On this episode of My Favorite Theorem, I had the immense pleasure of talking with Dusa McDuff in person! We met up in July at MathFest, the annual summer meeting of the Mathematical Association of America. I was too shy to introduce myself to her the first time I saw her give a talk, so I was glad to rectify that this time. You can listen to the episode here or at kpknudson.com.
McDuff is currently a professor at Barnard College. Before that, she was at Stony Brook University for many years. She has won major prizes, including the Satter Prize, Senior Berwick Prize, and Steele Prize, and received honorary doctorates from several universities. She was at MathFest to give the Hedrick lecture series and co-organize research talk sessions in symplectic geometry.
Symplectic geometry is a cousin of complex geometry. Complex here doesn’t mean complicated but refers to the complex plane, a two-dimensional plane with a real axis and an imaginary axis, where complex numbers like 2+3i live. There’s also a 4-dimensional version of the complex plane, and 6-dimensional, and 8-dimensional, and so on. Likewise, symplectic geometry can inhabit spaces of any even dimension.
A symplectic structure on an even-dimensional space is a way of getting measurements from pairs of coordinates. For, say, a 4-dimensional space, you take the two pairs of coordinates and get an “area” measurement for each pair. Then you add those areas. As McDuff says in the episode, it seems like a strange choice, but it ends up being “a very sensible thing to do.” The motivation to take such a measurement actually comes from physics, where the two quantities can represent the position and velocity of an object.
One topic of study in symplectic geometry is what kinds of transformations of a shape preserve its symplectic structure. The theorem McDuff chose as her favorite, the non-squeezing theorem, is a result in this direction. As Tara Holm describes in this math graduate student-level introductory article about symplectic geometry (pdf), the non-squeezing theorem can be interpreted as saying that a camel cannot fit through the eye of a needle, at least as long as camel and needle are symplectic. The theorem was proved by Russian mathematician Mikhail Gromov in the 1980s.
McDuff discussed the 4-dimensional version of the theorem: you can’t fit a (4-dimensional ball) into a (4-dimensional) cylinder while preserving the symplectic structure unless the radius of the cylinder is a least as big as than the radius of the ball. It’s hard to think in four dimensions, you can visualize it in three dimensions instead. (I won’t tell anyone.) In three dimensions, if you have a ball with a little give to it-maybe a bean bag or a basketball-you can fit it into a cylinder narrower than the diameter of the ball. If the ball is very soft-made of putty, perhaps-you can fit it into a very thin cylinder, as long as the cylinder is long enough, by stretching it. The non-squeezing theorem says that symplectic structures are more rigid than putty balls: you can’t squeeze them even a little bit and preserve the structure. Even though symplectic geometers are fond of pointing out how “floppy” symplectic structures are, this theorem shows that they’re much more rigid than something like volume.