Amie Wilkinson of the University of Chicago works in the rarefied area of mathematics called pure dynamics, studying how complex systems transform under the influence of simple rules. In this episode, she speaks with her fellow dynamicist, host Steven Strogatz, about the challenges of finding a place in mathematics as a woman, why groups can be understood as collections of moves, and what the recipe for puff pastry illustrates about chaos.
Steve Strogatz: What could be more fun? Two mathematicians.
Amie Wilkinson: Two dynamicists.
Strogatz: I know. So, I’ve been thinking about that. I mean, to me, it shows the vastness of the subject of math. That here’s dynamics: It’s a corner of the subject, and yet even within that corner, there’s your whole world. I haven’t really lived in your world, and I get the feeling you’ve only just recently started to put your toe into my world.
Wilkinson: Yeah. I kind of feel that way, too.
Strogatz: I’m thinking of myself as doing applications of dynamical systems to science, biology, physics, social science, and I think of you as doing dynamics within math itself.
Wilkinson: Right, right. You might say pure dynamics, I guess.
Strogatz: Yeah. Pure. I love it.
Steve Strogatz [narration]: From Quanta Magazine, this is The Joy of x. I’m Steve Strogatz. In this episode, Amie Wilkinson.
[INTRO MUSIC PLAYS]
Strogatz: I think one of the defining characteristics of Amie Wilkinson — she is so vigorous. She is a person of motion and energy, and it’s really fitting, because her subject is dynamics, which is the study of how things move. But in a very abstract sense, she’s a pure mathematician, a pure dynamicist, and it’s really interesting to me, because I also am interested in dynamics and motion, but from a very applied perspective. And so, sparks fly. We have some fun together.
One of the things that has obsessed Amie in her career is the study of chaos, which, as it sounds like, is unpredictable, seemingly random motion. But what’s so interesting in the kind of chaos that she studies is that it’s governed by perfectly predictable, deterministic rules. It’s just that the consequences of the rules may not be predictable in the long run. She looks at chaos very geometrically, very visually, and she uses iron-clad logic to extract predictions and conclusions about the things she studies that really often defy intuition. Some of her work is truly mind-bending, and yet it’s as tight as a proof from Euclidian geometry.
So, Amie and I had this conversation during the pandemic, which meant that we had to improvise a little bit, since we couldn’t go into the studio. I was upstairs in my attic. She was downstairs in her basement.
Wilkinson: The thing about math is, if it’s something that you’re good at and interested in, it often gets — although not always — but it often gets identified pretty early. So, I was a math kid starting, probably, in kindergarten.
Strogatz: No kidding. That’s interesting. How does that happen? Was someone giving you puzzles? Are your parents the type who would ask you to think about arithmetic when you were five years old?
Wilkinson: No, not at all, but I was in — no.
Strogatz: Not at all, huh?
Wilkinson: They weren’t really like that, but I was in a Montessori school for preschool and kindergarten. And I don’t know if you know about their methods, but there’s a lot of mathematics incorporated into the tools that they use, and it’s a play-based model. And I still remember the location of the various toys. I totally do. It’s funny.
Strogatz: Really? You can picture the box that had all the —
Strogatz: — I don’t know what. What kind of things?
Wilkinson: Well, there were the number chains. They were made of beads. It was sort of designed to teach children how to count in different bases and also how to visualize a number, a number squared, a number cubed. And so, I mean, it was almost like a logarithmic scale secretly being taught. There were these individual beads that you would line up ’til you got to the numbers. The 10-chain took the most work, so of course that was the one I liked. So, you would line up 10 beads until you got to 10, and then you would put a little marker that said “10,” and then there were beads arranged in a line of 10 on a piece of wire. And so, then you’d line up 10 — or nine more of those until you’d done 100 beads, and you put a little marker. And then there were squares of beads, and you’d line up the squares of beads. But you would use the beads to count up to 1,000, and at the very end, your prize was to put down a 10-by-10-by-10 cube with 1,000 beads on it.
Strogatz: Oh, 1,000. I see.
Wilkinson: And you would’ve in the process constructed this long chain of beads, maybe, I don’t know, 10 feet long or something.
Strogatz: So, you were curious about patterns, it sounds like, with these number chains. I mean, of course it’s always very pat to try to see the adult in the child, but then again there’s often some truth to it. Do you think that your five-year-old self was a harbinger of yourself today?
Wilkinson: I mean, in some sense, I’m still doing the 10-chain. I don’t know. Well, I mean, that’s a huge component of what I do as a mathematician. Not all pure mathematicians think that way, but that’s what sets me apart, I think, from maybe other scientists, or what applied mathematicians do.
Strogatz: So, you think of yourself as a builder. In what sense? You’re building imaginary worlds with mathematical patterns underlying them? Is that what you mean?
Wilkinson: Well, they don’t feel imaginary. So, it’s some cross between discovery and building that is my experience of doing math.
Strogatz: What’re you building?
Wilkinson: Well, I can be building an example of some mathematical phenomenon, an example of something that displays certain features that sort of exists in the mathematical universe. Or I could be building a proof. And usually, it’s kind of a cross between… I’m usually not saying, “I’m going to prove this thing,” and then I set about building the proof. I mean, what I prove sort of depends on what I discover along the way.
Strogatz: I was really interested in hearing about Amie’s own trajectory to becoming the mathematician that she is today. And that journey took her through what turned out to be something of a trial by fire as a university student at Harvard.
Wilkinson: A lot of people who go to Harvard experience this: just realizing you’re not the smartest person in the room. It’s where you have to learn how to get over yourself. It was a little hard being a woman in mathematics in the 1980s in college, in a university, and especially a university that didn’t have any women on the faculty, research faculty, someone to look up to, and just sort of say, “Oh, yeah, right, this is normal. I belong.”
Strogatz: So, at Harvard, though, I take it you were probably one of those whiz-kid freshmen, and you were put in some maybe rigorous linear algebra class or something like that.
Wilkinson: Yeah. I took the advanced linear algebra class and calculus class, and I did very well in that class, because I already knew all of that; I’d done it before. It wasn’t until my sophomore year when things started to fall apart. The director of undergraduate studies when I was there was — he probably was quite sexist, because I’ve heard stories of what he had told other people. For example, when I told him I wanted to go to graduate school, which I had decided at the very end of my experience at Harvard, and he launched into this whole description of how the world of mathematics is a giant pyramid, and there’s very few people at the top, and there are a lot of people at the bottom. And as long as you don’t mind being at the bottom, be my guest. [LAUGHS]
Oh, okay, yeah. Thanks for the encouragement. I have to say, though, that probably did spur me to go get a [BLEEP] Ph.D. and — sorry. I’ve got to watch my language, but that was pretty discouraging. But he had said, in fact, far more discouraging things to other women. I think he just was very old-school, “There’s a pyramid,” which is kind of not true.
Strogatz: I’m interested in that. You would say it’s not true. It doesn’t look like that to you today?
Wilkinson: Well, yes, in some ways it does, in the sense that the inequities you see in things like income are also seen in the kind of jobs that people have in academia. So, you have this huge number of people in untenured positions who are treated like [BLEEP], and then the very lucky few who have tenured professorships at Group One research institutions. So, in some sense, I belong to the lucky few, but I don’t think that this pyramid is so much based on merit. I mean, to some extent it is, but a lot of it is luck.
Strogatz: Yes. I have to agree with you on that one. Yeah.
Wilkinson: [BLEEP] I would never say something like that to an undergraduate. Saying there’s a pyramid implies that there’s sort of one linear direction that is up, and higher is good. But like any other discipline, it’s always multidimensional, the ways that you can measure people’s talents or gifts.
Strogatz: That’s so true. I love that vision of yours that — I mean, it’s very geometric, as suits you, I think, in some ways. That it’s such a simplistic thing, to think of it as a one-dimensional line with a direction from lower to higher. So, I was a freshman at Princeton, which would be pretty analogous to your being a freshman at Harvard, and —
Wilkinson: And what year was this, by the way?
Strogatz: So, I was a freshman in 1976-’77, and I took the whiz-kid linear algebra class. The book that we used for this subject of linear algebra didn’t have any pictures.
Strogatz: Yeah, I think literally the whole book had no pictures, and, yeah, that didn’t work for me as a visual kind of person. And the subject actually is extremely visual. Anyway, I was completely creamed in that course. I couldn’t understand anything. I was really intimidated, because I saw other kids raising their hand, and they seemed to know what the answers were. And so, I really got this feeling of imposter syndrome. Very interesting.
Wilkinson: I’m interested in that you said that, because what really kind of knocked me off my feet was abstract algebra. The teacher in the algebra course — so he was doing group theory. Okay, so, group theory to me is dynamics. It’s dynamical. Where do groups come from? Symmetry groups, motion. It’s extremely visual. But he taught it in a very symbolic way, and when he got to the idea of co-sets, I just … I couldn’t get it. I just didn’t get it. I couldn’t understand what kind of thing a co-set was, and I got knocked off my feet.
Strogatz: So, should we try this? Let’s do a little math over the airwaves here. What do you mean by group theory?
Wilkinson: Let’s give it a try. A group is an abstract mathematical object invented by mathematicians, but really, what it is, is it’s a collection of moves that you could do. And you can combine the moves and get a new move. So, a group is a collection of moves. Let’s say you’ve got a triangle. That’s —
Strogatz: Like an equilateral triangle.
Wilkinson: So, an equilateral triangle, right. So how can you move that triangle? So, you take those corners, which are called vertices, and you label them one, two, three. And you imagine all the ways that you can move that triangle to get another triangle. You might end up rearranging the numbers on the… For example, I could take that triangle, and I could rotate it by 60 degrees or — which one —
Strogatz: Yeah, right, 120, yeah.
Wilkinson: 120. You could rotate it by 120 and then line the new corners up to each other. And that would be a move, and it would change the numbers. Okay?
Strogatz: So, the only allowed moves are things that put the triangle back on top of itself.
Wilkinson: Back on top of itself. So, you could rotate the triangle by 240 degrees, and that would be a different move. But if you rotated the triangle by 360 degrees, you’d be back to the original triangle. So that’s a move, but it’s what you would call the identity. It’s a special move that doesn’t change anything. But I could also take a triangle, and I could flip it. So, I could sort of pick it up, turn it over and put it back down, and one of the numbers, the number at the top, stays the same, but the other two numbers switch. That’s another move. So, anyway, you could combine these moves to get other moves, and the set of all moves is called a group, and it has properties. The thing is that everyone sees groups. Even little grade-school kids see groups. The groups they see is numbers. So, you can think of numbers as being moves. For example, the number 1, you can think of as, “I take my number line, and I add one to everything in the line, and I move it to the right.”
Strogatz: Okay, so rather than thinking of 1 as a quantity, you think of 1 as, “Take a step to the right from wherever you are.”
Wilkinson: Take the whole line and move it to the right, yeah.
Strogatz: Yeah, and that’s interesting the way you say it. You think of it very holistically, that it’s not just like I started at 0, and now I take a step to the right. You think of every point, what —
Wilkinson: I just take —
Strogatz: — of the whole numbers?
Wilkinson: — all the numbers, and I’ve got my hands on that number line, and I move it to the right. That’s 1.
Strogatz: So, an infinite number of numbers all shift one step to the right.
Wilkinson: All shift. Yeah. And then zero is just — you don’t move it at all.
Strogatz: If you did the zero move.
Wilkinson: Zero move. And the -1 move is just “move everything to the left.” So, if you think of groups as moves, then they’re very natural things. If you start thinking in this way, it’s very easy to understand concepts and groups. If you think of groups as being generalizations of numbers, then a lot of things are very confusing. The thing about numbers is, they’re just — they represent so much more than just moves. And so, if you try to think of them as just a group, you don’t realize what the rules are. So —
Strogatz: You’re saying, because numbers are so overloaded already with other kinds of meanings and associations, whereas moves are just moves.
Wilkinson: Moves are just moves. If you always think of your group in terms of moves, then you don’t do stupid things. So, for example, numbers are what’s called an Abelian group: 2 + 9 equals 9 + 2.
Strogatz: Right, and this was the thing I was missing in multiplication. In my case, the multiplications of numbers also form a certain kind of group. So, 2 × 9, I didn’t realize, was 9 × 2.
Wilkinson: Right, and that kind of thing is just very rarely true with groups. If I take my triangle and I flip it, I rotate it and then I flip it again, that’s not the same as the rotating.
Strogatz: Yeah, it’s interesting. So, I mean, a different way to say what you just said is, you could flip and rotate, or you could rotate and flip, and the order matters.
Wilkinson: Yes. Elements of groups aren’t quantities. They don’t behave like quantities. They behave like moves, and everyone’s comfortable with the idea that if you do one thing and then you do another thing, it’s not the same as — you can’t switch the order. You put on your underwear.
Strogatz: Yeah, often.
Wilkinson: You put on your pants. It’s not the same as the pants and the underwear.
Strogatz: Yes, that’s a good one.
Strogatz: When we get back, some really flaky ideas about puff pastry. That’s ahead.
[MUSIC PLAYS FOR BREAK]
Strogatz: You are known for work in an area of math called dynamics or dynamical systems, which, it seems to me, has a lot to do with the notion that you’ve just been talking about with moves. But tell us. First of all, what do you mean by a dynamical system, and what’re the moves that would come into play there?
Wilkinson: Well, one kind of abstract concept I have to throw out there is the notion of a space, which could be just a collection of the vertices, the corners of a triangle, or it could be something very complicated like a collection of all configurations of the solar system in position and velocity, which would be a very high-dimensional space. So, there’s a notion of a space — or maybe just the surface of a ball could be a space. And then there’s usually just one move that you single out that you want to study. So, you could take the ball, and you could rotate it a little bit in some direction, and that’s your move, and then you’re stuck.
You want to study what that move does to that space. So, if you were studying the solar system, you want to say that the move is, “I’m just going to apply the law of physics, and I just want to —” and it’s in some configuration. The sun is here. This planet’s here. This planet’s here. They’re moving in this direction at this speed, and then the law of physics sort of tells you where that’s going to be, say, a minute later.
Strogatz: Right, that’s the move.
Wilkinson: That’s the move.
Strogatz: Just let the clock tick. Yeah, let the clock tick —
Wilkinson: Yeah, let the clock tick.
Strogatz: — and everything’ll move a little.
Wilkinson: Exactly. But, of course, then you’re in a different configuration. So, you’re going to move in a very different way, possibly, depending on where you are, what configuration you’re in. But that’s it. You’re given a space, and you’re given some move, and you want to see what happens in the long term.
Strogatz: This is such a fantastic mathematical concept, space, but when — I think when a lot of folks hear “space,” they’re thinking about outer space, or it could mean, “I’m walking around in this nice space in my basement.” But I think when you’re talking about space, you mean a collection of possibilities, in a certain sense, right? It’s sort of the totality of all the ways that something can be. That together, collectively, forms this collection as a space of possibilities, or something.
Wilkinson: Right. So, the collection of possibilities is a way of explaining “state space,” which is the type of space that would arise if your dynamical system came from physics. I really just mean “set,” and that set could be — have lots of structure, and be very complicated. It could be the three vertices of a triangle. It could be a circle, or it could be a collection of possibilities of a physical system. So, here are all the possible positions and velocities of these planets that are existing in some ideal system with no external forces you have to ignore, or else — you have to simplify. But —
Strogatz: Right. To say, “This is the thing I’m studying, and I’m going to ignore the effect of Alpha Centauri,” I mean, because everything in the universe influences everything else. But that’s too much to think about all at once.
Wilkinson: Right. Dynamics is just studying a single move that you do on a set, and seeing what happens when you do that move over and over and over and over. And if you think of doing that move over and over as being the evolution of time, then you can address a question like, “Given the current configuration of our solar system, how long will it take until Mercury goes retrograde?” or whatever. For how long will this be stable? So, that’s a long-term kind of dynamical question, not the questions that I — type of question I look at, but that’s the nature of the questions that I look at.
Strogatz: Usually, we hear about that in the world of horoscopes and stuff. For the astrologers, they’re very interested in, “where is Jupiter, and where is Mars, and where are they in relation to each other? Is this one in that one’s house?” All that stuff. But it’s like, “Where are all the planets?” But this is on time scales of human life, over years or months. You’re talking about after millions of years, will Jupiter just fly away and not be part of the solar system anymore? Or will Mercury go flying into the sun or something like that?
Wilkinson: It’s great. I can bring up these examples, because you’re an applied dynamicist, so you’re like my Google or my Wikipedia. I can pose these, and then you can answer what’s the state of the art.
Strogatz: Well, I wish.
Wilkinson: These are certainly questions people have worked on for ages, yeah.
Strogatz: So, what I love about what you’re doing is that, to me, you’re a kind of an artist who cleans away the… Who cares about Jupiter, really? I mean, yes, it’s interesting in some ways, if you’re interested in astronomy or astrology, but you — for you, the points are a little more disembodied. You’re interested in pure motion. It’s very stripped down, and yet it’s still so rich, because there’s a lot that can happen when you just repeat a motion again and again and again.
So, I have an example I think might be fun for us to talk about, that’s sort of in the spirit of what you were talking about with twisting the sphere. But I find it’s a visual that might be helpful here to get us into some of the nitty-gritty of your work, which is to take dough and then a rolling pin. Imagine I’m going to make pastry or — right? So, you want to take it from there?
Wilkinson: Well, sure. Yeah, I was thinking of the puff-pastry croissant. So, this is a beautiful way of describing a very strongly chaotic set of moves. The moves are very simple. You take your dough. You roll it out so that — it starts out in a square. You roll it out, so it’s a longer rectangle. You make it three times longer, and then you fold the top in, and the bottom in the way you would fold an envelope. But you can picture, if you made it three times longer, that when you fold the top and the bottom like an envelope, you get a square again.
Strogatz: Oh, that’s nice.
Wilkinson: So that’s a move on a square of dough.
Strogatz: Oh, that’s nice. That’s like the way you said with the equilateral triangle. When you rotate it, it still looks —
Wilkinson: Yeah, it still looks like a triangle.
Strogatz: — like you’ve just folded this thing, so that the square is still on top of itself, but you stretched it out by three and then folded the —
Strogatz: Okay. I like it. That’s a nice —
Wilkinson: So, you took a square of dough. You began with a square of dough, and you ended with a square of dough, the same square of dough. So, you could just do that over and over again, and, in fact, that is what pastry-makers do when they make puff pastry. They roll it out to three times the length. In the middle, you put a little square of butter, and then you fold the top and the bottom, and that’s the thing you start with. Okay, so it’s not quite just a piece of dough. It’s a piece of dough with a little packet of butter in the middle. And then to make it into puff pastry, you repeat this move of rolling it out, folding it in. Now you have a square again. Roll it out. Fold it in. You usually refrigerate it in between moves.
Strogatz: Wait. Do I put a piece of butter in every time or only the first time?
Wilkinson: No, just basically the first time. And so, what happens is, you always end up with a square of the same thickness, but it looks different if you look at it from the side. You started with just one chunk of butter, but the effect of these moves over and over is to distribute this butter. Think about it after one move. So, you have dough, butter, dough, dough or something. So basically, you have two pieces of dough and some butter in the middle. First, roll it out so you have a long piece of butter with dough on top and dough on bottom. But now when you fold it, that butter gets folded so there’s three layers of butter —
Strogatz: That’s great.
Wilkinson: — separated by dough.
Strogatz: Very delicious. This is good.
Wilkinson: It’s getting good, right? If you think about it, the next iteration, you’ll have 9 layers of butter separated by dough, and then 27. Each time you multiply the number of layers by three, and you stop at a certain point, and then you can roll it out and cut it into shapes, and what happens is in the oven, the butter melts, and it produces steam, and it separates the layers of the dough. And so, it puffs up.
Strogatz: Oh, that’s where the puff comes from in puff pastry is the steam?
Wilkinson: Yeah. The fat plays some important role in making the layers crispy as well, but it’s the steam from the liquid in the butter.
Strogatz: The end result of this fantastic process of the stretching by three and then repeated folding, and doing this over and over, is you get this very flaky structure made of very thin sheets. But there was something you said in an interview I read somewhere that I really liked, which was that one of the things you find beautiful about dynamics is that the shapes that result — in this case, the puff-pastry shape — contains … I don’t know quite how you said it. It was something like, the shape itself gives you a kind of history, or leaves a trace of what the process was that made it.
Wilkinson: Yeah, yeah, exactly.
Strogatz: I mean, this is actually a metaphor for some of the math that you think about, isn’t it?
Wilkinson: Yeah. It’s a good metaphor, because this move, this rolling up and folding, is a type of dynamical system that we call hyperbolic. And the word hyperbolic is used, because if you kind of look at what happens in cross-section to this square of dough, you have one direction that’s kind of being stretched out every time you do a move, and another that’s sort of getting skinnier, right? Because when you roll something out to three times its length, keeping the width fixed, it gets a third of the thickness, because you — conservation of dough.
And so, this combination of stretching in one direction and contracting in the other is an example of what we call hyperbolicity, and the main feature of a hyperbolic system is that you have this type of stretching and contracting behavior, kind of everywhere you look. And when you have that kind of hyperbolicity, it leads to what is popularly known as chaos.
Strogatz: You say popularly known —
Wilkinson: It’s chaos.
Strogatz: — like you don’t want to go slumming with us.
Wilkinson: Well, it’s a joke, because everyone uses the term “chaos,” but it’s not really defined. I would say there’s no fixed definition.
Strogatz: I guess that’s the problem, right?
Wilkinson: Then in the movie Jurassic Park, Jeff Goldblum described himself as a chaos theorist, which I don’t think exists. I mean, “chaos” is a decent term. What it’s indicating is that, when you apply your set of moves over and over again, there’s a high amount of unpredictability in the long term.
Strogatz: How so, in this case?
Wilkinson: In the case of the rolling dough? Yeah. I mean, if you think about where an individual molecule of butter ends up in the long term —
Strogatz: That’s it, right?
Wilkinson: Or I wish we had such things as units of butter, so we could say, “Well, where’s this little butter guy going to end up versus his neighbor?”
Strogatz: Oh, yes, I see.
Wilkinson: So, you could have two little pieces of butter that start out close together and end up in very different layers. So, you imagine each piece of butter has an address. If you do this 100 times, there’s going to be 2100 layers, right, in theory. So, each molecule of butter gets his own address for where he ends up, and what does chaos mean? It means where one molecule ends up and where his neighbor ends up are seemingly unrelated.
Strogatz: It’s wild.
Wilkinson: Isn’t that wild?
Strogatz: I mean, just this simple thing done over and over — yeah. You started with all the — it could be a very tiny pat of butter originally, so in that sense it’s like we know the state of where the butter is to a very good approximation. Yet after a small number of these operations, these moves, the butter could be essentially anywhere in the universe defined by the puff pastry. That’s pretty chaotic. And yet, the basic mechanism that generated it is very basic.
Wilkinson: It’s very simple.
Strogatz: It’s just repeated rolling and stretching and compression and keep doing it.
Wilkinson: That’s right.
Strogatz: I feel like that’s one of your signature moves, speaking of moves. The Amie Wilkinson move is to look for a general mechanism, sort of the hidden pattern underneath the chaos.
Strogatz: It’s really satisfying for me, as a person who also works in dynamical systems, to listen to Amie talk about this geometric picture that she has for the general mechanisms underlying chaos. We’re not really interested in puff pastry. Why are we talking about this? Because puff pastry is this metaphor that captures the essence of what we’re trying to understand when we simulate the turbulent motion of the atmosphere or the motion of hurricanes across the sea, and where are they going to hit land? All those things are very difficult to predict, and it seems puzzling, in a way, because we can predict eclipses. We can predict lots of other things that seem as complicated as the weather, but there’s something about chaotic systems that makes them hard to predict, and Amie gets at the essence of what that is with her geometric and visual pictures.
Think about this blob of dough that is being subjected to Amie’s imaginary rolling pin. When you push down on the dough with the rolling pin, part of the dough is going to get compressed. The part that’s under the pin is getting squashed. Part of it is getting stretched, the part that’s getting rolled out. And part of it, the third direction, nothing is really happening. That’s the direction along the length of the rolling pin. When you roll the rolling pin, you stretch out the dough in the rolling direction. But sideways along the pin, the dough doesn’t get pushed sideways to the left or right.
Wilkinson: The third direction: So, if it’s not stretched or contracted at all like in the perfect description I was giving, you stay in your lane. So, the butter doesn’t move. So, if there was any sort of unevenness when you started in the butter layer in that direction, it doesn’t get rectified by this rolling process. But if you allow yourself a little wiggle room — and this is a theme of my work. If you allow yourself a little wiggle room, you mix up that direction as well.
Strogatz: What does that mean, “allow yourself”? What is this theme?
Wilkinson: It’s a theme that I actually learned from my adviser, Charles Pugh, and his collaborator, Mike Shub, who was also a collaborator of mine. And that is, that a little hyperbolicity goes a long way to produce mixing and other kind of chaotic behaviors. So, partial hyperbolicity only has the stretching and the contracting in these two directions, and the idea is, well, it’s not always going to be mixed up in that third direction, but if you just are a little bit sloppy in how you roll in that third direction, that’ll also get mixed up, because just a little bit of sloppiness in the third direction allows the hyperbolicity in the other two directions to mix things up.
Strogatz: Can I try to say what I think I’m hearing?
Strogatz: I mean, I don’t know that I have the right picture. The picture I’m having in my head is, you could imagine a robotic cook, who has a rolling pin and always rolls it perfectly in one direction. That’s not what you’re going to do. You’re going to be sloppier than that. You’re going to have someone who’s trying to roll the pin in a straight line but who gets distracted by something on the television. And so, they roll kind of off in a sort of little bit sideways — I don’t know. Am I going in the right direction with this?
Wilkinson: Yes, you’re going in the right direction with this. Or if you look at how puff pastry is made industrially, they have this infinitely — this endless thin stream of dough that’s been rolled to a thinner length, and they roll it on top of itself. They kind of imagined having a very long piece of dough and just folding it, a continual sheet of thin dough. And you fold it on top of yourself, but you kind of go back and forth. Imagine you has as much dough as you wanted, and you took the thing, and you folded it back, but you kind of didn’t come back to where you started. You kind of shifted a little bit.
Strogatz: I see. So that’s the sloppiness.
Wilkinson: A little sloppy. You do it over to the side, and then you sort of do it —
Strogatz: Yeah, a little to the side.
Wilkinson: — over to the side again. Then you do it over to the side, and you do it — and so, you kind of — it kind of overlaps itself. So, it’s kind of stretching, but it’s also — there’s some shearing or kind of — and then you stop at a certain point, and you roll it thin again, and you cut it into a long sheet. I don’t know how these things work, but sort of, that moving things over a little to the side and not exact — that’s enough to mix things up in that third direction as well.
Strogatz: I see. I see. So that’s what you mean about the theme, that a little bit of hyperbolicity goes a long way. You don’t have to be wildly careful in that third direction. Just a little bit gets the job done.
Wilkinson: It gets the job done, because once you move something over a little bit, then you have this stretching, and you can sort of picture it. The chaotic motion in the other directions just spins it to some other random place, and then it gets moved to the next layer. And so, there’s some — yeah.
Strogatz: Does this have anything to do with this photograph I’ve seen of a pillow in your office, or is that something else?
Wilkinson: Yeah, it does.
Strogatz: So, what is that? Tell us about the pillow.
Wilkinson: Okay. I have to describe it. So, it looks like hair. It looks like I’ve taken a photograph of kind of mathematical hair, and it’s sort of spilling down from the top to the bottom of the pillow. And each strand of hair is like a curve, and so I call — so I wrote — I produced this weird mathematical hair with my friend/collaborator Mike Shub, and we wrote a paper about it, and we had some boring title for the paper, “A Stably Bernoullian Diffeomorphism That Is Not Anosov,” and the —
Strogatz: Yeah, that’s not catchy.
Wilkinson: Not catchy, and the referee kind of thought the interesting thing about this paper is not the stably Bernoullian that is not Anosov but this weird foliation that we produced that has these bizarre properties. And so, it suggested that we kind of change the focus, and so Mike came up with the title, “Pathological Foliations and Removable Zero Exponents.” So, he coined this term, pathological foliation.
Strogatz: Getting better. Yeah, that’s getting better.
Wilkinson: Much better. Much better. So, that is a picture of a pathological foliation.
Strogatz: So, when Amie speaks about pathological foliations, remember that in a foliation, we’re talking about leaves, and you shouldn’t think of leaves that fell off a tree. You should think of leaves of a book, sheets like the sheets in the puff pastry, the many flaky layers. And what’s pathological about pathological foliations is that the sheets don’t sit together nicely. They don’t behave like the pages in a book that’re all orderly. They get kind of crinkly and nasty and fractal and complicated in a way that captures the essence of the chaos that’s going on underneath, but that is very difficult to think about visually and geometrically. Yet, that’s exactly what Amie managed to solve with her math.
Strogatz: So, foliation here, the leaves are like the hairs?
Wilkinson: Yes. It doesn’t translate very well, does it? You don’t really think of leaves —
Strogatz: Not so well.
Wilkinson: — as being hairs. But maybe, better, is to think of the leaves of a book, okay? They stack one on top of the other.
Strogatz: That’s right.
Wilkinson: And if you were to actually look at a book sideways, you’d see lines, little lines, one for each page. So, that would be an example of a foliation by curves. There would be lines. So, now imagine taking your book and kind of warping it a little bit, and so the lines get all kind of — they’re warped, and they all fit together to make a book. But they’re no longer straight. Okay, now picture that on a pillow, and that’s the pathological foliation.
Strogatz: It’s a really nice-looking textile, actually. I mean, “pathological” makes it sound like something you wouldn’t want on a pillow, but it’s very pretty, I thought.
Wilkinson: I think it’s pretty, yeah.
Strogatz: Next time on The Joy of x, quantum physicist Charlie Marcus tries to demystify the wave/particle duality.
Charlie Marcus: Someone shows you a squirrel, and you say, “Oh, is it a rat, or is it a cat? I don’t understand. Is that a rat, or is it a cat?” The answer is, it’s not a rat or a cat. It’s a squirrel.
Strogatz: The Joy of x is a podcast project of Quanta Magazine. We’re produced by Story Mechanics. Our producers are Dana Bialek and Camille Peterson. Our music is composed by Yuri Weber and Charles Michelet. Ellen Horne is our executive producer. From Quanta, our editorial advisors are Thomas Lin and John Rennie. Our sound engineers are Charles Michelet and at the Cornell University Broadcast Studio, Glen Palmer and Bertrand Odom-Reed, who I like to call Bert.