others' expertise. Or collaborations may just be working with a student, pursuing my ideas (hopefully they contribute some of their own!) and imparting my knowledge in the process. This is why I do not think that encouraging people to predominantly cloister themselves away to work alone for a long time is the most productive or healthy one. 

The book also has a very narrow focus as to the goal of high-energy physics. For the author, the quest is a "the next theory," but in essence this means a theory of quantum gravity, which he acknowledges would be far from being able to be tested with any present or near-future data. Otherwise, we should look for a mathematically rigorous definition of quantum field theory; he hopes these will be one and the same thing. This latter problem has proven to be both very hard and not obviously useful -- it is certainly not obvious that the solution should even be unique, for example a theory of strings would cure ultra-violet divergences, and the question of whether strings should be necessary for such a theory is one that I know people have tried to explore. I also recently attended a talk by Michael Douglas where he reviewed recent attempts on rigorous QFT, so it is a subject that is regarded as important but very difficult, and still being explored by a small number of people. Regarding quantum gravity, some people in the community have taken the opinion that if you have no data, it is not a good problem, and are working on other things. Or people try to make contact with data using e.g. EFT approaches to measuring quantum effects of gravity. The string theory community might say that we do have a theory of quantum gravity, in fact we have a landscape of them, and try e.g. to use it to answer questions about black hole information. But at the same time some people then complain that the leading string theorists have moved on to other things: there are lots of important open fundamental problems, and we just do not know how they are interlinked, if at all!

Grimstrup's insistence that the solution to what he sees as problems is to shrink competition and also encourage research outside of academia, reminded me of another Dane, subject of another book I read recently: king Cnut, famous for (presumably apocryphally) standing on the beach in front of his ministers and commanding the tide to turn back. Otherwise Grimstrup hopes for a crisis, perhaps one provoked by his book. He explicitly states that he does not want to fuel the anti-establishment or ant-academic movements, but I suspect that the only crises we might suffer would not be good for the field.  Perhaps one is already taking place in the US; perhaps people will take his message to heart despite his protests and start a DOGE-style decimation of research. Necessarily, in science we mark our own homework: only other scientists are capable of judging the claims of their peers. If we start opening this up to question then we will only end with government appointees deciding what are acceptable topics and directions, or shutting public funding down altogether. What would be left over would surely be even greater competition for scarce resources.

For me, the solution to the problems in the book, to the extent that I agree with them, is to regularly remind ourselves that we should always maintain a childlike curiosity and not close our minds to new ideas and new possibilities. This is the message from the text of Hamming, and very well put in the writings of Feynman (who Grimstrup bizarrely dismisses as a technician compared to Bohr). Otherwise of course in science it is necessary to have a community spirit, to realise that we are all trying to make progress in the best way we know how, and to help each other do so; and it is necessary to maintain healthy competition as a motivator. But both conflicting instincts -- to compete and to group into communities -- are vital parts of human nature and denying this has been the mistake of utopians throughout history. 

I am also sure that many of the complaints that Grimstrup assigns to high-energy physics could also be applied to society more generally. So instead of trying to hold back or reverse the societal changes of the last century we should try to work with them as best we can. We have to accept that we live now in an attention economy; and this gives new opportunities: blogging, social media, writing articles in science magazines or popular press, etc. Since Grimstrup is now, interestingly, an independent scientist, perhaps tying his own research program so closely with his book is embracing the modern world at last, and creating a brand as a radical outside thinker, that will be attractive to private backers. He promotes the path that he has followed, crowdfunding his research or seeking support of patrons, as a possible path for the independently minded once they have completed their training in academia, and in this I wish him well: he is clearly serious, determined and sincere. But while this is now part of twenty-first century society, many people have noticed that this modern trend is a return to the nineteenth century (or even earlier, e.g. Leonardo da Vinci being invited to France by François 1) where a wealthy patron was the only source of funding. 

are the first names of eleven out of the first twenty players the Orioles drafted this year.

[More technical post follows.] I've been working on this project with (UCSB postdoc) Maciej Kolanowski on and off for a while now, but only in the last couple of weeks did I have the time to hunker down and help push the writing of the results to the finish. For your Sunday reading pleasure, it is already up on the arXiv here (it came out Thursday but I've been too busy to pause to post about it - partly because I've begun work on writing up the next paper in the backlog). The title is "Extended JT supergravity and random matrix models: The power of the string equation", and it is co-authored with Maciej Kolanowski.

In a way, it is a natural continuation of work I've described here from 2023 and 2024, described here and here. At a meeting at the Institute for Advanced Study in December 2023 I described in a talk (YouTube video here, look in particular from minute 35) something miraculous I'd discovered concerning capturing certain special supergravity (and black hole) behaviour using a random matrix model. The effective physics is [...] Click to continue reading this post →

The post The Power of the String Equation appeared first on Asymptotia.

A nice talk at MPIA by Hanna Ũbler (MPE) about very high-redshift (redshifts 8 to 14 even) galaxies and their black-hole contents started some nice discussions in the audience and afterwards about the formation of black holes. Because of the Eddington limit (which is a limit on luminosity), black-hole growth is probably limited. The limit is on luminosity, not mass accretion rate, and the relationship between these is the radiative efficiency. As the efficiency goes down, the mass accretion rate of an Eddington accretor goes up. So the question: When can you first form a super-massive (10 million solar masses, say) black hole is a question simultaneously about seed black holes and about radiative efficiency. Anyway, this is all unfortunate, because if the efficiency couldn't get very low, then the black holes we find with NASA JWST would already be putting very strong pressure on fundamental physics in the early Universe.

Coherent oscillators—astronomical sources that pulse or oscillate in a phase-stable way over long timescales—have been useful astrophysical tools. For two examples: Stably pulsing pulsars were used to discover gravitational radiation (and are being used to find the stochastic background). Delta-scuti star asteroseismic modes were used to find orbital companions. This led undergrad Nana Miller (NYU) and me and others to look for all the coherent modes we can find among all the stars in the NASA Kepler Mission data. We have have technology to find modes, and we have technology to test for coherence. Now we have to do injection–recovery tests to estimate our detection limits. Today I delivered a simple plan for doing injections.

Our plan is to produce a catalog, not of stars but of modes, every one of which has a coherence time that is longer than the lifetime (4 years) of the Kepler Mission. Then: What do we use them for?

Despite the “2” in the title, you can follow this post without having read part 1. The whole point is to sneak up on the metricky, analysisy stuff about potential functions from a categorical angle, by considering constructions that are categorically reasonable in their own right. Let’s go!

Let $V$ be a monoidal category, which I’ll assume is symmetric, closed, complete or cocomplete as necessary. Take a pair of $V$-categories $A$ and $B$, and a $V$-functor

$$F: A \to B.$$

Then we get a new functor

$$N_F: B \to [A^{op}, V]$$

defined by

$$(N_F(b))(a) = B(F a, b).$$

This is sometimes called the nerve functor induced by $F$, because of the first example in the following list.

Examples   In the first few examples, I’ll take the enriching category $V$ to be $Set$.

• Let $F$ be the inclusion $\Delta \hookrightarrow Cat$, which takes a finite nonempty totally ordered set $[n] = \{0, \ldots, n\}$ and realizes it as a category in the usual way. Then $N_F: Cat \to [\Delta^{op}, Set]$ sends a small category $D$ to its nerve, which is the simplicial set whose $n$-simplices are paths in $D$ of length $n$.

• Let $F$ be the functor $\Delta \to Top$ that sends $[n]$ to the topological $n$-simplex $\Delta^n$. Then $N_F: Top \to [\Delta^{op}, Set]$ sends a topological space $X$ to its “singular simplicial set”, whose $n$-simplices are the continuous maps $\Delta^n \to X$.

• Let $F: A \hookrightarrow B$ be an inclusion of partially ordered sets, viewed as categories in the usual way. Then $N_F : B \to [A^{op}, Set]$ is given by

  $$(N_F(b))(a) = \begin{cases} 1 &\text{if}   b \leq a \\ 0 &\text{otherwise}. \end{cases}$$

  For example, let $B$ be the set $\mathbb{Z}$ of integers, ordered by divisibility. Let $A \subseteq \mathbb{Z}$ be the set of (positive) primes. Then $N_F$ is the functor $\mathbb{Z} \to Set^{\{primes\}}$ that sends $n \in \mathbb{Z}$ to

  $$([p | n])_{\text{primes}   p}$$

  where $[p|n]$ is Iverson notation: it’s $1$ if $p$ divides $n$ and $0$ otherwise.

• If $F$ has a right adjoint $G$, then $N_F(b) = A(-, G b): A^{op} \to V$. In particular, $N_F(b)$ is representable for each $b \in B$.

• Take the inclusion $Field \hookrightarrow Ring$, where $Ring$ is the category of commutative rings. Now take opposites throughout to get $F: Field^{op} \hookrightarrow Ring^{op}$. The resulting nerve functor

  $$N_F: Ring^{op} \to [Field, Set]$$

  sends a ring $R$ to $Ring(R, -)|_{Field}$ (by which I mean the restriction of $Ring(R, -): Ring \to Set$ to the category of fields). More explicitly,

  $$N_F(R) = \sum_{p \in Spec(R)} Field(Frac(R/p), -).$$

  Here $Spec(R)$ is the set of prime ideals $p$ of $R$, and $Frac(R/p)$ is the field of fractions of the integral domain $R/p$. In particular, $N_F(b)$ is a coproduct of representables for each $b \in B$. (In the terminology of Diers, one therefore says that $Field \hookrightarrow Ring$ is a “right multi-adjoint”. A closely related statement is that it’s a parametric right adjoint. But that’s all just jargon that won’t matter for this post.)

Now let’s consider $V = (\mathbb{R}^+, \geq)$ with its additive monoidal structure, so that $V$-categories are generalized metric spaces and $V$-functors are the functions $f$ between metric spaces that are variously called short, distance-decreasing, contractive, or 1-Lipschitz: $d(f(a), f(a')) \leq d(a, a')$ for all $a, a'$. I’ll just call them “maps” of metric spaces.

• Take a map $f: A \to B$ of metric spaces. The induced map $N_f: B \to [A^{op}, \mathbb{R}^+]$ is given by

  $$(N_f(b))(a) = d(f(a), b).$$

  Unlike in the first post, I’m not going to assume here that our metric spaces are symmetric. so the “op” on the $A$ matters, and $d(a, f(b)) \neq d(f(b), a)$ in general.

  We’ll be particularly interested in the case where the $V$-functor $F$ is full and faithful. Most of the examples above have this property. In the metric case, $V = \mathbb{R}^+$, being full and faithful means being distance-preserving, or in other words, an inclusion of a metric subspace. In that case, it’s normal to drop the $f$. So we’d usually then write

  $$(N_f(b))(a) = d(a, b).$$

Next I’ll define the potential function of a $V$-functor. For this we need some of the apparatus of magnitude. Roughly speaking, this means we have a notion of the “size” of each object of our base monoidal category $V$.

More exactly, we take a field $k$ and a function $|\cdot|: ob(V) \to k$, to be thought of as assigning to each object $S$ of $V$ its size ${|S|} \in k$. And to make everything work, we assume our size function $|\cdot|$ has reasonable properties:

$$S \cong T \implies |S| = |T|, \quad |S \otimes T| = |S| \cdot |T|, \quad |I| = 1,$$

where $\otimes$ is the monoidal product on $V$ and $I$ is its unit.

The basic definitions are these. A weighting on a finite $V$-category $A$ is a function $w_A : ob(A) \to k$ such that

$$\sum_{b \in A} |A(a, b)| w_A(b) = 1$$

for all $a \in A$, and a coweighting $w^A$ on $A$ is a weighting on $A^{op}$. Although it’s not true that every finite $V$-category has a unique weighting, it’s a harmless enough assumption that I’ll make it here. The magnitude of a finite $V$-category $A$ is

$$|A| = \sum_{a \in A} w_A(a) = \sum_{a \in A} w^A(a) \in k.$$

It’s a tiny lemma that the total weight $\sum_{a \in A} w_A(a)$ is equal to the total coweight $\sum_{a \in A} w^A(a)$, so you can equivalently define the magnitude to be one or the other.

There’s a whole lot to say about why this is a worthwhile definition, and I’ve said it in detail many times here before. But here I’ll just say two short things:

• Taking $V = FinSet$, $k = \mathbb{Q}$, and $|\cdot|$ to be cardinality, the magnitude of a finite category is equal to the Euler characteristic of its classifying space, under suitable hypotheses guaranteeing that the latter is defined.

• Take $V = \mathbb{R}^+$, $k = \mathbb{R}$, and $|\cdot|$ to be $s \mapsto e^{-s}$ (because that’s just about the only function converting the tensor product of $V$ into addition, as our size functions are contractually obliged to do). Then we get a notion of the magnitude of a metric space, a real-valued invariant that turns out to be extremely interesting.

That’s the magnitude of enriched categories. But we can also talk about the magnitude of enriched presheaves. Take a finite $V$-category $A$ and a $V$-presheaf $X: A^{op} \to V$. Its magnitude is

$$|X| = \sum_{a \in A} w^A(a) |X(a)| \in k.$$

When I wrote about the magnitude of functors before, I concentrated on covariant functors $A \to V$, which meant that the weights in the weighted sum that defines $|X|$ were, well, the weights $w_A(a)$. But since we’ve now changed $A$ to $A^{op}$, the weights have become coweights.

Let me briefly recall why the magnitude of presheaves is interesting, at least in the case $V = FinSet$:

• If our presheaf $X$ is a coproduct of representables then $|X|$ is $|colim(X)|$, the cardinality of the colimit of $X$.

• The magnitude of presheaves generalizes the magnitude of categories. The magnitude of a category $A$ is equal to the magnitude of the presheaf $A^{op} \to Set$ with constant value $1$.

• In the other direction, the magnitude of categories generalizes the magnitude of presheaves: $|X| = |\mathbb{E}(X)|$ for all presheaves $X$. Here $\mathbb{E}(X)$ means the category of elements of $X$, also called the (domain of the) Grothendieck construction.

  It’s enlightening to think of this result as follows. If we extend the codomain of $X$ from $Set$ to $Cat$ then $\mathbb{E}(X)$ is the colax colimit of $X: A^{op} \to Cat$. So $|X| = |colaxcolim(X)|$. You can compare this to the more limited result that $|X| = |colim(X)|$, which only holds when $X$ has the special property that it’s a coproduct of representables.

  There’s also another closely related result, due to Ponto and Shulman: under mild further hypotheses, $|X| = \chi(hocolim(X))$. Mike and I had a little discussion about the relationship here.

  Now here’s something funny. Which do you think is a more refined invariant of a presheaf $X: A^{op} \to FinSet$: the magnitude $|X|$, which is equal to $|colaxcolim(X)|$ and $\chi(hocolim(X))$, or the cardinality $|colim(X)|$ of the strict colimit?

  From one perspective, it’s the magnitude. After all, we usually think of lax or pseudo 2-dimensional things as more subtle and revealing than strict 1-dimensional things. But from another, it’s the cardinality of the strict colimit. For instance, if $A$ is the one-object category corresponding to a group $G$ then $X$ is a right $G$-set, the cardinality of the colimit is the number of orbits (an interesting quantity), but the magnitude is always just the cardinality of the set $X$ divided by the order of $G$ (relatively boring and crude).

  In this post I’ll keep comparing these two quantities. Whichever seems “more refined”, the comparison is interesting in itself.

(I stuck to the case $V = FinSet$ here, because for an arbitrary $V$, there isn’t a notion of “colimit” as such: we’d need to think about weighted colimits. It’s also harder to say what the category of elements or homotopy colimit of an enriched presheaf should be.)

Now here’s something new. The potential function of a $V$-functor $F: A \to B$ is the function

$$h_F: \ob(B) \to k$$

defined by

$$h_F(b) = |N_F(b)|$$

for all $b \in B$. Expanding out the definitions, this means that

$$h_F(b) = |A(F-, b)| = \sum_{a \in A} w^A(a) |B(F a, b)|,$$

where $w^A$ is the notation I introduced for the coweighting on $A$. If $F$ is full and faithful then $h_F \equiv 1$ on the set of objects in the image of $F$.

Examples   The definition I just gave implicitly assumes that $\ob(A)$ is finite. But I’ll relax that assumption in some of these examples, at the cost of some things getting less than rigorous.

Again, I’ll begin with some examples where $V = FinSet$.

• What’s the potential function of the inclusion $\Delta \hookrightarrow Cat$? To make sense of this, we need a coweighting on $\Delta$, and I’m pretty sure there isn’t one. So let’s abandon $\Delta$ and instead use its subcategory $\Delta_{inj}$, consisting of all the finite nonempty totally ordered sets $[n] = \{0, \ldots, n\}$ but only the injective order-preserving maps between them. This amounts to considering only face maps, not degeneracies.

  The coweighting on $\Delta_{inj}$ is $[n] \mapsto (-1)^n$. So, the potential function $h$ of $\Delta_{inj} \hookrightarrow Cat$ is given, on a category $D$, by

  $$h(D) = \sum_{n \geq 0} (-1)^n |Cat([n], D)|.$$

  Here $Cat([n], D)$ is the set of paths of length $n$ in $D$, which we often write as $D_n$. So

  $$h(D) = \sum_{n \geq 0} (-1)^n |D_n|.$$

  Even if the category $D$ is finite, this is a divergent series. But morally, $h(D)$ is the alternating sum of the number of $n$-cells of $D$.

  In other words, the potential function of $\Delta_{inj} \to Cat$ is morally Euler characteristic.

  As I hinted above, it’s interesting to compare the potential function with what you get if you take the cardinality of the colimit of $B(F-, b)$ instead of its magnitude. I’ll call this the “strict potential function”. In this case, it’s

  $$D \mapsto |colim_{[n] \in \Delta_{inj}} D_n| = |\pi_0(D)|$$

  — the cardinality of the set of connected-components of $D$. So while the potential function gives Euler characteristic, the strict potential function gives the number of connected components.

• Similarly, the potential function of the usual functor $\Delta_{inj} \hookrightarrow Top$, sending $n$ to $\Delta^n$, is given by

  $$D \mapsto \sum_{n \geq 0} (-1)^n |Top(\Delta^n, D)|$$

  for topological spaces $D$. Again, this formally looks like Euler characteristic: the alternating sum of the number of cells in each dimension. And much as for categories, the strict potential function gives the number of path-components.

• For the inclusion $\{primes\} \hookrightarrow (\mathbb{Z}, |)$ that we saw earlier, the potential function $h: \mathbb{Z} \to \mathbb{Q}$ is what’s often denoted by $\omega$, sending an integer $n$ to the number of distinct prime factors of $n$. The strict potential function is the same.

• An easy example that works for any $V$: if $F: A \to B$ has a right adjoint then since $A(F-, b)$ is representable, its magnitude is equal to the cardinality of its colimit, which is $1$. So $h_F: \ob(B) \to k$ has constant value $1$, as does the strict potential function.

  More generally, if $A(F-, b)$ is a coproduct of $n$ representables then $h_F(b) = n$, and the same is true for the strict potential function.

• For the inclusion $F: Field^{op} \hookrightarrow Ring^{op}$, we saw that

  $$N_F(R) = \sum_{p \in Spec(R)} Field(Frac(R/p), -)$$

  for every ring $R$. This is a coproduct of representables, so by the previous example, $|N_F(R)| = |Spec(R)|$. That is, the potential function is given by

  $$h_F: R \mapsto |Spec(R)|$$

  (as is the strict potential function). Usually $Spec(R)$ is infinite, so this calculation is dodgy, but if we restrict ourselves to finite fields and rings then everything is above board.

• For an opfibration $F: A \to B$, one can show that the potential function $h_F$ is given by

  $$h_F(b) = |F^{-1}(b)|.$$

  I should explain the notation on the right-hand side. The category $F^{-1}(b)$ is the fibre over $b \in B$, consisting of the objects of $A$ that $F$ sends to $b$ and the maps in $A$ that $F$ sends to $1_b$. We can take its magnitude (at least if it’s finite and lightning doesn’t strike), which is what I’ve denoted by $|F^{-1}(b)|$.

  I won’t include the proof of this result, but I want to emphasize that it doesn’t involve finding a coweighting on $A$. You might think it would, because the definition of $h_F(b)$ involves the coweighting on $A$. But it turns out that it’s enough to just assume it exists.

• Now take $V = \mathbb{R}^+$, so that $V$-categories are generalized metric spaces. The potential function of a map $f: A \to B$ of metric spaces is the function $h_f : B \to \mathbb{R}$ given by

  $$h_f(b) = \sum_{a \in A} w^A(a) e^{-d(f(a), b)}.$$

  In particular, if $f$ is the inclusion of a subspace $A \subseteq B$, then it’s natural to write $h_A$ instead of $h_f$, and we have

  $$h_A(b) = \sum_{a \in A} w^A(a) e^{-d(a, b)}.$$

  This is equal to $1$ on $A$ but can take other values on $B \setminus A$. In suitable infinite contexts, sums become integrals and $w^A$ becomes something like a measure or distribution, in which case the formula becomes

  $$h_A(b) = \int_A e^{-d(a, b)} \, d w^A(a).$$

  This is exactly the formula for the potential function in Part 1, with one difference: there, I used weightings on $A$, and here, I’m using coweightings. It’s coweightings that one should use. In the previous post, I assumed that all metrics were symmetric, which means that weightings and coweightings are the same. So there’s no inconsistency.

  (Of course, one could take duals throughout and use weightings on $A$ instead. But we’ll see that whichever choice you make, you end up having to consider weightings on one of $A$ and $B$ and coweightings on the other.)

  What about the strict potential function, sending a point $b \in B$ to the “cardinality of the colimit of $B(f-, b): A \to \mathbb{R}^+$”? Well, I put that phrase in inverted commas because we’re now in an enriched context, so it needs a bit of interpretation. “Cardinality” is okay: it becomes the size function $|\cdot| = e^{-(\cdot)}: \mathbb{R}^+ \to \mathbb{R}$. “Colimit” wouldn’t usually make sense in an enriched world, but we’re saved by the fact that the monoidal unit of $V = \mathbb{R}^+$ (namely, $0$) is terminal. The colimit of a $V$-functor into $V$ is just its infimum. So the strict potential function of a map $f: A \to B$ of metric spaces is just

  $$b \mapsto e^{-\inf_{a \in A} d(f(a), b)} = e^{-d(f A, B)}.$$

  I showed you an example of the difference between the potential function and the strict potential function in Part 1, although not with those words. If we take $f$ to be the inclusion $\{-1, 1\} \hookrightarrow \mathbb{R}$ then the potential function is

  

  whereas the strict potential function is

  

  These two functions are identical on $(-\infty, -1] \cup [1, \infty)$, but different on the interval $(-1, 1)$. If you’ve ever wondered what the difference is between strict and lax colimits, here’s an example!

By definition, the potential function of an enriched functor $F: A \to B$ is given by

$$h_F(b) = \sum_{a \in A} w^A(a) |B(F(a), b)|,$$

but a slightly different viewpoint is sometimes helpful. We can take the function $w^A: ob(A) \to k$ and push it forward forward along $F$ to obtain a new function

$$\begin{array}{cccc} F_* w^A: &ob(B) &\to &k, \\ &b &\mapsto &\sum_{a \in F^{-1}(b)} w^A(a). \end{array}$$

Really it’s best to think of $w^A$ as not a function but a measure taking values in $k$ (although these are the same thing on a finite set). Then this is just the usual pushforward measure construction. In any case, the formula for the potential function now becomes

$$h_F(b) = \sum_{b' \in B} (F_* w^A)(b') |B(b', b)|,$$

which has the advantage that everything is taking place in $B$ rather than $A$. In the case where $f: A \to B$ is an embedding of a subspace of a metric space, $f_* w^A$ is just the extension of the measure $w^A$ on $A$ to all of $B$, which we’d usually just write as $w^A$ (with a wink to the audience). In integral notation, the last formula becomes

$$h_F(b) = \int_B e^{-d(x, b)} \, d w^A(x),$$

which is what we had in the last post.

I’ve now explained what potential functions are. But what are they good for?

Last time, I explained that they’re very good indeed for helping us to calculate the magnitude of metric spaces. The key was that

$$|A| = \int_B h_A \, d w_B$$

for a subspace $A$ of a metric space $B$. And as I recounted, that key unlocks the door to the world of PDE methods.

So you might hope something similar is true for enriched categories in general: that there’s a formula for the magnitude in terms of the potential function. And there is! For any $V$-functor $F: A \to B$, it’s a theorem that

$$|A| = \sum_{b \in B} h_F(b) w_B(b).$$

This is an easy calculation from the definitions.

(If you’re really paying attention, you’ll notice that we used the coweighting on $A$ to define the potential function, and now we’re using the weighting on $B$. That’s just how it turns out. One is a weighting, and the other is a coweighting.)

Examples

• For a $V$-functor $F: A \to B$ that has a right adjoint, we saw that the potential function $h_F$ has constant value $1$, so this formula tells us that

  $$|A| = \sum_{b \in B} w_B(b).$$

  But the right-hand side is by definition the magnitude of $B$, so what this formula is saying is that

  $$|A| = |B|.$$

  In other words, if there’s an adjunction between two categories, their magnitudes are equal!

  This has been known forever (Proposition 2.4), and is also intuitive from a homotopical viewpoint. But it’s nice that it just pops out.

• Less trivially, we saw above that for an opfibration $F: A \to B$, the potential function is $h_F: b \mapsto |F^{-1}(b)|$. So

  $$|A| = \sum_{b \in B} w_B(b) |F^{-1}(b)|.$$

  In other words, the magnitude of the “total” category $A$ is the weighted sum over the base $B$ of the magnitudes of the fibres. (Special case: the magnitude of a product is the product of the magnitudes.)

  Now this has been known forever too (Proposition 2.8), but I want to emphasize that the proof is fundamentally different from the one I just linked. That proof constructs the weighting on $A$ from the weightings on the base $B$ and the fibres. Now that’s an easy and informative proof, but what we’ve just done is different, because it didn’t involve figuring out the weighting or coweighting on $A$. So although the result isn’t new or difficult, it’s perhaps grounds for optimism that the method of potential functions will let us prove new things about enriched categories other than metric spaces.

What new things? I don’t know! This is where I’ve got to now. Maybe there are applications in the metric world in which $f: A \to B$ is nothing like an inclusion. Maybe there are applications to graphs, replacing the PDE methods used for subspaces of $\mathbb{R}^n$ by discrete analogues. Maybe the potential function method can be used to shed light on the tricky result that the magnitude of graphs is invariant under certain Whitney twists (Theorem 5.2), and more generally under the sycamore twists introduced by Emily Roff (Theorem 6.5). Let’s find out!

Next: Part 2

In the beginning, there were hardly any spaces whose magnitude we knew. Line segments were about the best we could do. Then Mark Meckes introduced the technique of potential functions for calculating magnitude, which was shown to be very powerful. For instance, Juan Antonio Barceló and Tony Carbery used it to compute the magnitude of odd-dimensional Euclidean balls, which turn out to be rational functions of the radius. Using potential functions allows you to tap into the vast repository of knowledge of PDEs.

In this post and the next, I’ll explain this technique from a categorical viewpoint, saying almost nothing about the analytic details. This is category theory as an organizational tool, used to help us understand how the various ideas fit together. Specifically, I’ll explain potential functions in terms of the magnitude of functors, which I wrote about here a few weeks ago.

Before I can describe this categorical viewpoint on potential functions, I have to explain what potential functions are in the magnitude context, and why they’re very useful. That’s what I’ll do today.

This part of the story is about metric spaces. For now I’ll assume they satisfy all the classical axioms, including symmetry of the metric, meaning that $d(a, a') = d(a', a)$ for all points $a$ and $a'$. When metric spaces are viewed as enriched categories, symmetry isn’t automatic — but we’ll come to that next time.

A weighting on a finite metric space $A$ is a function $w: A \to \mathbb{R}$ such that for all $a \in A$,

$$\sum_{a' \in A} e^{-d(a, a')} w(a') = 1.$$

Everyone who sees this formula for the first time asks where the exponential comes from. Ultimately it’s because of the enriched category viewpoint (which again we’ll come to next time), but the short story is that exponential is essentially the only reasonable function that converts addition into multiplication.

For simplicity, I’ll assume here that every finite metric space $A$ has a unique weighting, which I’ll call $w_A$. Since the definition of weighting involves the same number of equations as unknowns, this is generically true (and it’s always true for subspaces of $\mathbb{R}^n$), even though there are exceptions.

The magnitude of $A$ is

$$|A| = \sum_{a \in A} w_A(a) \in \mathbb{R}.$$

That’s for finite metric spaces. To extend the definition to compact metric spaces, there are various ideas you might try. You could define the magnitude of a compact space as the supremum of all the magnitudes of its finite subspaces. Or, you could take an ever-denser sequence of finite subsets of your compact space, then define its magnitude to be the limit of the magnitudes of the approximating subsets. Or, you could try replacing the sums in the formulas above by integrals, somehow.

Mark Meckes showed that all these approaches are equivalent. They all give the same definition of the magnitude of a compact space. (At least, this is true subject to a condition called “positive definiteness” which I won’t discuss and which always holds for subspaces of $\mathbb{R}^n$.)

How do we actually calculate magnitude, say for compact subspaces $A$ of $\mathbb{R}^n$?

In principle, you already know how do it. You run through all the finite subsets of $A$, calculate the magnitude of each using the definition above, and take the sup. The trouble is, this procedure is incredibly hard to work with. It’s completely impractical.

A slightly more practical approach is to look for a “weight measure”, that is, a signed measure $w$ on $A$ such that for all $a \in A$,

$$\int_A e^{-d(a, a')}\, d w(a') = 1.$$

As Mark showed, if $w$ is a weight measure then the magnitude of $A$ is given by $|A| = w(A)$. This is the analogue of the formula $|A| = \sum_a w(a)$ for finite spaces.

Example   Take an interval $[c, d]$ in $\mathbb{R}$. It’s an elementary exercise to check that

$$\tfrac{1}{2}(\delta_c + \lambda_{[c, d]} + \delta_d)$$

is a weight measure on $[c, d]$, where $\delta_c$ and $\delta_d$ denote the Dirac deltas at $c$ and $d$, and $\lambda_{[c, d]}$ is Lebesgue measure on $[c, d]$. It follows that $[c, d]$ is the total mass of this measure, which is

$$1 + \tfrac{1}{2}(d - c).$$

In other words, it’s $1$ plus half the length of the interval.

The trouble with weight measures is that very few spaces have them. Even Euclidean balls don’t, beyond dimension one. It turns out that we need something more general than a measure, something more like a distribution.

In another paper, Mark worked out exactly what kind of measure-like or distribution-like thing a weighting should be. The answer is very nice, but I won’t explain it here, because I want to highlight the formal aspects above all.

So I’ll simply write the weighting on a metric space $A$ as $w_A$, without worrying too much about what kind of thing $w_A$ actually is. All that matters is that it behaves something like a measure: we can pair it with “nice enough” functions $\phi: A \to \mathbb{R}$ to get a real number, which I’ll write as

$$\int_A \phi(a) \, d w_A(a)$$

or simply

$$\int_A \phi \, d w_A.$$

And I’ll assume that all the spaces $A$ that we discuss do have a unique weighting $w_A$.

That’s the background. But back to the question: how do we actually calculate magnitude?

First main idea   Don’t look at metric spaces $A$ in isolation. Instead, consider spaces embedded in some big space $B$ that we understand well.

Think of the big space $B$ as fixed. The typical choice is $\mathbb{R}^n$. One sense in which we “understand” $\mathbb{R}^n$ well is that we have a weight measure on it: it’s just Lebesgue measure divided by a constant factor $c_n$. This follows easily from the fact that $\mathbb{R}^n$ is homogeneous. (It doesn’t matter for anything I’m going to say, but the constant $c_n$ is $\int_{\mathbb{R}^n} e^{-\|x\|}\, d x$, for which there’s a standard formula involving $\pi$s and factorials.)

The potential function of a subspace $A \subseteq B$ is the function $h_A: B \to \mathbb{R}$ defined by

$$h_A(b) = \int_A e^{-d(a, b)} \, d w_A(a).$$

By definition of weighting, $h_A$ has constant value $1$ on $A$. But outside $A$, it could be anything, and it turns out that its behaviour outside $A$ is very informative.

Although $w_A$ is a measure-like thing on the subspace $A$ of $B$, we typically extend it by $0$ to all of $B$, and then the definition of the potential function becomes

$$h_A(b) = \int_B e^{-d(x, b)} \, d w_A(x)$$

Examples   In all these examples, I’ll take the big space $B$ to be $\mathbb{R}$.

• Let $A = \{0\}$. Then the weighting $w_A$ on $A$ is the Dirac delta at $0$, and the potential function $h_A: \mathbb{R} \to \mathbb{R}$ is given by

  $$h(b) = e^{-|b|}$$

  ($b \in \mathbb{R}$), whose graph looks like this:

  

• Let $A = [-2, 2]$. As we’ve already seen,

  $$w_A = \tfrac{1}{2}(\delta_{-2} + \lambda_{[-2, 2]} + \delta_2),$$

  and a little elementary work then reveals that the potential function $h_A: \mathbb{R} \to \mathbb{R}$ is given by

  $$h_A(b) = \begin{cases} e^{-(-2 - b)} &\text{if}   b \leq -2, \\ 1 &\text{if}   -2 \leq b \leq 2, \\ e^{-(b - 2)} &\text{if}   b \geq 2, \end{cases}$$

  which looks like this:

  

• In the two examples we just did, the potential function of $A$ is just the negative exponential of the distance between $A$ and the argument $b$. That’s not exactly coincidence: as we’ll see in the next part, the function just described corresponds to a strict colimit, whereas the potential function corresponds to a lax colimit. So it’s not surprising that they coincide in simple cases.

  But this only happens in the very simplest cases. It doesn’t happen for Euclidean balls above dimension $1$, or even for two-point spaces. For example, taking the subset $A = \{-1, 1\}$ of $B = \mathbb{R}$, we have

  $$h_A(x) = \frac{e^{-|x + 1|} + e^{-|x - 1|}}{1 + e^{-2}},$$

  which looks like this:

  

  whereas $b \mapsto e^{-d(A, b)}$ looks like this:

  

  Similar, but different!

And now we come to the:

Second main idea   Work with potential functions instead of weightings.

To explain what this means, I need to tell you three good things about potential functions.

• The potential function determines the magnitude:

  $$|A| = \int_B h_A \, d w_B.$$

  At the formal level, proving this is a one-line calculation: substitute the definition of $h_A$ into the right-hand side and follow your nose.

  For example, we saw that $\mathbb{R}^n$ has weighting $\lambda/c_n$, where $\lambda$ is Lebesgue measure and $c_n$ is a known constant. So

  $$|A| = \frac{1}{c_n} \int_{\mathbb{R}^n} h_A(x) \, d x$$

  for compact $A \subseteq \mathbb{R}^n$. Here $d x$ refers to ordinary Lebesgue integration.

  (For $\mathbb{R}^n$, and in fact a bit more generally, this result appears as Theorem 4.16 here.)

• You can recover the weighting from the potential function. So, you don’t lose any information by working with one rather than the other.

  How do you recover it? Maybe it’s easiest to explain in the case when the spaces are finite. If we write $Z_B$ for the $B \times B$ matrix $(e^{-d(b', b)})_{b', b \in B}$ then the definition of the potential function can be expressed very succinctly:

  $$h_A = Z_B w_A.$$

  Here $h_A$ and $w_A$ are viewed as column vectors with entries indexed by the points of $B$. (For $w_A$, those entries are $0$ for points not in $A$.) Assuming $Z_B$ is invertible, this means we recover $w_A$ from $h_A$ as $Z_B^{-1} h_A$. And something similar is true in the non-finite setting.

  However, what really makes the technique of potential functions sing is that when $B = \mathbb{R}^n$, there’s a much more explicit way to recover the weighting from the potential function:

  $$w_A = (I - \Delta)^{(n + 1)/2} h_A$$

  (up to a constant factor that I’ll ignore). Here $I$ is the identity and $\Delta$ is the Laplace operator, $\sum_{i = 1}^n \frac{\partial^2}{\partial x_i^2}$. This is Proposition 5.9 of Mark’s paper.

  How much more of this bullet point you’ll want to read depends on how interested you are in the analysis. The fundamental point is simply that $(I - \Delta)^{(n + 1)/2}$ is some kind of differential operator. But for those who want a bit more:

  • To make sense of everything, you need to interpret it all in a distributional sense. In particular, this allows one to make sense of the power $(n + 1)/2$, which is not an integer if $n$ is even.

  • Maybe you wondered why someone might have proved a result on the magnitude of odd-dimensional Euclidean balls only, as I mentioned at the start of the post. What could cause the odd- and even-dimensional cases to become separated? It’s because whether or not $(n + 1)/2$ is an integer depends on whether $n$ is odd or even. When $n$ is odd, it’s an integer, which makes $(I - \Delta)^{(n + 1)/2}$ a differential rather than pseudodifferential operator. Heiko Gimperlein, Magnus Goffeng and Nikoletta Louca later worked out lots about the even-dimensional case, but I won’t talk about that here.

  • Finally, where does the operator $(I - \Delta)^{(n + 1)/2}$ come from? Sadly, I don’t have an intuitive explanation. Ultimately it comes down to the fact that the Fourier transform of $e^{-\|\cdot\|}$ is $\xi \mapsto (1 + \|\xi\|)^{-(n + 1)/2}$ (up to a constant). But that itself is a calculation that’s really quite tricky (for me), and it’s hard to see anything beyond “it is what it is”.

• The third good thing about potential functions is that they satisfy a differential equation, in the situation where our big space $B$ is $\mathbb{R}^n$. Specifically:

  $$\begin{cases} h_A \equiv 1 &\text{on}   A \\ (I - \Delta)^{(n + 1)/2} h_A \equiv 0&\text{on}   \mathbb{R}^n \setminus A. \end{cases}$$

  Indeed, the definition of weighting implies that $h_A \equiv 1$ on $A$, and the “second good thing” together with the fact that $w_A$ is supported on $A$ give the second clause.

  Not only do we have a differential equation for $h_A$, we also have boundary conditions. There are boundary conditions at the boundary of $A$, because of something I’ve been entirely vague about: the functions we’re dealing with are meant to be suitably smooth. There are also boundary conditions at $\infty$, because our functions are also meant to decay suitably fast.

  Maybe Mark will read this and correct me if I’m wrong, but I believe there are exactly the right number of boundary conditions to guarantee that there’s (typically? always?) a unique solution. In any case, the following example — also taken from Mark’s paper — illustrates the situation.

Example   Let’s calculate the magnitude of a real interval $A = [c, d]$ using the potential function method.

Its potential function $h_A$ is a function $\mathbb{R} \to \mathbb{R}$ such that $h_A \equiv 1$ on $[c, d]$ and $h_A - h''_A = 0$ on the rest of the real line. That differential equation comes from taking $(I - \Delta)^{-(n + 1)/2}$ in the case $n = 1$.

The functions $f$ satisfying $f'' = f$ are those of the form $k e^{\pm x}$ for some constant $k$, and in our case we’re free to choose the constant and the $\pm$ sign differently on the two connected components of $\mathbb{R} \setminus A$. So there are lots of solutions. But $h_A$ is required to be continuous and to converge to $0$ at $\pm \infty$, and that pins it down uniquely: the one and only solution is

$$h_A(x) = \begin{cases} e^{-(c - x)} &\text{if}   x \leq c, \\ 1 &\text{if}   c \leq x \leq d, \\ e^{-(x - d)} &\text{if}   d \leq x. \end{cases}$$

I showed you the graph of this potential function above, in the case where $A = [-2, 2]$.

So the magnitude of $A = [c, d]$ is

$$\frac{1}{c_1} \int_\mathbb{R} h_A(x)\, d x,$$

where $c_1$ is the constant $\int_\mathbb{R} e^{-|x|} \, d x = 2$. This gives the answer:

$$|A| = 1 + \tfrac{1}{2}(d - c) = 1 + \tfrac{1}{2}length(A).$$

The crucial point is that in this example, we didn’t have to come up with a weighting on $A$. The procedure was quite mechanical. And that’s the attraction of the method of potential functions.

Next time, I’ll put all this into a categorical context using the notion of the magnitude of a functor, which I introduced here.

Buckminsterfullerene is a molecule shaped like a soccer ball, made of 60 carbon atoms. If one of the bonds between two hexagons rotates, we get a weird mutant version of this molecule:

This is an example of a Stone-Wales transformation: a 90° rotation in a so-called ‘π bond’ between carbon atoms. Here’s how it works in graphene:

Graphene is a sheet of carbon molecules arranged in hexagons. When they undergo a Stone–Wales transformation, we get a Stone–Wales defect with two pentagons and two heptagons, like this:

This picture is from Jacopo Bertolotti, based on a picture by Torbjörn Björkman which appeared in this paper:

• Torbjörn Björkman, Simon Kurasch, Ossi Lehtinen, Jani Kotakoski, Oleg V. Yazyev, Anchal Srivastava, Viera Skakalova, Jurgen H. Smet, Ute Kaiser and Arkady V. Krasheninnikov, Defects in bilayer silica and graphene: common trends in diverse hexagonal two-dimensional systems, Scientific Reports 3 (2013), 3482.

This paper also shows other interesting defects, and electron microscope pictures of how they actually look in graphene and hexagonal bilayer silica.

You’ll notice that in buckminsterfullerne the Stone–Wales transformation turned 2 hexagons and 2 pentagons into two pentagons and 2 hexagons, while in graphene it turned 4 hexagons into 2 pentagons and 2 heptagons. The general pattern is this:

I think it’s cool how a simple topological transformation akin to the Pachner moves shows up in chemistry!

I got the above image from here:

• Wei-Wei Wang, Jing-Shuang Dang, Jia Zheng, Xiang Zhao, Shigeru Nagase, Selective growth of fullerenes from C₆₀ to C₇₀: inherent geometrical connectivity hidden in discrete experimental evidence, The Journal of Physical Chemistry C 117 (2013), 2349–2357.

I got the image of a Stone–Wales transformation in buckminsterfullerene from here:

• L. A. Openov and Mikhail Maslov, On the vineyard formula for the pre-exponential factor in the Arrhenius law, Physics of the Solid State 56 6 (2014), 1239–1244.

The Arrhenius equation is a simple rule of thumb for the rates of chemical reactions. A lot of statistical mechanics gives simple laws for equilibrium behavior. For example, a state of energy will show up with probability proportional to in equilibrium at temperature where is Boltzmann’s constant. But dynamics is much harder, so the Arrhenius equation for the rates of transitions between states is precious, even though only approximate. I would like to understand this law better, and its range of approximate validity. The above paper digs into that. As an example, it studies the rate at which Stone–Wales transformations happen in buckminsterfullerene!

There should be a nice theory of topological transformations like Stone–Wales transformations or Pachner moves occurring randomly, following the laws of statistical mechanics. I guess the study of matrix models moves in this direction, but there’s no Arrhenius equation there, I don’t think!

Guest post by John Bourke.

The next International Category Theory Conference CT2025 will take place at Masaryk University (Brno, Czech Republic) from Sunday, July 13 and will end on Saturday, July 19, 2025.

Brno is a beautiful city surrounded by nature with a long tradition in category theory. If you are interested in attending, please read on!

Important dates

• April 2: talk submission
• April 18: early registration deadline
• May 7: notification of speakers
• May 23: registration deadline
• July 13-19: conference

In addition to 25 minute contributed talks, there will be speed talks replacing poster sessions, and we hope to accommodate as many talks as possible.

The invited speakers are:

• Clark Barwick (University of Edinburgh)
• Maria Manuel Clementino (University of Coimbra)
• Simon Henry (University of Ottawa)
• Jean-Simon Lemay (Macquarie University)
• Wendy Lowen (University of Antwerp)
• Maru Sarazola (University of Minnesota)

A couple of extra notes:

• We recommend that people book accommodation early as possible as there will be other events taking place in town that week, such as the Moto GP. We have booked a limited number of rooms in advance.
• In order to promote environmentally friendly travel within Europe, there will be a small prize for the person who travels the furthest from their starting point to the conference by train or bus. Obviously this will not be practical for most attendees, but if you are the sort of person who fancies an adventurous train trip, this could be your chance!

If you would like more details on any aspect of the conference, have a look at the website or send an email to the organizers at ct2025@math.muni.cz

We look forward to seeing as many of you in Brno as possible!

Recently, the South Pole Telescope’s SPT-3G collaboration released new measurements of the cosmic microwave background, the leftover light from the formation of the first atoms. By measuring this light, cosmologists can infer the early universe’s “shape”: how it rippled on different scales as it expanded into the universe we know today. They compare this shape to mathematical models, equations and simulations which tie together everything we know about gravity and matter, and try to see what it implies for those models’ biggest unknowns.

Some of the most interesting such unknowns are neutrino masses. We know that neutrinos have mass because they transform as they move, from one type of neutrino to another. Those transformations let physicists measure the differences between neutrino masses, but but themselves, they don’t say what the actual masses are. All we know from particle physics, at this point, is a minimum: in order for the neutrinos to differ in mass enough to transform in the way they do, the total mass of the three flavors of neutrino must be at least 0.06 electron-Volts.

(Divided by the speed of light squared to get the right units, if you’re picky about that sort of thing. Physicists aren’t.)

Neutrinos also influenced the early universe, shaping it in a noticeably different way than heavier particles that bind together into atoms, like electrons and protons, did. That effect, observed in the cosmic microwave background and in the distribution of galaxies in the universe today, lets cosmologists calculate a maximum: if neutrinos are more massive than a certain threshold, they could not have the effects cosmologists observe.

Over time as measurements improved, this maximum has decreased. Now, the South Pole Telescope has added more data to the pool, and combining it with prior measurements…well, I’ll quote their paper:

Ok, it’s probably pretty hard to understand what that means if you’re not a physicist. To explain:

1. There are two different hypotheses for how neutrino masses work, called “hierarchies”. In the “normal” hierarchy, the neutrinos go in the same order as the particles they interact with with the weak nuclear force: electron-neutrinos are lighter than muon neutrinos, which are lighter than tau neutrinos. In the “inverted” hierarchy, they come in the opposite order, and the electron neutrino is the heaviest. Both of these are consistent with the particle-physics data.
2. Confidence is a statistics thing, which could take a lot of unpacking to define correctly. To give a short but likely tortured-sounding explanation: when you rule out a hypothesis with a certain confidence level, you’re saying that, if that hypothesis was true, there would only be a 100%-minus-that-chance chance that you would see what you actually observed.

So, what are the folks at the South Pole Telescope saying? They’re saying that if you put all the evidence together (that’s roughly what that pile of acroynms at the beginning means), then the result would be incredibly uncharacteristic for either hypothesis for neutrino masses. If you had “normal” neutrino masses, you’d only see these cosmological observations 2.1% of the time. And if you had inverted neutrino masses instead, you’d only see these observations 0.01% of the time!

That sure makes it sound like neither hypothesis is correct, right? Does it actually mean that?

I mean, it could! But I don’t think so. Here I’ll start speculating on the possibilities, from least likely in my opinion to most likely. This is mostly my bias talking, and shouldn’t be taken too seriously.

5. Neutrinos are actually massless

This one is really unlikely. The evidence from particle physics isn’t just quantitative, but qualitative. I don’t know if it’s possible to write down a model that reproduces the results of neutrino oscillation experiments without massive neutrinos, and if it is it would be a very bizarre model that would almost certainly break something else. This is essentially a non-starter.

4. This is a sign of interesting new physics

I mean, it would be nice, right? I’m sure there are many proposals at this point, tweaks that add a few extra fields with some hard-to-notice effects to explain the inconsistency. I can’t rule this out, and unlike the last point there isn’t anything about it that seems impossible. But we’ve had a lot of odd observations, and so far this hasn’t happened.

3. Someone did statistics wrong

This happens more often. Any argument like this is a statistical argument, and while physicists keep getting better at statistics, they’re not professional statisticians. Sometimes there’s a genuine misunderstanding that goes in to testing a model, and once it gets resolved the problem goes away.

2. The issue will go away with more data

The problem could also just…go away. 97.9% confidence sounds huge…but in physics, the standards are higher: you need 99.99994% to announce a new discovery. Physicists do a lot of experiments and observations, and sometimes, they see weird things! As the measurements get more precise, we may well see the disagreement melt away, and cosmology and particle physics both point to the same range for neutrino masses. It’s happened to many other measurements before.

1. We’re reaching the limits of our current approach to cosmology

This is probably not actually the most likely possibility, but it’s my list, what are you going to do?

There are basic assumptions behind how most theoretical physicists do cosmology. These assumptions are reasonably plausible, and seem to be needed to do anything at all. But they can be relaxed. Our universe looks like it’s homogeneous on the largest scales: the same density on average, in every direction you look. But the way that gets enforced in the mathematical models is very direct, and it may be that a different, more indirect, approach has more flexibility. I’ll probably be writing about this more in future, hopefully somewhere journalistic. But there are some very cool ideas floating around, gradually getting fleshed out more and more. It may be that the answer to many of the mysteries of cosmology right now is not new physics, but new mathematics: a new approach to modeling the universe.

There are a ton of two-dimensional materials these days. You’ve probably heard of graphene, a single layer of carbon atoms arranged in a hexagonal grid.

In 2018, everything changed when two layers of graphene were twisted to reveal superconductivity! The twist itself is interesting (I briefly discussed it in a previous post), but the key takeaway is that these materials now come with an extra knob for accessing new phases of matter. It’s remarkable. We can first think of these materials like Lego blocks:

Each layer is a different material: mix and match, and you might discover an exotic new phase. This “Lego” idea had already been in the air before 2018, but the physics since then has shown that it’s not just about stacking—we can twist too, creating not just patterns, but new ways for electrons to move.

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We knew these patterns would occur, but we didn’t realize we could make it superconduct. Now we can stack and twist to great effect. Of course, twisted bilayer graphene isn’t about to revolutionize high-speed trains (it goes superconducting at only 4K1), but the way it goes superconducting is eerily reminiscent of higher-temperature superconductors. That means it might help us understand those other materials better.

And once people started twisting, they didn’t stop. We now have twisted multilayers of graphene, transition-metal dichalcogenide (TMD) bilayers2, and more. But it doesn’t end there; you can also apply magnetic fields, electric fields, and pattern the lattice in sophisticated ways. With all that in mind, here’s a short and incomplete survey of some of the exotic phases in these materials: