What Is the Positive Grassmannian and Why Does It Show Up Everywhere?

0:00All right. Okay, and now we're starting, starting.

0:06I'm Steve Strogatz. And I'm Jan 11. And this is The Joy of Why. A podcast from Quantum Magazine where we explore some of the biggest unanswered questions in math and science today. I can start us off with a discussion I had recently with a mathematician who works in the area of math that we call algebraic combinatorics. Oof. Yeah? What makes you say oof to that? Oh, I don't know. It sounds really hard. I mean, algebraic, we know what those two words mean. Combinatorics, I assume, some kind of discrete algebra? Yeah, something like that, right? It's all about discrete objects or structures. When I think of combinatorics, I sometimes think of the many ways in which you can organize something.

0:54Like this comes up in thermodynamics, where statistical thinking or in counting of entropy, What are the possible ways of organizing a sequence of numbers, for instance? And is that an element in combinatorics or is that too simplistic? No, that's exactly on the money. That's the kind of thing that we're thinking about. And for our listeners who may not be in the world of thermodynamics or statistical physics, we do combinatorics anytime we play card games. Right, exactly. So you can ask yourself with a deck of 52 cards in the four suits, how many ways are there to get three of a kind versus two of a kind or a straight?

1:30Those kinds of counting questions or probability questions, that's part of what gave rise to combinatorics. But now it's used all the time when we have to find clever, time-saving ways to count things or to organize things, as you said, in some very structured manner. Yeah, that's a great analogy. That's exactly how I think of it. If you're handed five cards, what are your possible combinatorics? Your first card is a king, then what are the possible combinations that follow with the next four cards and then the next three and then the next two and then the one. Right. And you've put your finger on the key word there, right?

2:03Combinatorics coming from the idea of many combinations of possibilities. Yeah. Fascinating. But one of the things that came up that I really would like to have your take on before we dive into the episode is the question of beauty. Because I know that in physics, you all use beauty occasionally as a... Oh, okay, So you're smirking at me here. I feel like physicists are the last ones with a straight face that really talk about beauty. It's a very serious criteria. You know, something looks really messy, ugly, long, not compact. You suspect you've done something wrong or there's something deeper worth pursuing.

2:41And that we might dismiss it, but it seems to work. That's the interesting and sometimes uncanny thing, right? That first of all, beauty, even just as an aesthetic issue, is hard to define. But then somehow we expect that the universe is going to have such good taste that it's going to, you know, that we can use beauty as a selection criterion for our theories. Yeah. And I don't know if that is still as forceful a kind of trend, but it has been incredibly successful. And we can quantify symmetries, you know, certain organizational structures, compact abstract expressions, grouping things together so that it seems to streamline.

3:22These are pretty compelling definitions of beauty, I think, in this context. Uh-huh. Well, so that's where we're going to go first. We want to hear Lauren Williams' take on the issue of beauty, not in physics, but in pure math. She is a recent MacArthur Genius Prize winner, a mathematician at Harvard University. and you'll see in our conversation that she seems to specialize in finding connections between fields or topics that don't seem like they're related. She finds a common thread, often not just with combinatorics, but with a very specific object in combinatorics that goes by the even more intimidating name of the positive Grossmanian.

4:03So get ready to go plunge into those waters. Well, Grossman was amazing, so I'm already interested. Let's dive in. Welcome to the Joy of Why. Thank you so much. It's great to be here. The thing that's got me really fired up to talk to you, Lauren, I have to say, besides all the beautiful math that you've done, I feel like your work touches on this uncanny aspect of math that has a whole philosophical dimension to it. If you don't mind, let's start with some personal stuff. I have two daughters, my wife and I, and if I understand right, you are one of four daughters? That's right. I'm the eldest of four girls.

4:42And so are your sisters also mathematically inclined? Yes. All of my sisters liked math as well. The age gaps between us are 2, 2, and 6. So when I was growing up, the house was pretty full. I remember teaching my youngest sister, Genevieve edition of numbers with carries when she was probably four or five. And I showed her how to do it. And then I gave her, you know, some homework. She came back and she had a big grin on her face. And she had done it a different way. She was doing things left to right instead of right to left. And she thought that was hysterical. So I don't want to put words in their mouths, but my recollection is that it seemed fun for them.

5:25And with my sister Eleanor, who's closest to me in age. When I was in fourth or fifth grade and, you know, she was two years younger, we decided that instead of learning another language, we could learn codes. And so we started learning about Caesar Cipher and then making up our own codes. We spent a lot of time doing that and it was a lot of fun. It was a secret language. I also saw some pictures of you looking on the internet playing the violin. And it makes me wonder if your interest in beautiful things, whether in music or in math. I mean, are you generally interested in beauty? Very much so.

6:00I started playing the violin when I was four. It was quite a big passion of mine. I was also writing poetry and stories as well. But for sure, it feels like the beauty that we see in mathematics has a lot to do with the beauty that we see in music or in poetry in terms of elegance or the symmetry, the harmony. There's a strong correlation there. I don't know if learning music early on helped me to appreciate math or made me more inclined to math, or if having a mathematical appreciation also went hand in hand with having an appreciation of music, I really don't know. Now, if I can fast forward a little bit, can you unpack for us what is the idea of combinatorics?

6:42What are the kinds of questions you would study in that field? Roughly speaking, combinatorics is the study of finite or discrete structures. When one starts out learning combinatorics, one tends to learn a lot of techniques for counting. We're studying different kinds of objects and counting them with the usual sort of non-negative integers, 0, 1, 2, and so on. Actually, probably 30, 40 years ago, combinatorics was looked down on as a field of math. There was a viewpoint among purists that combinatorics was sort of a bag of tricks. But in fact, my advisor, Richard Stanley, was one of the people who really made combinatorics a much more respectable field.

7:22He wrote these beautiful books. His work sort of pointed out that there was a lot more structure and deeper theory behind it all. And in particular, he has a lot of wonderful work in algebraic combinatorics, which is using techniques from algebra to study combinatorial problems. So I want to go back to something you said a minute ago. the idea that some parts of math are considered more peripheral or more marginal, like they can come in and out of fashion. And I'm just wondering if you could expand because it gets into the sociology of math. It is a sociological phenomenon. I think partly it has to do with the age of a field.

8:01So, for example, number theory is the study of numbers and their arithmetic properties, things like primes. It's a very, very old subject and had a very prestigious reputation. You know, Fermat's last theorem is one of the crowning achievements in number theory, whereas somehow combinatorics didn't really come into its present good reputation until fairly recently. I mean, in fact, Richard Stanley was a PhD student at Harvard in the 1960s. And at the time, Harvard had no professor doing combinatorics. He wanted to work with Giancarlo Rota, who was a combinatorialist working at MIT. And he had faculty members at Harvard telling him, you know, stop doing this Mickey Mouse mathematics.

8:49Fortunately, he persisted and he changed the field. And then, you know, more recently, in the last 10 years, the mathematician Jun He won a Fields Medal, largely for his work at the interface of combinatorics and algebraic geometry. And that, of course, it's maybe a sign that this has finally been recognized. You're really putting your finger on it there. This idea of the interconnectedness of math itself is part of a status of a subject. If a subject is seen as too far in the outskirts, not connecting to the other domains in math, that tends to lower its reputation. I agree. Right? Whereas the things that are in the core, whatever that means, have higher status, like algebraic geometry you mentioned.

9:32Right. So I began by asking you questions about beauty, the aesthetic dimension of math. And so I would like to try to connect that to a quote that I read of yours. It says, if you ask a question and the answer is not beautiful, that means you asked the wrong question. I don't know. I'd like to hear more. Yes. So starting from the beginning of my time as a graduate student, I was going to seminars. I was taking classes. And one of the things I was trying to figure out was what do different people find interesting in mathematics? Or what do they find beautiful in mathematics? What do I find interesting?

10:11What do I find? beautiful. And I picked up this aesthetic in part from my advisor, you know, the work should be beautiful. And as a pure mathematician, we're not beholden to particular questions or applications. And so what are the standards to which we should hold ourselves? Well, we should be looking for something beautiful. And we have complete freedom as pure mathematicians. And if we have complete freedom to ask whatever question we want. Well, what I want to do is find questions with beautiful answers. And I think those are the questions and answers that tend to wind up becoming the most important or having the most connections to other fields, whether it's within math or outside of math.

10:57In the popular mind, math is known as being a very black and white subject. There's a right answer. There's a wrong answer. When you start talking about beauty, it sounds very subjective. There's the old line about beauty being in the eye of the beholder. And it seems a little surprising that mathematicians would be so obsessed with beauty, given that it's so hard to define what beauty means. And I guess I'd like to know, first of all, what would it look like for math to be not beautiful? So a beautiful piece of math is short, clean, elegant, maybe a statement that connects to mathematical objects that you didn't expect to be connected.

11:35That is something that I would find more likely to be beautiful than if the theorem takes paragraphs or pages to even get to the statement. On the other hand, one could come up with a true theorem or a true fact that seems very arbitrary or specific. Maybe you ask like, okay, what if I take 73 random lines in the plane and then I look at the, I don't know, I look at some curve going through these four intersection points. What's going to happen? You know, that's less likely to be considered beautiful than a statement like there are infinitely many crimes. You know, a statement that is short and simple, clean, and feels more sort of universal is going to be considered more beautiful than the other kind of statement.

12:27So let me see if I follow you. The aesthetic you're talking about tends to like economy of means or minimalism. You like universality. You don't want a lot of very highly specific things. To the extent that it's general, that's a plus. To the extent that it's compact, that helps. I was hearing some disrespect towards certain types of numbers in favor of other types of numbers. Well, I am a combinatorialist after all. Okay. But it was more about questions where the answer is a theorem that holds true for infinitely many cases cases as opposed to a question with a single answer, which is some real specific number.

13:08So in terms of, like, if we were doing a self-help manual for mathematicians, and you suppose you're studying something and it's coming out not beautiful, according to these and maybe some other criteria, what do you do? Like, are there moves that you can make to push yourself on a more beautiful track? I guess I would try to tweak the question. If it seems like I'm arriving at the answer, but it's simply not elegant, doesn't seem to be interesting, then I would try to tweak the question. I mean, I might, in an extreme situation, just abandon that effort. But if I feel like there should be something interesting nearby, then I would try to adjust the question.

13:48And that's a nice gambit, because in a lot of fields, you're sort of stuck with the question. But in math, you mentioned the freedom to think about whatever they want. And so you're saying you can leverage that freedom to maybe change the question. Maybe your ugly answer is a sign that your question isn't quite right. Okay, so now I think I understand what you meant by that. Yeah, I would imagine that people in music or art could relate to this kind of statement. Maybe there's a theme that they're trying to play with in composing a piece of music and it doesn't have the right harmonies. I'm sure there's a lot of tweaking of the framework or tweaking of the approach that goes into art that results in a better end product.

14:27Well, so I want to dive into that now, this set of ideas related to a thing called the Grasmanian and the positive Grasmanian, which for a long time were a playground for pure mathematicians. So, okay, what the heck is this thing? For people who have never heard of the Grasmanian, start us off. How should we think about it in the simplest cases? Yeah. So the Grasmonian, it's a geometric object, and it's a bit like a library for keeping track of simpler objects. So, for example, the Grasmonian of one planes in two space is the set of all lines through the origin in the XY plane. And so we're thinking of this as a sort of one mathematical object that's keeping track of all of these lines.

15:22And so you can imagine all these lines passing through the origin in two-dimensional space. And then the positive part of this Grasmonian would be the set of those lines that go through the positive orthent. Let me think about these lines. If I were looking at a compass, there's a line that goes north and south. Yes. I think of that as one line. So north and south are both part of the same line. Yes, exactly. And east and west are also part of the same line. That's right. And then there's sort of infinitely many lines that we can get by tilting either of those lines. So I'm thinking about infinitely many different lines, and you're telling me there's a way I could think about them as somehow described by a single shape?

16:01Yes. And if you like, if you're thinking about this set of all lines through the origin in two-dimensional space, each one passes through the sort of northern half of your compass. And so you could actually just keep track of each of those lines by the intersection point with a circle. And then if you just think about this positive semicircle at the top, then each point on that sort of positive semicircle will be specifying uniquely a line that goes through the origin. Which on a compass would just have little markings on it that say north, northeast. We could even have north, north, north, northeast.

16:44Yes. I mean, that gets cumbersome to do it that way. But you're saying there's infinitely many points on this northern half circle. And then the totality of all those points makes what you call the semicircle. But do I consider what's going on with the equator or the east and western? That's right. We should only include one of them. Just one. Because there's only one. There's one line that passes through both the west and the east most points. So we should only keep one of them and throw the other point away. As far as a collection of all these lines, am I right in thinking that it's like a line segment that goes from east, but I don't include east, all the way around the top to west, but I do include west?

17:26Exactly. So is it like a line segment with one end point but not the other end point? Exactly. Yes. That would be the Grasmonian of one planes and two space. That's how we can think about it. You could think of it as being like a half open interval. Yes. Huh. Okay. That's a kind of a weird object. I mean, it feels kind of unsymmetrical, a half-open interval. Like, I'm dying to close that other end point, but I shouldn't. Not if I want the Grasmanian. Well, we sort of identify those two points. So it's almost like we could think about sort of gluing those together. Okay. And then for the positive Grasmanian, you only want the positive orthent.

18:04Right. So first, the Grasmanian of lines in three-dimensional space in this slightly higher dimensional setting would look more or less like the northern hemisphere. And then if we restrict to the positive part of the Grasmonian, then we're just looking at the intersection of that hemisphere with the positive orthent. And so it's actually going to look like a triangle, like a kind of a curvy triangle. Okay. And this word orthent, which isn't totally familiar, is the 3D version of quadrant. Exactly. Right. There's eight of them. That's why you're saying orthent. Yeah. Yeah. So So that positive coordinate would be where the X, Y, and Z coordinates are all positive or non-negative.

18:48Okay, and you say it looks sort of like a curvy triangle. And so at this point, if people are still with us, why would anyone think about this object? This doesn't seem like an obvious thing to think about. Yeah, yeah, it wasn't an obvious thing to think about. But back in the 1900s, mathematicians were studying certain kinds of matrices called totally positive matrices. And they had nice properties. They had some connections to different systems like oscillation. And then in the late 1990s, early 2000s, Lustig and Postnikov realized that there was a way to sort of generalize this notion of totally positive matrices to an object that lived inside the Grasmonian.

19:30And so it was just sort of a purely interesting mathematical idea to try to study total positivity, not just for matrices anymore, but for geometric objects like the Grasmonian. So there's all kinds of sets of matrices that describe motions and symmetries for the real world. And they come up in quantum theory. They're used all the time now in artificial intelligence. But even inside of math, as you say, they can act like machines that do things. to other mathematical objects. Yes. In math, one of the common themes is that we study not just the mathematical objects, but also the relationships between the objects.

20:12And matrices can give us a way to create or to analyze relationships between different mathematical objects. Now, one of the results that you are known for was this positive Grasmania that we talked about, you looked at in a combinatorial way in the very general case. So tell us a little bit of the flavor of what you did there. Yeah, absolutely. Back when I was a grad student or postdoc, I think I had a conversation with some other mathematicians about, you know, just what combinatorics is as a field. And one thing that we discussed at that dinner was that one can think of combinatorics not necessarily just as a field, but as an attitude.

20:54You know, we can go through life with a combinatorial attitude and take a combinatorial approach to different problems. And the positive Grasmonian can be divided into pieces of different dimensions. An analogy I like to use is that of the cube, say the three-dimensional cube. If a combinatorialist looks at it, they may come away saying, well, it has six two-dimensional faces, these squares on the different sides, and it has 12 one-dimensional edges, and it also has eight zero-dimensional pieces, the eight vertices. And so you can associate these numbers, 6, 12, and 8, to a cube. That's what a combinatorialist might do.

21:38Now, there are infinitely many positive Gressmonians, and they can have arbitrarily high dimension, but the first problem that I worked on in graduate was coming up with an explicit formula for how many pieces there are of each dimension. So I wrote down a polynomial that for any K and N tells you how many pieces there are of each dimension in that positive Grasmonian. Okay, so let's now make a little swerve from this pretty abstract realm of matrices and Grasmanians and positive Grasmanians to the much more mundane world of traffic and waves on the ocean and proteins being made inside of cells.

22:19Because it turns out all those things can be viewed as part of one story. That's right. There have been sort of three different areas that I've had personal experience with where the positive Grasmonian got connected. during my postdoc, I learned that another mathematician, Sylvie Cortiel, had written a paper which said that my polynomials that were counting pieces of the positive gristmonian according to dimension were also computing probabilities in a model that had been introduced to study translation in protein synthesis and was also used as a model for traffic flow. I was floored when I saw this paper to think that my polynomials had to do with a sort of quote-unquote real world.

23:03Her result was quite beautiful. Basically, she was saying that my polynomials were giving the probability that in a lattice with n sites or in a road with space for n cars, there are exactly k cars present. That's what my polynomials were computing. So great. So we know the probability that in a road with space for cars, there's exactly K present. Well, what if we want to know the probability that the cars are present in positions one, four, five, eight? You know, what if you want to know all the probabilities that any given configuration of cars is there? And so that was the natural question to ask.

23:42That actually kicked off a decades-long collaboration with Sylvie. I mean, we've written a number of papers together by now. Hmm. So there's a lot to unpack there, Janet. Did that wash over you? Well, I mean, I grasp some of it, right? This idea that you can cut some mathematical object into pieces, find some mathematical rule for the number of pieces occurring of a certain variety. She was saying dimensionality. So having heard all of that, which is very intriguing, just give me the bird's eye view of what a Grasmanian is. Okay, fair enough. Right. It's not a concept we run into every day. So here's what it is.

24:23There's a technical way we could define it. But before I give you that, can I give you what's it going to do for us? How is it helpful? It's a really nice meta concept. It's a shape that tells us about other shapes. It's a shape that can be used as a library or a catalog for other kinds of structures or shapes. So it's one shape or it's a family of shapes? It's a family of shapes. There's different Grasmanians. I mean, here's the technical definition, which may work for you, but I don't want to linger on it too long because I don't think it's the most helpful way to think about it. Technically, it has to do with thinking about all the different ways that k-dimensional spaces, linear spaces, like a two-dimensional space would be a plane.

25:11A one-dimensional space would be a line. A three-dimensional space is what we're used to for ordinary 3D space. Yeah. So you're trying to think about the totality of all k-dimensional linear spaces through the origin of n-dimensional space. Okay. So there's two parameters, k and n. Now, the simplest case would be think about lines through the origin. That'd be one-dimensional spaces through the origin in a plane. So n is 2, k is 1. Right. That would be the 1, 2 Grasmanian or something like that. Okay. Okay? And so there's infinitely many, a whole continuum of lines. But if you wanted to parametrize them in our language, if you wanted to catalog them, you could do it by saying, what's their compass direction?

26:04Like there's the line that goes north-south, or there's the line that goes north-northeast and south-southwest, or something like that, right? So if I listed all of those possible lines, it could be the whole upper semicircle. So that's a shape. And so people study different Grasmanians, some high dimensional space and some lower dimensional. Exactly. Now, Lauren specializes in this piece of it that's called the positive Grasmanian, which in our little example with lines through the origin in the plane would be like only considering the ones that have positive slope. And that turns out to have extra structure that makes it more helpful in lots of applications.

26:46At this point, it seems like something that pure geometers would think about. This is about a shape that classifies other shapes. The spooky thing is that this pops up all over the place in real-world settings. So like she mentions, traffic flow. I want you to have not an image of cars motoring down the highway because she doesn't really mean that kind of traffic. Think of back when COVID was rampant and we had to stand in line at the checkout for the supermarket and you had to stay six feet behind the person in front of you, right? So imagine you had something like 10 spots available that you could stand on.