Tonight is “Culture Night” in Copenhagen, the night when the city throws open its doors and lets the public in. Museums and hospitals, government buildings and even the Freemasons, all have public events. The Niels Bohr Institute does too, of course: an evening of physics exhibits and demos, capped off with a public lecture by Denmark’s favorite bow-tie wearing weirder-than-usual string theorist, Holger Bech Nielsen. In between, there are a number of short talks by various folks at the institute, including yours truly.
In my talk, I’m going to try and motivate the audience to care about math. Math is dry of course, and difficult for some, but we physicists need it to do our jobs. If you want to be precise about a claim in physics, you need math simply to say what you want clearly enough.
Since you guys likely don’t overlap with my audience tonight, it should be safe to give a little preview. I’ll be using a few examples, but this one is the most complicated:
I’ll be telling a story I stole from chapter seven of the web serial Almost Nowhere. (That link is to the first chapter, by the way, in case you want to read the series without spoilers. It’s very strange, very unique, and at least in my view quite worth reading.) You follow a warrior carrying a spear around a globe in two different paths. The warrior tries to always point in the same direction, but finds that the two different paths result in different spears when they meet. The story illustrates that such a simple concept as “what direction you are pointing” isn’t actually so simple: if you want to think about directions in curved space (like the surface of the Earth, but also, like curved space-time in general relativity) then you need more sophisticated mathematics (a notion called parallel transport) to make sense of it.
It’s kind of an advanced concept for a public talk. But seeing it show up in Almost Nowhere inspired me to try to get it across. I’ll let you know how it goes!
By the way, if you are interested in learning the kinds of mathematics you need for theoretical physics, and you happen to be a Bachelor’s student planning to pursue a PhD, then consider the Perimeter Scholars International Master’s Program! It’s a one-year intensive at the Perimeter Institute in Waterloo, Ontario, in Canada. In a year it gives you a crash course in theoretical physics, giving you tools that will set you ahead of other beginning PhD students. I’ve witnessed it in action, and it’s really remarkable how much the students learn in a year, and what they go on to do with it. Their early registration deadline is on November 15, just a month away, so if you’re interested you may want to start thinking about it.
What’s the difference between a black hole and a neutron star?
When a massive star nears the end of its life, it starts running out of nuclear fuel. Without the support of a continuous explosion, the star begins to collapse, crushed under its own weight.
What happens then depends on how much weight that is. The most massive stars collapse completely, into the densest form anything can take: a black hole. Einstein’s equations say a black hole is a single point, infinitely dense: get close enough and nothing, not even light, can escape. A quantum theory of gravity would change this, but not a lot: a quantum black hole would still be as dense as quantum matter can get, still equipped with a similar “point of no return”.
A slightly less massive star collapses, not to a black hole, but to a neutron star. Matter in a neutron star doesn’t collapse to a single point, but it does change dramatically. Each electron in the old star is crushed together with a proton until it becomes a neutron, a forced reversal of the more familiar process of Beta decay. Instead of a ball of hydrogen and helium, the star then ends up like a single atomic nucleus, one roughly the size of a city.
Now, let me ask a slightly different question: how do you tell the difference between a black hole and a neutron star?
Sometimes, you can tell this through ordinary astronomy. Neutron stars do emit light, unlike black holes, though for most neutron stars this is hard to detect. In the past, astronomers would use other objects instead, looking at light from matter falling in, orbiting, or passing by a black hole or neutron star to estimate its mass and size.
Now they have another tool: gravitational wave telescopes. Maybe you’ve heard of LIGO, or its European cousin Virgo: massive machines that do astronomy not with light but by detecting ripples in space and time. In the future, these will be joined by an even bigger setup in space, called LISA. When two black holes or neutron stars collide they “ring” the fabric of space and time like a bell, sending out waves in every direction. By analyzing the frequency of these waves, scientists can learn something about what made them: in particular, whether the waves were made by black holes or neutron stars.
One big difference between black holes and neutron stars lies in something called their “Love numbers“. From far enough away, you can pretend both black holes and neutron stars are single points, like fundamental particles. Try to get more precise, and this picture starts to fail, but if you’re smart you can include small corrections and keep things working. Some of those corrections, called Love numbers, measure how much one object gets squeezed and stretched by the other’s gravitational field. They’re called Love numbers not because they measure how hug-able a neutron star is, but after the mathematician who first proposed them, A. E. H. Love.
What can we learn from Love numbers? Quite a lot. More impressively, there are several different types of questions Love numbers can answer. There are questions about our theories, questions about the natural world, and questions about fundamental physics.
You might have heard that black holes “have no hair”. A black hole in space can be described by just two numbers: its mass, and how much it spins. A star is much more complicated, with sunspots and solar flares and layers of different gases in different amounts. For a black hole, all of that is compressed down to nothing, reduced to just those two numbers and nothing else.
With that in mind, you might think a black hole should have zero Love numbers: it should be impossible to squeeze it or stretch it. This is fundamentally a question about a theory, Einstein’s theory of relativity. If we took that theory for granted, and didn’t add anything to it, what would the consequences be? Would black holes have zero Love number, or not?
It turns out black holes do have zero Love number, if they aren’t spinning. If they are, things are more complicated: a few calculations made it look like spinning black holes also had zero Love number, but just last year a more detailedproof showed that this doesn’t hold. Somehow, despite having “no hair”, you can actually “squeeze” a spinning black hole.
(EDIT: Folks on twitter pointed out a wrinkle here: more recentpapers are arguing that spinning black holes actually do have zero Love number as well, and that the earlier papers confused Love numbers with a different effect. All that is to say this is still very much an active area of research!)
The physics behind neutron stars is in principle known, but in practice hard to understand. When they are formed, almost every type of physics gets involved: gas and dust, neutrino blasts, nuclear physics, and general relativity holding it all together.
Because of all this complexity, the structure of neutron stars can’t be calculated from “first principles” alone. Finding it out isn’t a question about our theories, but a question about the natural world. We need to go out and measure how neutron stars actually behave.
Love numbers are a promising way to do that. Love numbers tell you how an object gets squeezed and stretched in a gravitational field. Learning the Love numbers of neutron stars will tell us something about their structure: namely, how squeezable and stretchable they are. Already, LIGO and Virgo have given us some information about this, and ruled out a few possibilities. In future, the LISA telescope will show much more.
Returning to black holes, you might wonder what happens if we don’t stick to Einstein’s theory of relativity. Physicists expect that relativity has to be modified to account for quantum effects, to make a true theory of quantum gravity. We don’t quite know how to do that yet, but there are a few proposals on the table.
Asking for the true theory of quantum gravity isn’t just a question about some specific part of the natural world, it’s a question about the fundamental laws of physics. Can Love numbers help us answer it?
Maybe. Some theorists think that quantum gravity will change the Love numbers of black holes. Fewer, but still some, think they will change enough to be detectable, with future gravitational wave telescopes like LISA. I get the impression this is controversial, both because of the different proposals involved and the approximations used to understand them. Still, it’s fun that Love numbers can answer so many different types of questions, and teach us so many different things about physics.
Unrelated: For those curious about what I look/sound like, I recently gave a talk of outreach advice for the Max Planck Institute for Physics, and they posted it online here.
Last month, our local nest of science historians at the Niels Bohr Archive hosted a Zoom talk by Jed Z. Buchwald, a Newton scholar at Caltech. Buchwald had a story to tell about experimental uncertainty, one where Newton had an important role.
If you’ve ever had a lab course in school, you know experiments never quite go like they’re supposed to. Set a room of twenty students to find Newton’s constant, and you’ll get forty different answers. Whether you’re reading a ruler or clicking a stopwatch, you can never measure anything with perfect accuracy. Each time you measure, you introduce a little random error.
Textbooks worth of statistical know-how has cropped up over the centuries to compensate for this error and get closer to the truth. The simplest trick though, is just to average over multiple experiments. It’s so obvious a choice, taking a thousand little errors and smoothing them out, that you might think people have been averaging in this way through history.
They haven’t though. As far as Buchwald had found, the first person to average experiments in this way was Isaac Newton.
What did people do before Newton?
Well, what might you do, if you didn’t have a concept of random error? You can still see that each time you measure you get a different result. But you would blame yourself: if you were more careful with the ruler, quicker with the stopwatch, you’d get it right. So you practice, you do the experiment many times, just as you would if you were averaging. But instead of averaging, you just take one result, the one you feel you did carefully enough to count.
Before Newton, this was almost always what scientists did. If you were an astronomer mapping the stars, the positions you published would be the last of a long line of measurements, not an average of the rest. Some other tricks existed. Tycho Brahe for example folded numbers together pair by pair, averaging the first two and then averaging that average with the next one, getting a final result weighted to the later measurements. But, according to Buchwald, Newton was the first to just add everything together.
Even Newton didn’t yet know why this worked. It would take later research, theorems of statistics, to establish the full justification. It seems Newton and his later contemporaries had a vague physics analogy in mind, finding a sort of “center of mass” of different experiments. This doesn’t make much sense – but it worked, well enough for physics as we know it to begin.
So this Newtonmas, let’s thank the scientists of the past. Working piece by piece, concept by concept, they gave use the tools to navigate our uncertain times.
A big theme of this conference, as in the past few years, was gravitational waves. From LIGO’s first announcement of a successful detection, amplitudeologists have been developing new methods to make predictions for gravitational waves more efficient. It’s a field I’ve dabbledin a bit myself. Last year’s QCD Meets Gravity left me impressed by how much progress had been made, with amplitudeologists already solidly part of the conversation and able to produce competitive results. This year felt like another milestone, in that the amplitudeologists weren’t just catching up with other gravitational wave researchers on the same kinds of problems. Instead, they found new questions that amplitudes are especially well-suited to answer. These included combining two pieces of these calculations (“potential” and “radiation”) that the older community typically has to calculate separately, using an old quantum field theory trick, finding the gravitational wave directly from amplitudes, and finding a few nice calculations that can be used to “generate” the rest.
A large chunk of the talks focused on different “squaring” tricks (or as we actually call them, double-copies). There were double-copies for cosmology and conformal field theory, for the celestial sphere, and even some version of M theory. There were new perspectives on the double-copy, new building blocks and algebraic structures that lie behind it. There were talks on the so-called classical double-copy for space-times, where there have been some strange discoveries (an extra dimension made an appearance) but also a more rigorous picture of where the whole thing comes from, using twistor space. There were not one, but two talks linking the double-copy to the Navier-Stokes equation describing fluids, from two different groups. (I’m really curious whether these perspectives are actually useful for practical calculations about fluids, or just fun to think about.) Finally, while there wasn’t a talk scheduled on this paper, the authors were roped in by popular demand to talk about their work. They claim to have made progress on a longstanding puzzle, how to show that double-copy works at the level of the Lagrangian, and the community was eager to dig into the details.
From there, a grab-bag of talks covered other advancements. There were talks from string theorists and ambitwistor string theorists, from Effective Field Theorists working on gravity and the Standard Model, from calculations in N=4 super Yang-Mills, QCD, and scalar theories. Simon Caron-Huot delved into how causality constrains the theories we can write down, showing an interesting case where the common assumption that all parameters are close to one is actually justified. Nima Arkani-Hamed began his talk by saying he’d surprise us, which he certainly did (and not by keeping on time). It’s tricky to explain why his talk was exciting. Comparing to his earlier discovery of the Amplituhedron, which worked for a toy model, this is a toy calculation in a toy model. While the Amplituhedron wasn’t based on Feynman diagrams, this can’t even be compared with Feynman diagrams. Instead of expanding in a small coupling constant, this expands in a parameter that by all rights should be equal to one. And instead of positivity conditions, there are negativity conditions. All I can say is that with all of that in mind, it looks like real progress on an important and difficult problem from a totally unanticipated direction. In a speech summing up the conference, Zvi Bern mentioned a few exciting words from Nima’s talk: “nonplanar”, “integrated”, “nonperturbative”. I’d add “differential equations” and “infinite sums of ladder diagrams”. Nima and collaborators are trying to figure out what happens when you sum up all of the Feynman diagrams in a theory. I’ve made progress in the past for diagrams with one “direction”, a ladder that grows as you add more loops, but I didn’t know how to add “another direction” to the ladder. In very rough terms, Nima and collaborators figured out how to add that direction.
I’ve probably left things out here, it was a packed conference! It’s been really fun seeing what the community has cooked up, and I can’t wait to see what happens next.
Two big physics experiments consistently make the news. The Large Hadron Collider, or LHC, and the Laser Interferometer Gravitational-Wave Observatory, or LIGO. One collides protons, the other watches colliding black holes and neutron stars. But while this may make the experiments sound quite similar, their goals couldn’t be more different.
The goal of the LHC, put simply, is to discover the rules that govern reality. Should the LHC find a new fundamental particle, it will tell us something we didn’t know about the laws of physics, a newly discovered fact that holds true everywhere in the universe. So far, it has discovered the Higgs boson, and while that particular rule was expected we didn’t know the details until they were tested. Now physicists hope to find something more, a deviation from the Standard Model that hints at a new law of nature altogether.
LIGO, in contrast, isn’t really for discovering the rules of the universe. Instead, it discovers the consequences of those rules, on a grand scale. Even if we knew the laws of physics completely, we can’t calculate everything from those first principles. We can simulate some things, and approximate others, but we need experiments to tweak those simulations and test those approximations. LIGO fills that role. We can try to estimate how common black holes are, and how large, but LIGO’s results were still a surprise, suggesting medium-sized black holes are more common than researchers expected. In the future, gravitational wave telescopes might discover more of these kinds of consequences, from the shape of neutron stars to the aftermath of cosmic inflation.
There are a few exceptions for both experiments. The LHC can also discover the consequences of the laws of physics, especially when those consequences are very difficult to calculate, finding complicated arrangements of known particles, like pentaquarks and glueballs. And it’s possible, though perhaps not likely, that LIGO could discover something about quantum gravity. Quantum gravity’s effects are expected to be so small that these experiments won’t see them, but some have speculated that an unusually large effect could be detected by a gravitational wave telescope.
As scientists, we want to know everything we can about everything we find. We want to know the basic laws that govern the universe, but we also want to know the consequences of those laws, the story of how our particular universe came to be the way it is today. And luckily, we have experiments for both.
In physics, what you don’t know can absolutely hurt you. If you ignore that planets have their own gravity, or that metals conduct electricity, you’re going to calculate a lot of nonsense. At the same time, as physicists we can’t possibly know everything. Our experiments are never perfect, our math never includes all the details, and even our famous Standard Model is almost certainly not the whole story. Luckily, we have another option: instead of ignoring what we don’t know, we can parametrize it, and estimate its effect.
Estimating the unknown is something we physicists have done since Newton. You might think Newton’s big discovery was the inverse-square law for gravity, but others at the time, like Robert Hooke, had also been thinking along those lines. Newton’s big discovery was that gravity was universal: that you need to know the effect of gravity, not just from the sun, but from all the other planets as well. The trouble was, Newton didn’t know how to calculate the motion of all of the planets at once (in hindsight, we know he couldn’t have). Instead, he estimated, using what he knew to guess how big the effect of what he didn’t would be. It was the accuracy of those guesses, not just the inverse square law by itself, that convinced the world that Newton was right.
If you’ve studied electricity and magnetism, you get to the point where you can do simple calculations with a few charges in your sleep. The world doesn’t have just a few charges, though: it has many charges, protons and electrons in every atom of every object. If you had to keep all of them in your calculations you’d never pass freshman physics, but luckily you can once again parametrize what you don’t know. Often you can hide those charges away, summarizing their effects with just a fewnumbers. Other times, you can treat materials as boundaries, and summarize everything beyond in terms of what happens on the edge. The equations of the theory let you do this, but this isn’t true for every theory: for the Navier-Stokes equation, which we use to describe fluids, it still isn’t known whether you can do this kind of trick.
Parametrizing what we don’t know isn’t just a trick for college physics, it’s key to the cutting edge as well. Right now we have a picture for how all of particle physics works, called the Standard Model, but we know that picture is incomplete. There are a million different theories you could write to go beyond the Standard Model, with a million different implications. Instead of having to use all those theories, physicists can summarize them all with what we call an effective theory: one that keeps track of the effect of all that new physics on the particles we already know. By summarizing those effects with a few parameters, we can see what they would have to be to be compatible with experimental results, ruling out some possibilities and suggesting others.
In a world where we never know everything, there’s always something that can hurt us. But if we’re careful and estimate what we don’t know, if we write down numbers and parameters and keep our options open, we can keep from getting burned. By focusing on what we do know, we can still manage to understand the world.
Two weeks ago, I told you that Andrew and Michèle and I had written a paper, predicting what gravitational wave telescopes like LIGO see when black holes collide. You may remember that LIGO doesn’t just see colliding black holes: it sees colliding neutron stars too. So why didn’t we predict what happens when neutron stars collide?
Actually, we did. Our calculation doesn’t just apply to black holes. It applies to neutron stars too. And not just neutron stars: it applies to anything of roughly the right size and shape. Black holes, neutron stars, very large grapefruits…
That’s the magic of Effective Field Theory, the “zoom lens” of particle physics. Zoom out far enough, and any big, round object starts looking like a particle. Black holes, neutron stars, grapefruits, we can describe them all using the same math.
Ok, so we can describe both black holes and neutron stars. Can we tell the difference between them?
In our last calculation, no. In this one, yes!
Effective Field Theory isn’t just a zoom lens, it’s a controlled approximation. That means that when we “zoom out” we don’t just throw out anything “too small to see”. Instead, we approximate it, estimating how big of an effect it can have. Depending on how precise we want to be, we can include more and more of these approximated effects. If our estimates are good, we’ll include everything that matters, and get a good approximation for what we’re trying to observe.
At the precision of our last calculation, a black hole and a neutron star still look exactly the same. Our new calculation aims for a bit higher precision though. (For the experts: we’re at a higher order in spin.) The higher precision means that we can actually see the difference: our result changes for two colliding black holes versus two colliding grapefruits.
So does that mean I can tell you what happens when two neutron stars collide, according to our calculation? Actually, no. That’s not because we screwed up the calculation: it’s because some of the properties of neutron stars are unknown.
The Effective Field Theory of neutron stars has what we call “free parameters”, unknown variables. People have tried to estimate some of these (called “Love numbers” after the mathematician A. E. H. Love), but they depend on the details of how neutron stars work: what stuff they contain, how that stuff is shaped, and how it can move. To find them out, we probably can’t just calculate: we’ll have to measure, observe an actual neutron star collision and see what the numbers actually are.
That’s one of the purposes of gravitational wave telescopes. It’s not (as far as I know) something LIGO can measure. But future telescopes, with more precision, should be able to. By watching two colliding neutron stars and comparing to a high-precision calculation, physicists will better understand what those neutron stars are made of. In order to do that, they will need someone to do that high-precision calculation. And that’s why people like me are involved.
I am an “amplitudeologist”. I work on particle physics calculations, computing “scattering amplitudes” to find the probability that fundamental particles bounce off each other. This sounds like the farthest thing possible from black holes. Nevertheless, the two are tightly linked, through the magic of something called Effective Field Theory.
Effective Field Theory is a kind of “zoom knob” for particle physics. You “zoom out” to some chosen scale, and write down a theory that describes physics at that scale. Your theory won’t be a complete description: you’re ignoring everything that’s “too small to see”. It will, however, be an effective description: one that, at the scale you’re interested in, is effectively true.
Particle physicists usually use Effective Field Theory to go between different theories of particle physics, to zoom out from strings to quarks to protons and neutrons. But you can zoom out even further, all the way out to astronomical distances. Zoom out far enough, and even something as massive as a black hole looks like just another particle.
In this picture, the force of gravity between black holes looks like particles (specifically, gravitons) going back and forth. With this picture, physicists can calculate what happens when two black holes collide with each other, making predictions that can be checked with new gravitational wave telescopes like LIGO.
Researchers have pushed this technique quite far. As the calculations get more and more precise (more and more “loops”), they have gotten more and more challenging. This is particularly true when the black holes are spinning, an extra wrinkle in the calculation that adds a surprising amount of complexity.
That’s where I came in. I can’t compete with the experts on black holes, but I certainly know a thing or two about complicated particle physics calculations. Amplitudeologists, like Andrew McLeod and me, have a grab-bag of tricks that make these kinds of calculations a lot easier. With Michèle Levi’s expertise working with spinning black holes in Effective Field Theory, we were able to combine our knowledge to push beyond the state of the art, to a new level of precision.
This project has been quite exciting for me, for a number of reasons. For one, it’s my first time working with gravitons: despite this blog’s name, I’d never published a paper on gravity before. For another, as my brother quipped when he heard about it, this is by far the most “applied” paper I’ve ever written. I mostly work with a theory called N=4 super Yang-Mills, a toy model we use to develop new techniques. This paper isn’t a toy model: the calculation we did should describe black holes out there in the sky, in the real world. There’s a decent chance someone will use this calculation to compare with actual data, from LIGO or a future telescope. That, in particular, is an absurdly exciting prospect.
Because this was such an applied calculation, it was an opportunity to explore the more applied part of my own field. We ended up using well-known techniques from that corner, but I look forward to doing something more inventive in future.
At this year’s conference, gravitational waves have grown from a promising new direction to a large fraction of the talks. While there were still the usual talks about quantum field theory and string theory (everything from bootstrap methods to a surprising application of double field theory), gravitational waves have clearly become a major focus of this community.
This was highlighted before the first talk, when Zvi Bern brought up a recent paper by Thibault Damour. Bern and collaborators had recently used particle physics methods to pushbeyond the state of the art in gravitational wave calculations. Damour, an expert in the older methods, claims that Bern et al’s result is wrong, and in doing so also questions an earlier result by Amati, Ciafaloni, and Veneziano. More than that, Damour argued that the whole approach of using these kinds of particle physics tools for gravitational waves is misguided.
There was a lot of good-natured ribbing of Damour in the rest of the conference, as well as some serious attempts to confront his points. Damour’s argument so far is somewhat indirect, so there is hope that a more direct calculation (which Damour is currently pursuing) will resolve the matter. In the meantime, Julio Parra-Martinez described a reproduction of the older Amati/Ciafaloni/Veneziano result with more Damour-approved techniques, as well as additional indirect arguments that Bern et al got things right.
Before the QCD Meets Gravity community worked on gravitational waves, other groups had already built a strong track record in the area. One encouraging thing about this conference was how much the two communities are talking to each other. Several speakers came from the older community, and there were a lot of references in both groups’ talks to the other group’s work. This, more than even the content of the talks, felt like the strongest sign that something productive is happening here.
Many talks began by trying to motivate these gravitational calculations, usually to address the mysteries of astrophysics. Two talks were more direct, with Ramy Brustein and Pierre Vanhove speculating about new fundamental physics that could be uncovered by these calculations. I’m not the kind of physicist who does this kind of speculation, and I confess both talks struck me as rather strange. Vanhove in particular explicitly rejects the popular criterion of “naturalness”, making me wonder if his work is the kind of thing critics of naturalness have in mind.
It’s that time of year again, and I’m at Amplitudes, my field’s big yearly conference. This year we’re in Dublin, hosted by Trinity.
Increasingly, the organizers of Amplitudes have been setting aside a few slots for talks from people in other fields. This year the “closest” such speaker was Kirill Melnikov, who pointed out some of the hurdles that make it difficult to have useful calculations to compare to the LHC. Many of these hurdles aren’t things that amplitudes-people have traditionally worked on, but are still things that might benefit from our particular expertise. Another such speaker, Maxwell Hansen, is from a field called Lattice QCD. While amplitudeologists typically compute with approximations, order by order in more and more complicated diagrams, Lattice QCD instead simulates particle physics on supercomputers, chopping up their calculations on a grid. This allows them to study much stronger forces, including the messy interactions of quarks inside protons, but they have a harder time with the situations we’re best at, where two particles collide from far away. Apparently, though, they are making progress on that kind of calculation, with some clever tricks to connect it to calculations they know how to do. While I was a bit worried that this would let them fire all the amplitudeologists and replace us with supercomputers, they’re not quite there yet, nonetheless they are doing better than I would have expected. Other speakers from other fields included Leron Borsten, who has been applying the amplitudes concept of the “double copy” to M theory and Andrew Tolley, who uses the kind of “positivity” properties that amplitudeologists find interesting to restrict the kinds of theories used in cosmology.
The biggest set of “non-traditional-amplitudes” talks focused on using amplitudes techniques to calculate the behavior not of particles but of black holes, to predict the gravitational wave patterns detected by LIGO. This year featured a record six talks on the topic, a sixth of the conference. Last year I commented that the research ideas from amplitudeologists on gravitational waves had gotten more robust, with clearer proposals for how to move forward. This year things have developed even further, with several initial results. Even more encouragingly, while there are several groups doing different things they appear to be genuinely listening to each other: there were plenty of references in the talks both to other amplitudes groups and to work by more traditional gravitational physicists. There’s definitely still plenty of lingering confusion that needs to be cleared up, but it looks like the community is robust enough to work through it.
I’m still busy with the conference, but I’ll say more when I’m back next week. Stay tuned for square roots, clusters, and Nima’s travel schedule. And if you’re a regular reader, please fill out last week’s poll if you haven’t already!