This transcript is provided as a courtesy and may contain errors.
Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I’m a freelance math and science writer based in Salt Lake City, but I’m currently recording in Chicago at the Mathematical Association of America’s annual summer meeting MathFest. Because I am on location here, I am not joined by our cohost Kevin Knudson, but I’m very honored to be in the same room as today’s guest, Dusa McDuff. I’m very grateful she took the time to talk with me today because she’s pretty busy at this meeting. She’s been giving the Hendrick Lecture Series and been organizing some research talk sessions. So I’m very grateful that she can be here. The introductions at these talks have been very long and full of honors and accomplishments, and I’m not going to try to go through all that, but maybe you can just tell us a little bit about yourself.
Dusa McDuff: OK. Well, I’m British, originally. I was born in London and grew up in Edinburgh, where I spent the first twenty years or so of my life. I was an undergraduate at Edinburgh and went to graduate study at Cambridge, where I was working in some very specialized area, but I happened to go to Moscow in my third year of graduate study and studied with a brilliant mathematician called Gelfand [spelling], who opened my eyes to lots of interesting mathematics, and when I came back, he advised that I become a topologist, so I tried to become a topologist. So that’s more what I’ve been doing recently, gradually moving my area of study. And now I study something called symplectic topology, or symplectic geometry, which is the study of space with a particular structure on it which comes out of physics called a symplectic structure.
EL: OK. And what is your favorite theorem?
DM: My favorite theorem at the moment has got to do with symplectic geometry, and it’s called the nonsqueezing theorem. This is a theorem that was discovered in the mid-80s by a brilliant mathematician called Gromov who was trying to understand. A symplectic structure is a strange structure you can put on space that really groups coordinates in pairs. You take two coordinates (x1,y1) and another two coordinates (x2,y2), and you measure an area with respect to the first pair, an area with respect to the second pair, and add them. You get this very strange measurement in four-dimensional space, and the question is what are you actually measuring? The way to understand that is to try to see it visually. He tried to explore it visually by saying, “Well, let’s take a round ball in four-dimensional space. Let’s move it so we preserve this strange structure, and see what we end up with.” Can we end up with arbitrary curly shapes? What happens? One thing you do know is that you have to preserve volume, but apart from that, nothing else was known.
So his nonsqueezing theorem says that if you took a round ball, say the radii were 1 in every direction, it’s not possible to move it so that in two directions the radii are less than 1 and in the other directions it’s arbitrary, as big as you want. The two directions where you’re trying to squeeze are these paired directions. It’s saying you can’t move it in such a way.
I’ve always liked this theorem. For one thing, it’s very important. It characterizes the structure in a way that’s very surprising. And for another thing, it’s so concrete. It’s just about shapes in four dimensions. Now four dimensions is not so easy to understand.
EL: No, not for me, at least!
DM: Thinking in four dimensions is tricky, and I’ve spent many, many years trying to understand how you might think about moving things in four dimensions, because you can’t do that.
EL: And to back up a little bit, when you say a round ball, are you talking about a two-dimensional ball that’s embedded in four-dimensional space, or a four-dimensional ball?
DM: I’m talking about a four-dimensional ball.
DM: It’s got radius 1 in all directions. You’ve got a center point and move in distance 1 in every direction, that gives you a four-dimensional shape, it’s boundary is a three-dimensional sphere, in fact.
EL: Right, OK.
DM: Then you’re trying to move that, preserving this rather strange structure, and trying to see what happens.
EL: Yeah, so this is saying that the round ball is very rigid in some way.
DM: It’s very round and rigid, and you can’t squeeze it in these two related directions.
EL: At least to preserve the symplectic structure. Of course, you can do this and preserve the volume.
EL: This is saying that symplectic structures are
DM: Different, intrinsically different, in a very direct way.
EL: I remember one of the pictures in your talk kind of shows this symplectic idea, where you’re basically projecting some four-dimensional thing onto two different two-dimensional axes. It does seem like a very strange way to get a volume on something.
DM: It’s a strange measurement. Why you have that, why are you interested in two directions? It’s because they’re related. This structure came from physics, elementary physics. You’re looking at the movement, say, of particles, or the earth around the sun. Each particle has got a position coordinate and a velocity coordinate. It’s a pairing of position and velocity for each degree of freedom that gives this measurement.
EL: And somehow this is a very sensible thing to do, I guess.
DM: It’s a very sensible thing to do, and people have used the idea that the symplectic form is fundamental in order to calculate trajectories, say, of rockets flying off. You want to send a probe to Mars, you want to calculate what happens. You want to have accurate numerical approximations. If you make your numerical approximations preserve the underlying symplectic structure, they just do much better than if you just take other approximation methods.
DM: That was another talk, that was a fascinating talk at this year’s MathFest telling us about this, showing even if you’re trying to approximate something simple like a pendulum, standard methods don’t do it very well. If you use these other methods, they do it much better.
EL: Oh wow, that’s really interesting. So when did you first learn about the nonsqueezing theorem?
DM: Well I learned about it essentially when it was discovered in the mid-1980s.
DM: I happened to be thinking about some other problem, but I needed to move these balls around preserving the symplectic structure. I just realized there was this question and I couldn’t necessarily do this when Gromov showed that one really could not do this, that there’s a strict limit. So I’ve always been interested in questions, many other questions coming from that.
EL: Another part of this podcast is that we like to ask our guests to pair their theorem with another delight in life, a food, beverage, piece of art or music, so what have you chosen to pair with the nonsqueezing theorem?
DM: Well you asked me this, and I decided I’d pair it with an avocado because I like avocados, and they have a sort of round, pretty spherical big seed in the middle. The seed is sort of inside the avocado, which surrounds it.
EL: OK. I like that. And the seed can’t be squeezed. The avocado’s seed cannot be squeezed. Is there anything else you’d like to say about the nonsqueezing theorem?
DM: Only that it’s an amazing theorem, that it really does underlie the whole of symplectic geometry. It’s led to many, many interesting questions. It seems to be a simple-minded thing, but it means that you can define what it means to preserve a symplectic structure without using derivatives, which means you can try and understand much more general kinds of motions, which are not differentiable but which preserve the symplectic structure. That’s a very little-understood area that people are trying to explore. What’s the difference between having a derivative and not having a derivative? It’s a sort of geometric thing. You actually see surprising differences. That’s amazing to me.
EL: Yeah. That’s a really interesting aspect to this that I hadn’t thought about. In the talk that you gave today was that the ball can’t be squeezed but the ellipsoids can. It’s this really interesting difference, also, between the ellipsoids and the ball.
DM: Right. So you have to think that somehow an ellipsoid, which is like a ball, but one direction is stretched, it’s got certain planes, there are certain discrete things you can do. You can slice it and then fold it along that slice. It’s a discrete operation somehow. That gives these amazing results about bending these ellipsoids.
EL: That’s another fascinating aspect to it. You I’m sure don’t remember this, but we actually met nine years ago when I was at the Institute for Advanced Study’s summer program for women in math. I’m pretty sure you don’t remember because I was too shy to actually introduce myself, but I remember you gave a series of lectures there about symplectic geometry. I studied Teichmüller theory, something pretty far away from that, and so I didn’t know if I was going to be interested in those. I remember that you really got me very interested in doing that many years ago. I was really excited when I saw that you were here and I’d be able to not be quite so shy this year and actually get to talk to you.
DM: That’s the thing, overcoming shyness. I used to be very shy and didn’t talk to people at all. But now I’m too old, I’ve given it all up.
EL: Well thank you very much for being on this podcast, and I hope you have a good rest of MathFest.
DM: Thank you.