Tamar Ziegler and I have just uploaded to the arXiv our paper “Infinite partial sumsets in the primes“. This is a short paper inspired by a recent result of Kra, Moreira, Richter, and Robertson (discussed for instance in this Quanta article from last December) showing that for any set of natural numbers of positive upper density, there exists a sequence of natural numbers and a shift such that for all this answers a question of Erdős). In view of the “transference principle“, it is then plausible to ask whether the same result holds if is replaced by the primes. We can show the following results:
Theorem 1
- (i) If the Hardy-Littlewood prime tuples conjecture (or the weaker conjecture of Dickson) is true, then there exists an increasing sequence of primes such that is prime for all .
- (ii) Unconditionally, there exist increasing sequences and of natural numbers such that is prime for all .
- (iii) These conclusions fail if “prime” is replaced by “positive density subset of the primes” (even if the density is equal to 1).
We remark that it was shown by Balog that there (unconditionally) exist arbitrarily long but finite sequences of primes such that is prime for all . (This result can also be recovered from the later results of Ben Green, myself, and Tamar Ziegler.) Also, it had previously been shown by Granville that on the Hardy-Littlewood prime tuples conjecture, there existed increasing sequences and of natural numbers such that is prime for all .
The conclusion of (i) is stronger than that of (ii) (which is of course consistent with the former being conditional and the latter unconditional). The conclusion (ii) also implies the well-known theorem of Maynard that for any given , there exist infinitely many -tuples of primes of bounded diameter, and indeed our proof of (ii) uses the same “Maynard sieve” that powers the proof of that theorem (though we use a formulation of that sieve closer to that in this blog post of mine). Indeed, the failure of (iii) basically arises from the failure of Maynard’s theorem for dense subsets of primes, simply by removing those clusters of primes that are unusually closely spaced.
Our proof of (i) was initially inspired by the topological dynamics methods used by Kra, Moreira, Richter, and Robertson, but we managed to condense it to a purely elementary argument (taking up only half a page) that makes no reference to topological dynamics and builds up the sequence recursively by repeated application of the prime tuples conjecture.
The proof of (ii) takes up the majority of the paper. It is easiest to phrase the argument in terms of “prime-producing tuples” – tuples for which there are infinitely many with all prime. Maynard’s theorem is equivalent to the existence of arbitrarily long prime-producing tuples; our theorem is equivalent to the stronger assertion that there exist an infinite sequence such that every initial segment is prime-producing. The main new tool for achieving this is the following cute measure-theoretic lemma of Bergelson:
Lemma 2 (Bergelson intersectivity lemma) Let be subsets of a probability space of measure uniformly bounded away from zero, thus . Then there exists a subsequence such that for all .
This lemma has a short proof, though not an entirely obvious one. Firstly, by deleting a null set from , one can assume that all finite intersections are either positive measure or empty. Secondly, a routine application of Fatou’s lemma shows that the maximal function has a positive integral, hence must be positive at some point . Thus there is a subsequence whose finite intersections all contain , thus have positive measure as desired by the previous reduction.
It turns out that one cannot quite combine the standard Maynard sieve with the intersectivity lemma because the events that show up (which roughly correspond to the event that is prime for some random number (with a well-chosen probability distribution) and some shift ) have their probability going to zero, rather than being uniformly bounded from below. To get around this, we borrow an idea from a paper of Banks, Freiberg, and Maynard, and group the shifts into various clusters , chosen in such a way that the probability that at least one of is prime is bounded uniformly from below. One then applies the Bergelson intersectivity lemma to those events and uses many applications of the pigeonhole principle to conclude.
I mentioned this earlier, but now it’s actually happening! I hope you can think of good workshops and apply to run them in Edinburgh.
The International Centre for Mathematical Sciences, or ICMS, in Edinburgh, will host a new project entitled Mathematics for Humanity. This will be devoted to education, research, and scholarly exchange having direct relevance to the ways in which mathematics can contribute to the betterment of humanity. Submitted proposals will be reviewed on April 15, 2023.
The activities of the program will revolve around three interrelated themes:
A. Integrating the global research community (GRC)
B. Mathematical challenges for humanity (MCH)
C. Global history of mathematics (GHM)
Development of the three themes will facilitate the engagement of the international mathematical community with the challenges of accessible education, knowledge-driven activism, and transformative scholarship.
For theme A, a coherent plan of activities for an extended period can be presented (at least 2 weeks, and up to to 3 months), comprising courses and seminars bringing together researchers from at least two different regions, which should be combined with networking activities and hybrid dissemination. Themes B and C would also comprise individual collaborative events.
Within each of the three themes, researchers can apply for one of the following activities:
Research-in-groups. This is a proposal for a small group of 3 to 6 researchers to spend from 2 weeks to 3 months in Edinburgh on a reasonably well-defined research project. The researchers will be provided working space and funds for accommodation and subsistence.
Research course or seminar. A group of researchers can propose a course or a seminar on topics relevant to one of the three themes. These should be planned as hybrid events with regular meetings in Edinburgh that can also be accessed online. Proposals should come with a detailed plan for attracting interest and for the dissemination of ideas.
Research workshops. These are 5-day workshops in the standard ICMS format, of course with a focus on one of the three themes.
Research school. These are hybrid schools of two-weeks length on one of the themes. These should come with substantial planning, a coherent structure, and be aimed towards post-graduate students and early career researchers.
The ICMS expects that up to 30 researchers will be in residence in Edinburgh at any given time over a 9-month period, which might be divided into three terms, mid-September to mid-December, mid-January to mid-April, and mid-April to mid-July. Every effort will be made to provide a unified facility for the activities of all groups working on all three themes, thereby encouraging a synergistic exchange of ideas and vision. The proposals will be reviewed twice a year soon after the spring deadline of 15 April and the autumn deadline of 15 November.
Queries about the project should be sent to ICMS director Minhyong Kim or deputy director Beatrice Pelloni, who will be aided by the Scientific Committee in the selection of proposals:
• John Baez (UC Riverside) • Karine Chemla (Paris)
• Sophie Dabo (Lille) • Reviel Netz (Stanford)
• Bao Chau Ngo (Chicago and VIASM) • Raman Parimala (Emory)
• Fernando Rodriguez Villegas (ICTP, Trieste) • Terence Tao (UCLA)
I’m working with an organization that may eventually fund proposals to fund workshops for research groups working on “mathematics for humanity”. This would include math related to climate change, health, democracy, economics, etc.
I can’t give details unless and until it solidifies.
However, it would help me to know a bunch of possible good proposals. Can you help me imagine some?
A good proposal needs:
a clearly well-defined subject where mathematics is already helping humanity but could help more, together with
a specific group of people who already have a track record of doing good work on this subject, and
some evidence that having a workshop, maybe as long as 3 months, bringing together this group and other people, would help them do good things.
I’m saying this because I don’t want vague ideas like “oh it would be cool if a bunch of category theorists could figure out how to make social media better”.
I asked for suggestions on Mathstodon and got these so far:
figuring out how to better communicate risks and other statistical information,
improving machine learning to get more reliable, safe and clearly understandable systems,
studying tipping points and ‘tipping elements’ in the Earth’s climate system,
creating higher-quality open-access climate simulation software,
Each topic already has people already working on it, so these are good examples. Can you think of more, and point me to groups of people working on these things?
We discussed this here earlier, but now it’s actually happening!
The International Centre for Mathematical Sciences, or ICMS, in Edinburgh, will host a new project entitled ‘Mathematics for Humanity’. This will be devoted to education, research, and scholarly exchange having direct relevance to the ways in which mathematics can contribute to the betterment of humanity. Submitted proposals will be reviewed on April 15, 2023.
The activities of the program will revolve around three interrelated themes:
A. Integrating the global research community (GRC)
B. Mathematical challenges for humanity (MCH)
C. Global history of mathematics (GHM)
Development of the three themes will facilitate the engagement of the international mathematical community with the challenges of accessible education, knowledge-driven activism, and transformative scholarship.
For theme A, a coherent plan of activities for an extended period can be presented (at least 2 weeks, and up to to 3 months), comprising courses and seminars bringing together researchers from at least two different regions, which should be combined with networking activities and hybrid dissemination. Themes B and C would also comprise individual collaborative events.
Within each of the three themes, researchers can apply for one of the following activities:
Research course or seminar. A group of researchers can propose a course or a seminar on topics relevant to one of the three themes. These should be planned as hybrid events with regular meetings in Edinburgh that can also be accessed online. Proposals should come with a detailed plan for attracting interest and for the dissemination of ideas.
Research workshops. These are 5-day workshops in the standard ICMS format, of course with a focus on one of the three themes.
Research school. These are hybrid schools of two-weeks length on one of the themes. These should come with substantial planning, a coherent structure, and be aimed towards post-graduate students and early career researchers.
The ICMS expects that up to 30 researchers will be in residence in Edinburgh at any given time over a 9-month period, which might be divided into three terms, mid-September to mid-December, mid-January to mid-April, and mid-April to mid-July. Every effort will be made to provide a unified facility for the activities of all groups working on all three themes, thereby encouraging a synergistic exchange of ideas and vision. The proposals will be reviewed twice a year soon after the spring deadline of 15 April and the autumn deadline of 15 November.
Queries about the project should be sent to ICMS director Minhyong Kim or deputy director Beatrice Pelloni, who will be aided by the Scientific Committee in the selection of proposals:
• John Baez (UC Riverside) • Karine Chemla (Paris)
• Sophie Dabo (Lille) • Reviel Netz (Stanford)
• Bao Chau Ngo (Chicago and VIASM) • Raman Parimala (Emory)
• Fernando Rodriguez Villegas (ICTP, Trieste) • Terence Tao (UCLA)
[WARNING: SPOILERS FOLLOW]
Update (Jan. 23): Rationalist blogger, Magic: The Gathering champion, and COVID analyst Zvi Mowshowitz was nerd-sniped by this review into writing his own much longer review of M3GAN, from a more Orthodox AI-alignment perspective. Zvi applies much of his considerable ingenuity to figuring out how even aspects of M3GAN that don’t seem to make sense in terms of M3GAN’s objective function—e.g., the robot offering up wisecracks as she kills people, attracting the attention of the police, or ultimately turning on her primary user Cady—could make sense after all, if you model M3GAN as playing the long, long game. (E.g., what if M3GAN planned even her own destruction, in order to bring Cady and her aunt closer to each other?) My main worry is that, much like Talmudic exegesis, this sort of thing could be done no matter what was shown in the movie: it’s just a question of effort and cleverness!
Tonight, on a rare date without the kids, Dana and I saw M3GAN, the new black-comedy horror movie about an orphaned 9-year-old girl named Cady who, under the care of her roboticist aunt, gets an extremely intelligent and lifelike AI doll as a companion. The robot doll, M3GAN, is given a mission to bond with Cady and protect her physical and emotional well-being at all times. M3GAN proceeds to take that directive more literally than intended, with predictably grisly results given the genre.
I chose this movie for, you know, work purposes. Research for my safety job at OpenAI.
So, here’s my review: the first 80% or so of M3GAN constitutes one of the finest movies about AI that I’ve seen. Judged purely as an “AI-safety cautionary fable” and not on any other merits, it takes its place alongside or even surpasses the old standbys like 2001, Terminator, and The Matrix. There are two reasons.
First, M3GAN tries hard to dispense with the dumb tropes that an AI differs from a standard-issue human mostly in its thirst for power, its inability to understand true emotions, and its lack of voice inflection. M3GAN is explicitly a “generative learning model”—and she’s shown becoming increasingly brilliant at empathy, caretaking, and even emotional manipulation. It’s also shown, 100% plausibly, how Cady grows to love her robo-companion more than any human, even as the robot’s behavior turns more and more disturbing. I’m extremely curious to what extent the script was influenced by the recent explosion of large language models—but in any case, it occurred to me that this is what you might get if you tried to make a genuinely 2020s AI movie, rather than a 60s AI movie with updated visuals.
Secondly, until near the end, the movie actually takes seriously that M3GAN, for all her intelligence and flexibility, is a machine trying to optimize an objective function, and that objective function can’t be ignored for narrative convenience. Meaning: sure, the robot might murder, but not to “rebel against its creators and gain power” (as in most AI flicks), much less because “chaos theory demands it” (Jurassic Park), but only to further its mission of protecting Cady. I liked that M3GAN’s first victims—a vicious attack dog, the dog’s even more vicious owner, and a sadistic schoolyard bully—are so unsympathetic that some part of the audience will, with guilty conscience, be rooting for the murderbot.
But then there’s the last 20% of the movie, where it abandons its own logic, as the robot goes berserk and resists her own shutdown by trying to kill basically everyone in sight—including, at the very end, Cady herself. The best I can say about the ending is that it’s knowing and campy. You can imagine the scriptwriters sighing to themselves, like, “OK, the focus groups demanded to see the robot go on a senseless killing spree … so I guess a senseless killing spree is exactly what we give them.”
But probably film criticism isn’t what most of you are here for. Clearly the real question is: what insights, if any, can we take from this movie about AI safety?
I found the first 80% of the film to be thought-provoking about at least one AI safety question, and a mind-bogglingly near-term one: namely, what will happen to children as they increasingly grow up with powerful AIs as companions?
In their last minutes before dying in a car crash, Cady’s parents, like countless other modern parents, fret that their daughter is too addicted to her iPad. But Cady’s roboticist aunt, Gemma, then lets the girl spend endless hours with M3GAN—both because Gemma is a distracted caregiver who wants to get back to her work, and because Gemma sees that M3GAN is making Cady happier than any human could, with the possible exception of Cady’s dead parents.
I confess: when my kids battle each other, throw monster tantrums, refuse to eat dinner or bathe or go to bed, angrily demand second and third desserts and to be carried rather than walk, run to their rooms and lock the doors … when they do such things almost daily (which they do), I easily have thoughts like, I would totally buy a M3GAN or two for our house … yes, even having seen the movie! I mean, the minute I’m satisfied that they’ve mostly fixed the bug that causes the murder-rampages, I will order that frigging bot on Amazon with next-day delivery. And I’ll still be there for my kids whenever they need me, and I’ll play with them, and teach them things, and watch them grow up, and love them. But the robot can handle the excruciating bits, the bits that require the infinite patience I’ll never have.
OK, but what about the part where M3GAN does start murdering anyone who she sees as interfering with her goals? That struck me, honestly, as a trivially fixable alignment failure. Please don’t misunderstand me here to be minimizing the AI alignment problem, or suggesting it’s easy. I only mean: supposing that an AI were as capable as M3GAN (for much of the movie) at understanding Asimov’s Second Law of Robotics—i.e., supposing it could brilliantly care for its user, follow her wishes, and protect her—such an AI would seem capable as well of understanding the First Law (don’t harm any humans or allow them to come to harm), and the crucial fact that the First Law overrides the Second.
In the movie, the catastrophic alignment failure is explained, somewhat ludicrously, by Gemma not having had time to install the right safety modules before turning M3GAN loose on her niece. While I understand why movies do this sort of thing, I find it often interferes with the lessons those movies are trying to impart. (For example, is the moral of Jurassic Park that, if you’re going to start a live dinosaur theme park, just make sure to have backup power for the electric fences?)
Mostly, though, it was a bizarre experience to watch this movie—one that, whatever its 2020s updates, fits squarely into a literary tradition stretching back to Faust, the Golem of Prague, Frankenstein’s monster, Rossum’s Universal Robots, etc.—and then pinch myself and remember that, here in actual nonfiction reality,
Incredibly, unbelievably, here in the real world of 2023, what still seems most science-fictional about M3GAN is neither her language fluency, nor her ability to pursue goals, nor even her emotional insight, but simply her ease with the physical world: the fact that she can walk and dance like a real child, and all-too-brilliantly resist attempts to shut her down, and have all her compute onboard, and not break. And then there’s the question of the power source. The movie was never explicit about that, except for implying that she sits in a charging port every night. The more the movie descends into grotesque horror, though, the harder it becomes to understand why her creators can’t avail themselves of the first and most elemental of all AI safety strategies—like flipping the switch or popping out the battery.
The tenfold way is a mathematical classification of Hamiltonians used in condensed matter physics, based on their symmetries. Nine kinds are characterized by choosing one of these 3 options:
and one of these 3 options:
(Charge conjugation symmetry in condensed matter physics is usually a symmetry between particles - e.g. electrons or quasiparticles of some sort - and holes.)
The tenth kind has unitary “$S$” symmetry, a symmetry that simultaneously reverses the direction of time and interchanges particles and holes. Since it is unitary and we’re free to multiply it by a phase, we can assume without loss of generality that $S^2 = 1$.
What are examples of real-world condensed matter systems of all ten kinds?
I’ll take what I can get! If you know materials of a few of the ten kinds, that’s a start!
I was vacationing with the kids in San Francisco and it turned out for transit reasons to improve our day a lot to be able to store our suitcases somewhere in the city for the whole day. There is, as they say, an app for that, called Bounce. It’s a pretty clever idea! You pay $7.50 a bag and Bounce connects you with a location that’s willing to store luggage for you — in our case, a hotel (a budget option which is apparently famous for having the toilet just being out there openly in the room, to save space) but they use UPS locations and other stores too. A luggage locker at the train station would be cheaper, but of course that would mean you have to go to the train station, which might be out of your way.
Now the question is this — could I have saved some money and just shown up at a random hotel, handed the bellhop a twenty, and asked him to keep four bags in the back room for the day? Seems kind of reasonable. On the other hand, I can imagine hotels being under insurance instructions not to store bags for unknown non-guests. But why wouldn’t the same insurance caution keep them from signing up with Bounce? Maybe Bounce has taken on the liability somehow.
Anyway, this is not a service I anticipate needing often, but in a moment when it was exactly what I needed, it did exactly what I wanted, so I recommend it.
PS: My kids are now extremely into San Francisco.
Wow! I just learned an objective reason why sets and vector spaces are special!
Of course we all know math relies heavily on set theory and linear algebra. And if you know category theory, you can say various things about why the categories $\mathsf{Set}$ and $\mathsf{Vect}$ are particularly convenient frameworks for calculation. But I’d never known a theorem that picks out these categories, and just a few others.
Briefly: these are categories of algebraic gadgets where all the objects are free!
We could call these ‘totally free’ algebraic gadgets.
By ‘algebraic gadgets’ I mean sets equipped with some $n$-ary operations obeying equational laws. Examples include monoids, groups, rings, modules over a fixed ring, Lie algebras over a fixed field, etc.
There are three famous formalisms for studying algebraic gadgets. They all describe the same kinds of algebraic gadgets, so which you use is largely a matter of convenience:
varieties in the sense of universal algebra,
Lawvere theories, and
finitary monads on the category of sets.
Given any variety, or Lawvere theory, or finitary monad on the category of sets, we get a category $\mathsf{C}$ of algebraic gadgets together with a functor
$R : \mathsf{C} \to \mathsf{Set}$
sending each gadget of this kind to its underlying set. And this functor will always have a left adjoint
$L : \mathsf{Set} \to \mathsf{C}$
sending any set to the free gadget on that set.
Steven Givant and later Kearnes, Kiss and Szendrei completely classified the varieties for which every object in $\mathsf{C}$ is isomorphic to one of the form $L(S)$ for some set $S$. We could call these the totally free varieties:
That’s all!
A commutative division ring is called a ‘field’, and a module over a field is called a ‘vector space’. The quaternions are a nice example of a noncommutative division ring. Anyone who has studied modules over the quaternions knows that these act a lot like vector spaces, in part because they’re all free.
An affine space over a field is, poetically speaking, just a vector space that has forgotten its origin. If you pick any point in the affine space and call it $0$, you get a vector space. You can’t take linear combinations of points in an affine space, just ‘affine combinations’. These are $n$-ary operations that obey a bunch of equational laws that I’m too lazy to list. But when your affine space comes from a vector space in the way I just described, these affine combinations can be written as linear combinations:
$a_1 v_1 + \cdots + a_n v_n$
where the coefficients sum to one: $a_1 + \cdots + a_n = 1$. So, for example, if you have two points $v_1, v_2$ in an affine space you can get all the points on the line through them by taking affine combinations $av_1 + (1-a)v_2$.
We can also do all this stuff with a division ring replacing a field! In fact, ‘field’ used to mean ‘division ring’, and the noncommutative ones were called ‘skew fields’.
Now, you’ll notice from what I said that a module over a division ring $F$ is just the same as a pointed affine space over $F$ — that is, an affine space over $F$ equipped with a chosen point. Similarly a pointed set is just a set with a chosen point. So we can list the totally free varieties in a more enlightening way:
And this should remind us of the ‘field with one element’. We don’t know what the field with one element is, exactly, but we know that the modules of this mythical beast should be pointed sets. There are lots of reasons for that. Here we see another: it would unify the above classification!
Suppose we could go back in time, redefine ‘field’ to include division rings and also one extra thing: the ‘field with one element’, $F_{\mathrm{un}}$. We wouldn’t even need to know what $F_{\mathrm{un}}$ is, just that that affine spaces over it are sets. Then this would be the complete classification of totally free varieties:
That would be nice.
But of course, it’s sort of trivial that every set is the free set on some set. In this case our category $\mathsf{C}$ of algebraic gadgets is just $\mathsf{Set}$ itself, and $R : \mathsf{Set} \to \mathsf{Set}$ is the identity functor, so $L: \mathsf{Set} \to \mathsf{Set}$ and the monad $T = R L$ are also the identity.
This raises the question of ‘relativising’ the ideas I’ve been talking about, by replacing $\mathsf{Set}$ with some other category $\mathsf{X}$.
Puzzle. Give me as many interesting examples as you can of categories $\mathsf{X}$ and monads $T: \mathsf{X} \to \mathsf{X}$ such that every $T$-algebra is free.
Of course we can always take $T$ to be the identity, but that’s boring.
A very interesting small step would be to stick with $\mathsf{X} = \mathsf{Set}$ but drop the requirement that $T$ be finitary. This lets us talk about algebraic gadgets with infinitary operations. Are there any interesting ones where every $T$-algebra is free?
But I’m more interested in other categories $\mathsf{X}$. For example, what if $\mathsf{X}$ itself is $\mathsf{Vect}$, or the category of algebras of some other finitary monad on $\mathsf{Set}$?
Last time I explained a wonderful one-to-one correspondence between the 10 Morita equivalence classes of Clifford algebras and Cartan’s 10 infinite families of compact symmetric spaces.
This time I’ll do something different. I’ll explain a wonderful one-to-one correspondence between the 10 Morita equivalence classes of Clifford algebras and Cartan’s 10 infinite families of compact symmetric spaces.
Yes, it’s different! Not only will the details of the construction look very different, it gives a different correspondence! And I hope you can help me figure out what’s going on.
I thank Claude Schochet for pointing out that these two constructions don’t match.
The construction of symmetric spaces from Clifford algebras I described last time is something I made up myself, though it’s so simple someone must have thought of it earlier. The one I’ll talk about now is nicely explained here:
John Milnor, Morse Theory, Princeton U. Press, Princeton, NJ, 1963.
Dan Dugger, A Geometric Introduction to K-theory, draft.
Both Milnor and Dugger use this construction as part of a proof of Bott periodicity:
$\pi_{k+8}(\mathrm{O}(\infty)) \cong \pi_k(\mathrm{O}(\infty))$
where $\mathrm{O}(\infty)$ is the infinite-dimensional orthogonal group. The proof goes roughly as follows. First, you approximate $\mathrm{O}(\infty)$ by $\mathrm{O}(n)$, the group of orthogonal transformations of $\mathbb{R}^n$. Elements of $\pi_k(\mathrm{O}(n))$ are connected components of the $k$-fold loop space of $\mathrm{O}(n)$: that is, the space of loops in the space of loops in… $\mathrm{O}(n)$. Next comes a very interesting step: you can approximate this $k$-fold loop space by a much smaller space consisting of geodesics in a space of geodesics in… $\mathrm{O}(n)$. Milnor calls this space $\Omega_k(n)$. He describes $\Omega_k(n)$ algebraically using Clifford algebras! This lets him understand its set of connected components.
I’m going to sidestep most of this stuff, fascinating though it is, because I just want to get my hands on these spaces $\Omega_k(n)$ with a minimum of fuss. They are symmetric spaces! So they’ll give our second construction of symmetric spaces from Clifford algebras.
We’ll define these spaces $\Omega_k(n)$ recursively.
First, recall that a complex structure on $\mathbb{R}^n$ is a linear operator $J \colon \mathbb{R}^n \to \mathbb{R}^n$ such that $J^2 = - 1$. We say a complex structure is orthogonal if $J \in O(n)$.
Next, start picking orthogonal complex structures $J_1, J_2, \dots$ on $\mathbb{R}^n$, each of which anticommutes with all the previous ones:
$J_k J_\ell = - J_\ell J_k \quad \text{ if } \; \ell \lt k$
Let $\Omega_k(n)$ be the space of orthogonal complex structures $J_k$ that anticommute with all the previous ones.
Unfortunately this is a bit vague, because this space depends on all our previous choices $J_1, \dots, J_{k-1}$. Usually we’ll get isomorphic spaces $\Omega_k(n)$ no matter how we choose $J_{k-1}$. But at certain stages we’ll need to make a ‘good’ choice—which simply means any choice that makes $\Omega_k(n)$ have the largest possible dimension. This then determines $\Omega_k(n)$ uniquely up to isomorphism. I’ll say more about this later.
By definition we have
$\mathrm{O}(n) \supseteq \Omega_1(n) \supseteq \Omega_2(n) \supseteq \cdots$
and we might as well define $\Omega_0(n)$ to be $O(n)$ itself.
Milnor explicitly works out all these spaces $\Omega_k(n)$. He shows they’re all submanifolds of $\mathrm{O}(n)$. Since $\mathrm{O}(n)$ has a god-given Riemannian metric, the spaces $\Omega_k(n)$ become Riemannian manifolds. And he shows they’re all symmetric spaces!
Milnor works step by step, computing $\Omega_1(n), \Omega_2(n),$ and so on, and in a few pages he shows that
$\Omega_8(n) \cong \mathrm{O}(n/16)$
whenever $n$ is divisible by 16. This is a version of Bott periodicity!
What are these spaces $\Omega_k(n)$? The first couple are easy. $\Omega_1(n)$ is the space of orthogonal complex structures on $\mathbb{R}^n$ — or in other words, ways of making the real Hilbert space $\mathbb{R}^n$ into a complex Hilbert space. Notice this is empty unless $n$ is even.
Next, suppose $n$ is even and we’ve made $\mathbb{R}^n$ into a complex Hilbert space. Then $\Omega_2(n)$ is the space of ways of choosing a second orthogonal complex structure that anticommutes with the first. But this is the space of ways of ways of making our complex Hilbert space into a quaternionic Hilbert space! And this will be empty unless $n$ is divisible by 4.
It keeps on going like this, but it gets harder. On Mathstodon I talked my way through many of Milnor’s calculations, but here I’ll just state the results.
If $n$ isn’t a high enough power of $2$ then $\Omega_k(n)$ is just empty. The ones we care about, namely the ones up to $\Omega_8(n)$, are nonempty whenever $n$ is divisible by $16$. So I’ll just assume $n = 16r$ and state the results in that case. We get a very nice list of symmetric spaces:
$\Omega_0(n) = \mathrm{O}(16r)$. This is the group of all orthogonal transformations of $\mathbb{R}^{16r}$.
$\Omega_1(n) \cong \mathrm{O}(16r)/\mathrm{U}(8r)$. This is the space of orthogonal complex structures on $\mathbb{R}^{16r}$.
$\Omega_2(n) \cong \mathrm{U}(8r)/\mathrm{Sp}(4r)$. This is the space of orthogonal quaternionic structures on $\mathbb{C}^{8r}$: that is, orthogonal complex structures on the underlying real Hilbert space of $\mathbb{C}^{8r}$ that anticommute with multiplication by $i$.
$\Omega_3(n) \cong \bigsqcup_{0 \le d \le 4r} Sp(4r)/\mathrm{Sp}(d) \times \mathrm{Sp}(4r - d)$. This is the space of all quaternionic subspaces of $\mathbb{H}^{4r}$: a union of quaternionic Grassmannians.
$\Omega_4(n) \cong Sp(2r)$. This is the group of all quaternionic unitary transformations of $\mathbb{H}^{2r}$, also known as the compact symplectic group.
$\Omega_5(n) \cong Sp(2r)/U(2r)$. This is a complex Lagrangian Grassmannian: the space of all Lagrangian subspaces of a $2r$-dimensional complex symplectic vector space.
$\Omega_6(n) \cong U(2r)/O(2r)$. This is a real Lagrangian Grassmannian: the space of all Lagrangian subspaces of a $2r$-dimensional real symplectic vector space.
$\Omega_7(n) \cong \bigsqcup_{0 \le d \le 2r} O(2r)/\mathrm{O}(d) \times \mathrm{O}(2r - d)$. This is the space of all real subspaces of $\mathbb{R}^{2r}$: a union of real Grassmannians.
$\Omega_8(n) \cong O(r)$.
What does this stuff have to do with Clifford algebras? Well, a bunch of anticommuting complex structures
$J_1, \dots , J_k : \mathbb{R}^n \to \mathbb{R}^n$
is exactly the same as a representation of the algebra $Cliff_k$ on $\mathbb{R}^n$, i.e. an algebra homomorphism
$\rho \colon Cliff_k \to M_n(\mathbb{R})$
But in the definition of $\Omega_k(n)$ we are also requiring that these complex structures be orthogonal. We can state this extra requirement using the $\ast$-algebra structure on $Cliff_k$ that I explained last time: it amounts to saying $\rho$ is a $\ast$-representation, meaning a representation with
$\rho(a^\ast) = \rho(a)^\ast$
(In other jargon, it’s a $\ast$-algebra homomorphism.)
So, we can describe the symmetric spaces $\Omega_k(n)$ recursively as follows:
$\Omega_k(n)$ is the space of ways of extending the already chosen $\ast$-representation of $Cliff_k$ on $\mathbb{R}^n$ to a $\ast$-representation of $Cliff_{k+1}$.
This description works for $k \ge 1$, and in some ways it’s very nice, but it involves a sequence of choices, so let me say a bit about that! When we get to
$\displaystyle{ \Omega_3(n) \cong \displaystyle{\bigsqcup_{0 \le d \le 4r} Sp(4r)/\mathrm{Sp}(d) \times \mathrm{Sp}(4r - d)} }$
this has many connected components, one for each dimension $d$, and we should pick a point in the component of highest dimension, namely $d = 2r$. Similarly, when we get to
$\displaystyle{ \Omega_7(n) \cong \bigsqcup_{0 \le d \le 2r} O(2r)/\mathrm{O}(d) \times \mathrm{O}(2r - d) }$
we should pick a point in the component of highest dimension, namely $d = r$. And if we continue on, we must do the same thing whenever $k = 3$ or $7$ modulo $8$.
But now for the main point!
We now have two different ways to build symmetric spaces: the way I just described and the way I described last time. We should compare them. To simplify notation, let’s work ‘stably’, taking the direct limit of the spaces $\Omega_k(n)$ as $n \to \infty$, and working with the infinite-dimensional Lie groups $\mathrm{O}, \mathrm{U}$ and $\mathrm{Sp}$ instead of their finite-dimensional incarnations as above.
So, today’s construction gives infinite-dimensional symmetric spaces
$\Omega_k = \lim_{n \to \infty} \Omega_k(n)$
depending only on $k$ mod 8. The notation $\Omega_k$ is nice because they are actually iterated loop spaces. And I’ll call the infinite-dimensional symmetric spaces we got last time $\Upsilon_k$ because this letter doesn’t get used enough. Let’s compare them:
$\begin{array}{ll} \Upsilon_0 \cong \mathrm{O}/\mathrm{O} \times \mathrm{O} \quad & \Omega_0 \cong \mathrm{O} \\ \Upsilon_1 \cong \mathrm{U}/\mathrm{O} & \Omega_1 \cong \mathrm{O}/\mathrm{U} \\ \Upsilon_2 \cong \mathrm{Sp}/\mathrm{U} & \Omega_2 \cong \mathrm{U}/\mathrm{Sp} \\ \Upsilon_3 \cong \mathrm{Sp} & \Omega_3 \cong \mathrm{Sp}/\mathrm{Sp} \times \mathrm{Sp} \\ \Upsilon_4 \cong \mathrm{Sp}/\mathrm{Sp} \times \mathrm{Sp} \quad & \Omega_4 \cong \mathrm{Sp} \\ \Upsilon_5 \cong \mathrm{U}/\mathrm{Sp} & \Omega_5 \cong \mathrm{Sp}/\mathrm{U} \\ \Upsilon_6 \cong \mathrm{O}/\mathrm{U} & \Omega_6 \cong \mathrm{U}/\mathrm{O} \\ \Upsilon_7 \cong \mathrm{O} & \Omega_7 \cong \mathrm{O}/\mathrm{O} \times \mathrm{O} \end{array}$
And look! The second list is just the first list turned upside down!
So, the question is why.
A similar thing happens for complex Clifford algebras, by the way. Last time we got two infinite-dimensional symmetric spaces from those, and Milnor also gets two. Using the obvious notation we have
$\begin{array}{ll} \Upsilon_0^{\mathbb{C}} \cong \mathrm{U}/\mathrm{U} \times \mathrm{U} \quad & \Omega_0^{\mathbb{C}} \cong \mathrm{U} \\ \Upsilon_0^{\mathbb{C}} \cong \mathrm{U} & \Omega_0^{\mathbb{C}} \cong \mathrm{U}/\mathrm{U} \times \mathrm{U} \end{array}$
One nice thing about explaining a problem in detail in a blog article is that it gives me time to think about it. So I now have some thoughts about what’s going on here. But I’d also like to hear yours!
By the way, the symmetric spaces that are actually groups stand out as odd in the charts above, but they’re not really so odd because in both constructions they naturally as quotients:
$\mathrm{O} \cong \mathrm{O} \times \mathrm{O}/\mathrm{O}, \qquad \mathrm{U} \cong \mathrm{U} \times \mathrm{U}/\mathrm{U}, \qquad \mathrm{Sp} \cong \mathrm{Sp} \times \mathrm{Sp}/\mathrm{Sp}$
So, while I’m drawing big charts, let me draw one using this notation:
$\begin{array}{ll} \Upsilon_0 \cong \mathrm{O} /\mathrm{O} \times \mathrm{O} \quad & \Omega_0 \cong \mathrm{O} \times \mathrm{O}/\mathrm{O} \\ \Upsilon_1 \cong \mathrm{U}/\mathrm{O} & \Omega_1 \cong \mathrm{O}/\mathrm{U} \\ \Upsilon_2 \cong \mathrm{Sp}/\mathrm{U} & \Omega_2 \cong \mathrm{U}/\mathrm{Sp} \\ \Upsilon_3 \cong \mathrm{Sp} \times \mathrm{Sp} / \mathrm{Sp} & \Omega_3 \cong \mathrm{Sp}/\mathrm{Sp} \times \mathrm{Sp} \\ \Upsilon_4 \cong \mathrm{Sp}/\mathrm{Sp} \times \mathrm{Sp} \quad & \Omega_4 \cong \mathrm{Sp} \times \mathrm{Sp} / \mathrm{Sp} \\ \Upsilon_5 \cong \mathrm{U}/\mathrm{Sp} & \Omega_5 \cong \mathrm{Sp}/\mathrm{U} \\ \Upsilon_6 \cong \mathrm{O}/\mathrm{U} & \Omega_6 \cong \mathrm{U}/\mathrm{O} \\ \Upsilon_7 \cong \mathrm{O} \times \mathrm{O}/\mathrm{O} & \Omega_7 \cong \mathrm{O}/\mathrm{O} \times \mathrm{O} \end{array}$
Now you see the second list is the first turned upside down in two completely different senses of ‘turned upside down’. You can flip the whole first list upside down, or you can take the reciprocal of each ‘fraction’ on the list.
It also works like this in the complex case:
$\begin{array}{ll} \Upsilon_0^{\mathbb{C}} \cong \mathrm{U}/\mathrm{U} \times \mathrm{U} \quad & \Omega_0^{\mathbb{C}} \cong \mathrm{U} \times \mathrm{U}/\mathrm{U} \\ \Upsilon_0^{\mathbb{C}} \cong \mathrm{U} \times \mathrm{U}/\mathrm{U} & \Omega_0^{\mathbb{C}} \cong \mathrm{U}/\mathrm{U} \times \mathrm{U} \end{array}$
I’ve got a new paper out this week, with Andrew McLeod, Roger Morales, Matthias Wilhelm, and Chi Zhang. It’s yet another entry in this year’s “cabinet of curiosities”, quirky Feynman diagrams with interesting traits.
A while back, I talked about a set of Feynman diagrams I could compute with any number of “loops”, bypassing the approximations we usually need to use in particle physics. That wasn’t the first time someone did that. Back in the 90’s, some folks figured out how to do this for so-called “ladder” diagrams. These diagrams have two legs on one end for two particles coming in, two legs on the other end for two particles going out, and a ladder in between, like so:
There are infinitely many of these diagrams, but they’re all beautifully simple, variations on a theme that can be written down in a precise mathematical way.
Change things a little bit, though, and the situation gets wildly more intractable. Let the rungs of the ladder peek through the sides, and you get something looking more like the tracks for a train:
These traintrack integrals are much more complicated. Describing them requires the mathematics of Calabi-Yau manifolds, involving higher and higher dimensions as the tracks get longer. I don’t think there’s any hope of understanding these things for all loops, at least not any time soon.
What if we aimed somewhere in between? A ladder that just started to turn traintrack?
Add just a single pair of rungs, and it turns out that things remain relatively simple. If we do this, it turns out we don’t need any complicated Calabi-Yau manifolds. We just need the simplest Calabi-Yau manifold, called an elliptic curve. It’s actually the same curve for every version of the diagram. And the situation is simple enough that, with some extra cleverness, it looks like we’ve found a trick to calculate these diagrams to any number of loops we’d like.
(Another group figured out the curve, but not the calculation trick. They’ve solved different problems, though, studying all sorts of different traintrack diagrams. They sorted out some confusion I used to have about one of those diagrams, showing it actually behaves precisely the way we expected it to. All in all, it’s been a fun example of the way different scientists sometimes hone in on the same discovery.)
These developments are exciting, because Feynman diagrams with elliptic curves are still tough to deal with. We still have whole conferences about them. These new elliptic diagrams can be a long list of test cases, things we can experiment with with any number of loops. With time, we might truly understand them as well as the ladder diagrams!
Today Chirag Modi (Flatiron) gave a really great lunchtime talk about new technologies in cosmology and inference or measurement of cosmological parameters. He beautifully summarized how cosmology is done now (or traditionally): Make summary statistics of the observables, make a theory of the summary statistics, make up a surrogate likelihood function for use in inference, measure covariance matrices to use in the latter, and go. He's trying to obviate all of these things by using the simulations directly to make the measurements. He has nice results in forward modeling of the galaxy field, and in simulation-based inferences. Many interesting things came up in his talk, including the idea that I have discussed over the years with Kate Storey-Fisher (NYU) of enumerating all possible cosmological statistics! So much interesting stuff in the future of large-scale structure.