Pod Insights The Joy of Why What Is the Positive Grassmannian and Why Does It Show Up Everywhere? About this episode What links certain mathematical models of traffic flow, shallow-water waves, and quantum particle scattering? The surprising answer lies in a corner of the algebraic combinatorics world that goes by the name of positive Grassmannian. In simple terms, the positive Grassmannian is a shape that classifies other shapes. Remarkably, pieces of the positive Grassmannian can be reassembled in forms that reveal shared structures in these and many other seemingly unrelated mathematical systems.
That we know the positive Grassmannian crops up in many real-world settings is largely down to the theoretical work of Lauren Williams at Harvard University. In this latest episode of The Joy of Why, Williams talks to co-host Steven Strogatz about her work, how she realized the surprising pervasiveness of the positive Grassmannian, and how she has made a career of finding connections among fields that don’t at first sight seem connected. The conversation then switches to another project Williams is working on, called First Proof, which is trying to measure objectively how good AI systems are at coming up with proofs of research-level mathematical statements, and which leads to an exploration of whether AI may or may not take over mathematics.
Note: Since this conversation was recorded, results from the First Proof Second Batch project were released on June 10, 2026 https://1stproof.org/assets/docs/report.pdf.
Transcript 0:00 Speaker 00 Hi, I'm Terry Tao. My new book, Six Math Essentials, explores six concepts that have guided mathematicians like me throughout history. With a simple introduction to numbers, algebra, geometry, probability, analysis, and dynamics, I share how these ideas helped me think mathematically and can help everyone better understand our complex world. Mathematics isn't magic. It's a way of thinking anyone can learn. Please visit quantobooks.org to learn more.
0:25 Speaker 01 All right.
0:26 Speaker 02 Okay, and now we're starting, starting?
0:32 Speaker 01 I'm Steve Strogatz.
0:33 Speaker 02 And I'm Jan Levin.
0:35 Speaker 01 And this is The Joy of Why.
0:37 Speaker 02 A podcast from Quantum Magazine, where we explore some of the biggest unanswered questions in math and science today.
0:44 Speaker 01 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.
0:55 Speaker 02 Oof.
0:59 Speaker 01 Yeah? What makes you say oof to that?
1:01 Speaker 02 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?
1:08 Speaker 01 Yeah, something like that, right? It's all about discrete objects or structures.
1:13 Speaker 02 When I think of combinatorics, I sometimes think of the many ways in which you can organize something. Like, 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?
1:35 Speaker 01 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.
1:47 Speaker 02 Right, exactly.
1:49 Speaker 01 So you can ask yourself, with a deck of 52 cards and the four suits, how many ways are there to get three of a kind versus two of a kind or a straight? Those 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.
2:13 Speaker 02 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?
2:26 Speaker 01 Uh huh. Right, and you've put your finger on the key word there, right? Combinatorics coming from the idea of many combinations of possibilities.
2:33 Speaker 02 Yeah, fascinating.
2:34 Speaker 01 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 criti-- oh, okay, so you're smirking at me here.
2:50 Speaker 02 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. And that, we might dismiss it, but it seems to work.
3:11 Speaker 01 That's the interesting and sometimes uncanny thing, right?
3:14 Speaker 02 Yeah.
3:15 Speaker 01 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 gonna, you know, that we can use beauty as a selection criterion for our theories.
3:29 Speaker 02 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 expressions, grouping things together so that it seems to streamline. These are pretty compelling definitions of beauty, I think, in this context.
3:52 Speaker 01 Uh huh. Well, so that's where we're gonna go first. We wanna 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.
4:26 Speaker 01 Of the positive Grassmannian. So get ready to go plunge into those waters.
4:34 Speaker 02 Well, Grossman was amazing, so I'm already interested.
4:37 Speaker 01 Let's dive in.
4:40 Speaker 01 Welcome to the joy of why.
4:41 Speaker 03 Thank you so much. It's great to be here.
4:43 Speaker 01 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?
5:06 Speaker 03 That's right. I'm the eldest of four girls.
5:08 Speaker 01 And so are your sisters also mathematically inclined?
5:13 Speaker 03 Yes, all of my sisters liked math as well. The age gaps between us are two, two and six. So when I was growing up, the house was pretty full. I remember teaching my youngest sister Genevieve addition 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. And with my sister Eleanor, who's closest to me in age,
5:55 Speaker 03 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.
6:12 Speaker 01 I also saw some pictures of you looking on the internet, playing the violin. And it makes me wonder if you're interested in beautiful things, whether in music or in math. I mean, are you generally interested in beauty?
6:25 Speaker 03 Very much so. I 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.
7:01 Speaker 01 Now, if I can fast forward a little bit, can you unpack for us, what is the idea of combinatorics? What are the kinds of questions you would study in that field?
7:11 Speaker 03 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, zero, one, two, and so on. Actually, probably thirty, forty 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. He wrote these beautiful books. His work sort of pointed out that there was a lot more structure and deeper theory behind it all.
7:56 Speaker 03 And in particular, he has a lot of wonderful work in algebraic combinatorics, which is using techniques from algebra to study combinatorial problems.
8:04 Speaker 01 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.
8:19 Speaker 03 It is a sociological phenomenon. I think partly it has to do with the age of a field. So 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 combinator-
8:46 Speaker 03 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 John Carlo Rota, who was a combinatorist working at MIT. And he had faculty members at Harvard telling him, you know, stop doing this Mickey Mouse mathematics. Fortunately, he persisted and he changed the field. And then, you know, more recently, in the last 10 years, the mathematician Jun Ha won a Fields Medal, largely for his work at the interface of combinatorics and algebraic geometry.
9:29 Speaker 03 And that, of course, it's maybe a sign that this has finally been recognized.
9:34 Speaker 01 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.
9:51 Speaker 03 I agree.
9:52 Speaker 01 Whereas the things that are in the core, whatever that means, have higher status, like algebraic geometry you mentioned.
9:58 Speaker 03 Right.
9:59 Speaker 01 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.
10:18 Speaker 03 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? What 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
10:59 Speaker 03 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.
11:23 Speaker 01 In 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?
11:51 Speaker 03 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. That 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
12:36 Speaker 03 considered beautiful than a statement like, there are infinitely many primes. You know, a statement that is short and simple, clean, and feels
12:48 Speaker 03 More sort of universal is going to be considered considered more beautiful than the other kind of statement.
12:53 Speaker 01 So 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.
13:18 Speaker 03 Well, I am a combinatorialist after all.
13:19 Speaker 01 Okay.
13:20 Speaker 03 But but it was more about questions where the answer is a theorem that holds true for infinitely many cases as opposed to a question with a single answer, which is some real specific number.
13:34 Speaker 01 So 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?
13:53 Speaker 03 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.
14:14 Speaker 01 And 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.
14:34 Speaker 03 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.
14:52 Speaker 01 Well, so I want to dive into that now. This set of ideas related to a thing called the Grassmannian and the positive Grassmannian, 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 Grassmannian, Start us off, how should we think about it in the simplest cases?
15:18 Speaker 03 Yeah, so the the Grassmannian, it's a geometric object. And it's a bit like a library for keeping track of simpler objects. So for example, the Grassmannian of one planes in two space is the set of all lines through the origin in the X Y plane. And so we're we're thinking of this as a sort of one mathematical object that's keeping track of all of these lines. And so you can imagine all these lines passing through the origin in two dimensional space. And then the positive part of this Grassmannian would be the set of those lines that go through the positive orthant.
16:00 Speaker 01 Let me think about these lines. If I were to get a compass, there's a line that goes north and south.
16:05 Speaker 03 Yes.
16:05 Speaker 01 I think of that as one line. So north and south are both part of the same line.
16:10 Speaker 03 Yes, exactly.
16:11 Speaker 01 And east and west are also part of the same line.
16:13 Speaker 03 That's right. And then there's sort of infinitely many lines that we can get by tilting either of those lines.
16:19 Speaker 01 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?
16:28 Speaker 03 Yes. 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.
17:04 Speaker 01 Which on a compass would just have little markings on it that say north, northeast. We could even have north, north, north, northeast.
17:10 Speaker 03 Yes.
17:10 Speaker 01 I mean, that's 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?
17:26 Speaker 03 That's right, we should only include one of them. 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.
17:39 Speaker 01 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?
17:52 Speaker 03 Exactly.
17:53 Speaker 01 So is it like a line segment with one endpoint, but not the other endpoint?
17:56 Speaker 03 Exactly. Yes, that would be the Grassmannian of one plane in two space. That's how we can think about it.
18:02 Speaker 01 You could think of it as being like a half-open interval?
18:05 Speaker 03 Yes.
18:06 Speaker 01 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 endpoint, but I shouldn't. Not if I want the Grassmannian.
18:16 Speaker 03 Well, we sort of identify those two points. So it's almost like, we could think about sort of gluing those together.
18:25 Speaker 01 Okay. And then for the positive Grassmannian, you only want the positive orthant?
18:30 Speaker 03 Right. So first, the Grassmannian 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 Grassmannian, then we're just looking at the intersection of that hemisphere with the positive orthant. And so it's actually going to look like a triangle, like a kind of a curvy triangle.
18:58 Speaker 01 Okay. And this word orthant, which isn't totally familiar, is the 3D version of quadrant.
19:02 Speaker 03 Exactly.
19:03 Speaker 01 Right? There's eight of them, so that's why you're saying orthant.
19:06 Speaker 03 Yeah, yeah. So that positive orthant would be where the x, y, and z coordinates are all positive or non-negative.
19:14 Speaker 01 Okay. 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.
19:25 Speaker 03 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, Loestig 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 Grassmannian. And so it was just sort of a purely interesting mathematical idea to try to study total positivity,
20:06 Speaker 03 not just for matrices anymore, but for geometric objects like the Grassmannian. So there's all kinds of sets of matrices that describe motions and symmetries for the real world.
20:18 Speaker 01 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.
20:29 Speaker 03 Yes. In math, one of the common themes is that we study not just the mathematical objects, but also the relationships between the objects. And matrices can give us a way to create or to analyze relationships between different mathematical objects.
20:47 Speaker 01 Now, one of the results that you are known for was this positive Grassmannian 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.
21:02 Speaker 03 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. You know, we can go through life with a combinatorial attitude and take a combinatorial approach to different problems. And the positive Grassmannian 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
21:44 Speaker 03 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, six, 12, and eight, to a cube. That's what a combinatorial list might do. Now, there are infinitely many positive Grassmannians, and they can have arbitrarily high dimension. But the first problem that I worked on in graduate school 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 KNN, tells you how many pieces there are of each dimension in that positive Grassmannian.
22:28 Speaker 01 Okay, so let's now make a little swerve from this pretty abstract realm of matrices and Grassmannians and positive Grassmannians to the much more mundane world of traffic and waves on the ocean and proteins being made inside of cells, because it turns out all those things can be viewed as part of one story.
22:49 Speaker 03 That's right. There have been sort of three different areas that I've had personal experience with where the positive Grassmannian got connected. During my postdoc, I learned that another mathematician, Sylvie Courteille, had written a paper which said that my polynomials that were counting pieces of the positive Grassmannian 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 "real world." Her result was quite beautiful.
23:30 Speaker 03 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 N 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. That actually kicked off a decades-long collaboration with Sylvie. I mean, we've written a number of papers together by now.
24:18 Speaker 01 So there's a lot to unpack there, Janet. Did that wash over you?
24:22 Speaker 02 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. Should we say dimensionality? So having heard all of that, which is very intriguing, just give me the bird's eye view of what a Grassmannian is.
24:44 Speaker 01 Okay, fair enough. Right, it's not a concept we run into every day. So here's what it is. There'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?
24:57 Speaker 02 Yeah.
24:57 Speaker 01 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.
25:12 Speaker 02 So it's one shape or it's a family of shapes?
25:15 Speaker 01 It's a family of shapes. There's different Grassmannians. 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, a 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
25:51 Speaker 01 Linear spaces through the origin of n-dimensional space.
25:58 Speaker 02 Okay.
25:58 Speaker 01 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.
26:09 Speaker 02 So n is 2, k is 1?
26:12 Speaker 01 Right. That would be the 1-2 Grassmannian, or something like that.
26:15 Speaker 02 Okay.
26:16 Speaker 01 Okay? And so there's infinitely many, a whole continuum of lines. But if you wanted to parameterize them in our language, if you wanted to catalog them, you could do it by saying, what's their compass direction? Like, 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.
26:47 Speaker 02 And so people study different Grassmannians, some high-dimensional space and some lower-dimensional?
26:54 Speaker 01 Exactly. Now, Lauren specializes in this piece of it that's called the positive Grassmannian, 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. At 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
27:37 Speaker 01 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. That would be like our n. And now people start arriving to get in line, and also people at the front of the line can leave. And the rules of the game are that whatever spot you're on, you have some probability of moving forward one spot, except not if someone's standing there. There's a constraint. If you let this whole thing run for a long time with people arriving at random and leaving at random and moving forward one spot at random when they can, you could classify all the possible ways that these 10 spots could be occupied. By four people, let's say, that would be the K.
28:22 Speaker 01 It turns out, the four-ten Grassmannian tells me something about the likelihood of seeing a particular number of people in this queue.
28:32 Speaker 02 Now, I'm curious, I can imagine during COVID, as you said, having to solve this problem. Right? It's a problem that has to be solved because we now have distribution centers for vaccines, and this is happening, or something like that. How does somebody notice that the polynomial that they've generated to answer this practical question happens to be the same as a polynomial a very abstract mathematician has found for a Grassmannian on the positive, with positive Grassmannian? I mean, how do they even notice this correlation?
29:05 Speaker 01 That might be the unique genius of Lauren Williams and her collaborator Sylvie Courteille. And it's not just about the queues. If you think about ribosomes moving down an mRNA molecule as they're doing protein synthesis, it also pops up in that setting. You see what I'm gutting at? There's a really fun, diverse set of applications, all mysteriously falling under the heading of the positive Grassmannian. But after the break, Lauren Williams will walk us through why this phenomenon might be happening, why it's happening so pervasively, and also, how artificial intelligence may or may not take over mathematics.
30:02 Speaker 01 Welcome back to The Joy of Why. We're speaking with Harvard mathematician Lauren Williams about algebraic combinatorics and the positive Grassmannian.
30:13 Speaker 01 I would love to ask about why these connections to the much more mundane world happen. I know that no one knows.
30:22 Speaker 03 Well, you know, what's really interesting to me is that, you know, with the model of traffic flow, it's this model of particles that repel each other. And with the shallow water waves, these are waves that are sort of coming together and interacting. And with the scattering amplitudes, it's about particles that are being thrown together and interacting. And somehow, it's always about particles or waves that are being flung together and then they sort of repel in some way. And, you know, the coordinates one uses for the Grassmannian are Plucker coordinates. And if you have your K-Bijou matrix, you think of this as a, as a list of column vectors. Well, if two, two vectors get so close that they're actually on top of each other, your Plucker coordinate vanishes. So there's something
31:06 Speaker 03 in the nature of the Plucker coordinates on the Grassmannian that build in this repelling property. And so I've always wondered if it could be possible to connect these three different settings, whether it's the particles repelling each other, or the waves, or the scattering amplitudes. But somehow, I think it all comes down to Plucker coordinates on the Grassmannian.
31:25 Speaker 01 That's nice. That's a very, very nice answer. And, I mean, it feels like a more refined way of saying what you said at first, because I've seen diagrams, like when you look at the particles in the Feynman diagrams, veering towards each other and then bouncing off. Or if you look at the diagrams of the water waves, where you're just looking at, I don't know what, the crests or something, there's ways of drawing the pictures that it almost looks like you're drawing the same picture over and over.
31:52 Speaker 03 Right.
31:52 Speaker 01 Yeah. So, I mean, it would be weird, but then again, maybe not so surprising that if, at a very deep level, we're drawing this same picture over and over, and nature is interpreting it, or math is interpreting it, in different ways, in different settings, but it's kind of the same mechanism. But your thing with the Plucker coordinates and the, does the zero mean that that's the analog of repulsion? They won't go through each other because of that zero?
32:16 Speaker 03 Well, they could, but then there's a sign change. And then somehow like with the shallow water wave stuff, I was thinking about why positivity comes into the picture, you know, to analyze these solutions, to analyze these water waves, you use soliton solutions to the KP equation. And that involves a tau function in which you take the log of a certain function. And this function is built out of the Plucker coordinates in some way. And as long as the Plucker coordinates are all non-negative, you're taking the log of something that's always positive. But if you lose this positivity, if you now are talking about all points in the Grassmannian, not just the positive part, you might at some point be taking the log of zero or something really close to zero.
32:59 Speaker 03 But then what happens is that your model for shallow water waves goes off to plus or minus infinity, which obviously does not represent the real world. And so there's something about, you know, if you want to stay in the real world, you have to stay away from this zero. And it means restricting to the positive Grassmannian. So, yeah.
33:17 Speaker 01 But if we were to just get a little sloppier, but I think maybe more understandable about it, is it that there are sort of a bank of possible patterns that can happen in our minds or in nature. And sometimes those patterns just, you know, if they're fundamental enough, they will show up in many parts of our thought and in our observations. So like, there's a certain family of patterns that you are swirling around and this positive Grassmannian story is encoding them. And they have different manifestations in math and in the world, but it's kind of the same pattern over and over.
33:51 Speaker 03 Yeah, maybe that's right. Maybe that's right. I mean, the Grassmannian is so universal. And then there's something about positivity that just captures properties of the real world for some reason.
34:01 Speaker 01 This story is not over, because then somehow you get entangled, naturally I use that word, with things happening in quantum physics, specifically with things related to a very beautiful quantum field theory. N equals four supersymmetrics Yang Mills theory.
34:20 Speaker 03 Yes, yes.
34:21 Speaker 01 If I've got that right. But anyway, Nima Arkani-Hamed and other collaborators are looking at this fantastic model, and somehow you connect to them. You want to build that bridge for us?
34:33 Speaker 03 Yes. So they started to realize that somehow the structure of the positive Grassmannian was helping to understand scattering amplitudes. So scattering amplitudes are basically probabilities that tell you what you might expect would happen if you throw a bunch of particles with given momenta together and more particles come out. Well, I guess that sort of classical approach to scattering amplitudes was to use some complicated diagrams called Feynman diagrams. But the physicists, Nima Arkani-Hamed and collaborators, realized that there were more compact ways to understand these scattering amplitudes, and they involved
35:18 Speaker 03 a lot of the machinery of the positive Grassmannian. Yeah. And then this in turn led to a beautiful geometric object that they call the amplitude hedron, whose volume computes scattering amplitudes.
35:33 Speaker 01 Before we start delving into the amplitude hedron, if I'm saying that right, there was one question I had about something in doing a little background reading that you mentioned these Feynman diagrams. It's a wonderful technique for calculating the kinds of information that physicists need to try to match what they see in their experiments or to make predictions about future experiments. But it can be very arduous. There could be thousands of diagrams, sometimes even more, that they have to calculate on computers. And the crazy thing that seems to have come out in this amplitude hedron story, as done by the physicists, is that
36:11 Speaker 01 The thousands of calculations can be reduced sometimes to one calculation. That is, when you mentioned calculating a volume, it's analogous to finding a volume of a shape. And it seems like a miracle. How could a thousand or a million things be replaced by one thing? And it reminded me of cancellations that I teach when I teach calculus. There's something we teach, and you probably have to teach calculus from time to time, too. We call it telescoping series, where there's a series of terms and then on the inside there's a lot of things being added and then subtracted again, and added and subtracted, and they all collapse. And I feel like, from what I read, that in your picture, because we talked about positive
36:54 Speaker 01 as an adjective applied to the Grassmannian, that when you have this positivity extrus thing thrown in there, it somehow gives rise to this kind of, it's not the same cancellation as in a telescoping series, but it feels like it has that flavor. A lot of internal cancellation, simplifying a big messy thing to something much simpler. Am I on the right track with that? I mean, even morally, if not in detail?
37:19 Speaker 03 So there are many cancellations that occur when one goes from kind of Feynman diagram expressions to the sort of most compact expressions that we know. A big advance in this area was the recurrence of BCFW, Brito, Kachazo, Fang, and Witten. And they wrote down this beautiful and much more compact recurrence for computing scattering amplitudes. And then what was noticed a few years later by a physicist named Hodges was that in some special cases, if you take the recurrence and you express your amplitude as a sum of terms, it looked like the sum of terms was computing the volume of some geometric object by cutting it into pieces and adding up the volumes of those pieces.
38:04 Speaker 03 So this was an observation of Hodges in a few very special cases. And then he asked the question, is this true in general? Can we write all of these scattering amplitudes as computing volumes of some geometric object by cutting them up into pieces and summing them up? So Neymar, Konohamed, and Yaroslav Trinka. Invented slash discovered the amplitude hedron as the answer to this question. So they defined this object and it's closely related to the positive Gresmonian. And they proposed in their 2013 paper that this was the answer to Hodges' question. The volume of this object is indeed computing the scattering amplitudes in question.
38:47 Speaker 01 So maybe we should close our discussion here by just going into a little bit of what you've been doing very recently in connection with a project known as First Proof. Can you fill us in on what this project is about and what you're trying to do with it?
39:01 Speaker 03 Yeah, so First Proof is a project that we initiated in the fall and the motivation and the idea was to try to come up with an objective measure of how good AI systems are at coming up with proofs of mathematical statements. There's been a lot of noise in the media, either sort of hyping up the ability of AI or denigrating it. And we thought mathematicians themselves should try to figure out how best we can use AI in our own research. And in particular, to figure out how good AI is it coming up with proofs of statements.
39:39 Speaker 03 But this is a very tricky thing to test because LLMs, AI models, are extremely good at searching the literature. So if you ask your favorite AI model to come up with a proof of a mathematical statement, if that statement and proof are on the internet somewhere, it's going to find it. So we wanted to know how good is it at coming up with new proofs that aren't already out there. And so what we decided we needed to do was take mathematical statements, lemmas, say, from our own research where we had proved the lema or the statement, but we had not released the solution on the internet anywhere, and propose these kinds of statements as problems, as a challenge for AI systems.
40:24 Speaker 03 So a group of 11 of us got together and produced These kinds of problems from our work and put them out on the internet in a paper on February 6 as a challenge for AI systems.
40:37 Speaker 01 That's February 6, 2026, for people in the future listening to this.
40:41 Speaker 03 That's right, yes. And then what we did at the time was we wanted to sort of make clear that we had solved these problems ourselves. We encrypted our solutions. We put the encrypted solutions on the internet. And then we said that we would release the key to the encryption. We'd release the solutions publicly in one week's time. And so during that time, we were really gratified to see that there was just an incredible amount of interest, both from the mathematical community, like professional mathematicians or math aficionados, but also from the big AI companies, you know, jumping on the challenge and seeing what they could do.
41:15 Speaker 01 Yeah, because these are not like the Olympiad problems or the high school math contest problems or anything like that. These are really research level questions, but bite sized.
41:26 Speaker 03 That's right.
41:26 Speaker 01 Right? As you say, they're lemas.
41:28 Speaker 03 That's right.
41:28 Speaker 01 Not the whole paper.
41:29 Speaker 03 Right, right, right. So this was a new kind of challenge because, as you said, most previous benchmarks consisted of problems with numerical answers as opposed to answers that consisted of proofs. So with all of our problems, we made sure that we had proofs that were roughly five pages in length or less.
41:49 Speaker 01 And how did the AIs do? Is it possible to assess?
41:54 Speaker 03 Yeah, so we did our own private assessments at the time that we came up with these 10 questions. And actually, designing the protocols around testing is also a tricky thing to do because you could give an AI model one shot to answer the question. You know, you could just give it the problem and see how it does. Or one could have an extended conversation with the model and try to coax it to give a better answer. But so in our private tests that we did beforehand, we just gave each AI model one shot to answer the question. We didn't have any back and forth. And what we found at that time was that the models could solve two of our 10 questions.
42:36 Speaker 01 Oh, okay. That's not bad.
42:37 Speaker 03 Yeah.
42:37 Speaker 01 These are hard questions.
42:38 Speaker 03 Yeah, no, yeah, not bad.
42:40 Speaker 01 Two out of, okay.
42:40 Speaker 03 And during that week. Various individuals and also people with the companies were working on the problems and coming up with solutions. And if you sort of put together the best efforts from all of the different people and groups who submitted answers, we did get perhaps correct solutions to six of the 10. But we are trying to shy away from making any formal statements about how people or groups did because we didn't lay any ground rules. Since different people in different groups and different companies would have had different procedures and different amounts of feedback, it's hard to sort of compare how the models did.
43:23 Speaker 01 And so now you have, very recently, it was only a few days before our conversation right now, you released what you're calling, what are you calling it?
43:34 Speaker 03 The second batch.
43:35 Speaker 01 The second batch?
43:37 Speaker 03 Yes. First proof is a baking pun. It's about proofing the dough before you bake it. And so we, we put out, you know, our first batch of problems back in February. And just a few days ago, on March 14, 2026, on Pi Day, we put out an announcement that we will release a second batch of problems sometime later in the spring. They will similarly be sort of bite sized problems from different areas of mathematics, coming from research of mathematicians. But this time, we mean for our problems to be a more formal benchmark. And we do intend to get the solutions graded at the end.
44:16 Speaker 01 Okay. Well, this will be interesting to see. Are there any discoveries about either of the things we really talked about, the Gromovian and its relatives, or this AI work you most hope to see, say, 10 years from now?
44:32 Speaker 03 As far as the Gromovian goes, I'm hopeful that maybe there's even more exciting connections to other parts of the real world. And as far as the AI models go, it's very hard for me to predict. You know, it feels like the ecosystem in which we're doing math is being upended. And we're trying to figure out how best to adapt, how we can use these new tools. I would hope that 10 years from now, they would be sort of research partners with a sort of higher level of reliability and confidence than we have at the moment.
45:08 Speaker 01 All right. And the last thing, is there Something you could put your finger on that particularly is a source of joy for you as a mathematician, what brings you joy in your work?
45:18 Speaker 03 I think it's identifying connections between things I didn't expect to be connected. You know, just finding these kinds of connections, whether it's to the traffic flow or to the shallow water waves or to the scattering amplitudes. I have so many stories from my research where I might have a conversation with another mathematician and they show me some, some numbers of something that they were computing and then I recognize them as having come up before. It's always so exciting and intriguing. I mean, it's the sort of mystery and then we have to do the detective work of figuring out how these objects are connected. Yeah. So I think that's the thing that I find most exciting.
46:00 Speaker 03 And then, of course, the joy is when you realize you make that connection, you understand, you have this realization of how they are secretly connected and how you can sort of make that rigorous.
46:09 Speaker 01 Well, very, very good. It's really been fun. Thank you, Lauren.
46:13 Speaker 03 Thank you, Stephen.
46:14 Speaker 02 Wow. So this is terrifying, right? I really do wonder, hey, have I written the last of my very, very technical papers? But then I also remember the time that computers were first invented and everybody was saying that's not, I don't remember what computers were first invented. But you know what I'm saying? When they got cheaper, more readily available, large processing machines could do huge data sets and the same kind of thing was said, well, now the technical people are obsolete. I don't know. What do you think?
46:43 Speaker 01 Well, I'm confused about it. I really am of two minds and, you know, we still have CAPTCHA, that thing where you have to identify that you're not a robot by doing some little image processing. Apparently that's still hard for the AIs. So yes, they're very good at certain things, but there's still a way to go. They also seem to lack common sense in a lot of domains. But still, back to your question, though. I mean, will they make you and me and people like us obsolete? Because what we do isn't exactly the realm of common sense. We're in a, as my wife would be the first to tell you.
47:19 Speaker 02 Right, exactly. And she'd be right.
47:23 Speaker 01 No, but you know, like a lot of the work we do involves very explicit rules. You could imagine we might be at risk more than the people who do plumbing or caregiving, who will be the last to be superseded by robots and AIs.
47:36 Speaker 02 Well, just to play devil's advocate, I think the idea of these machines as thought partners is closer to what I'm imagining is going to happen. Because I still don't see the machine asking the questions.
47:48 Speaker 01 Not yet, no. Do you think in the era when AI starts doing math alongside us, or maybe even instead of us, will beauty play the same role then? Like a guide to what you should think about, what questions you should ask, how to judge whether you're on the right track with the theorems you can obtain?
48:07 Speaker 02 Gosh, it's a really profound question, because one of the roles beauty might be playing is rendering some very complex subject comprehensible to us. Which I really need, because I don't have infinite compute. So I need to have a more aesthetic approach. So, I mean, you could kind of say, in a way, nature already has all the answers. The whole game is discovering what nature already knows. So if the AI just simply has this infinite list of things it knows, you know, if we don't understand it, I don't know that the game has changed that much. I don't know, what do you think?
48:42 Speaker 01 I always wonder about, is understanding overrated?
48:48 Speaker 01 So here's what I mean, that we might be confusing means and ends. Like if the end is to predict nature, to be able to find formulas and theorems that are true, understanding may be a crutch. It helps us get good answers. It helps us get more control over the universe. But it's not the game. Like if you're trying to save someone's life, you may have to come up with a medical therapy that you don't understand that works. And so it's not always so clear to me that understanding is the goal in itself. But on the other hand, there are people who say it's not science without understanding. It's something less than science. It's like a degradation of the human spirit. Why even do it if you're not understanding?
49:33 Speaker 01 I don't know what to think about that. I can see both sides of that argument. But what's really interesting in what Lauren Williams and her colleagues are doing is they are giving these secret problems from research-level math that haven't been published, so the AI can't look them up on the internet, and asking them, how many of our 10 problems can you solve? It's just an interesting benchmark, different methodology than we're seeing elsewhere.
49:57 Speaker 02 Yeah. Yeah, it's amazing because it means they're not just regurgitating, culling a human response.
50:03 Speaker 01 So far, they're not mastering that. They're not climbing the whole mountain.
50:07 Speaker 02 But right now, in the context of what people are doing, is it possible to have a machine that says, you know, here's an interesting idea? Or, you know, I'm bored today, I'm gonna try this, or...
50:19 Speaker 01 We'll really know that they've arrived when they're a guest on The Joy of Why.
50:23 Speaker 02 Yeah, when we have Claude on.
50:25 Speaker 01 Yeah, when we have Claude, and of course, by then, maybe we won't be the hosts anymore.
50:29 Speaker 02 Yeah.
50:32 Speaker 02 Oh, man.
50:33 Speaker 01 But until then.
50:34 Speaker 02 Until then.
50:35 Speaker 01 See you later, Jenna.
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50:55 Speaker 01 The Joy of Why is a podcast from Quantum Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests, or other editorial decisions in this podcast or in Quantum Magazine. The Joy of Why is produced by PRX Productions. The production team is Caitlin Falls, Jade Abdul Malik, Genevieve Sponsler, and Merritt Jacob. The executive producer of PRX Productions is Jocelyn Gonzalez. Edwin Ochoa is our project manager. From Quanta Magazine, Simon France and Samir Patel provided editorial guidance,
51:35 Speaker 01 with support from Samuel Velasco, Simone Barr, and Michael Kenyon Golo. Samir Patel is Quanta's editor in chief. The episode art is by Chanel Nibbelink, and our logo is by Jackie King and Christina Armitage. Special thanks to Garth Avery at the Cornell Broadcast Studio. I'm your host, Steve Strogatz. If you have any questions or comments, please email us at [email protected] .
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