Did it feel like a given that you would go to college?
It felt kind of given, in that I always did well in school. And my parents were very supportive. It happened that in Nevada, they created a program called the Millennium scholarship, so students who went to Nevada high schools and then Nevada colleges would get a large portion of their tuition covered. So that made it easy. But the majority of my high school friends didn’t go to college.
By the end of high school, I knew mathematics was what really tickled me. So I was very much looking forward to a place where instead of having one calculus class, I could take four math classes. I didn’t have any picture in terms of a career. I knew I enjoyed learning math in the moment, so I went to college because I could continue learning it.
Did you feel like you fit in at college, as the first person in your family to go there?
I was a commuter student for the first half of college, and then the second half I just had an apartment. So I never had that college dorm experience. And most of my social life was spent with this group of friends I had through high school.
I think the much larger cultural adjustment, which was pretty difficult at the time, was starting grad school at Stanford, because UNR [University of Nevada, Reno] and Stanford are such different worlds. Stanford was an exposure to that world of generations of professors — my classmates whose father is a professor and whose grandfather was a professor. I didn’t feel like there was ever any hostility directed at me; it was just a foreign environment.
That’s when you started working in symplectic and contact geometry. What drew you to that field?
You can think of the different fields of geometry as existing along a spectrum from the most rigid to the most flexible. And what really attracted me to symplectic and contact geometry is that it is somewhere in the middle. I find that middle intriguing, because it’s very mysterious. And it’s also where a lot of the most visual geometry happens. When you go to a totally flexible world, it’s hard to explain why, but everything becomes algebra in a sense. And when you go into an extremely rigid world, so much depends on precise measurements. Whereas in between, visual thinking is more useful.
Something else I like is that it’s a very young field. Symplectic geometry has only been seriously studied for maybe 35 years, so people don’t know what’s going on all that well. Because of that, it brings in all these other fields and throws them in a mixing pot. And that makes it compelling.
What kind of structures does symplectic geometry deal with?
The roots are in classical mechanics. And one of the most important aspects of classical mechanics is that if I have some system, maybe a pendulum or the motion of the planets, then as long as I understand the energy for all possible configurations, I can deduce how that system evolves with time.
If we abstract that into a geometric structure, energy is just a function from the space to the real numbers, whereas the time evolution is a symmetry of the space. Classical mechanics gives you a way that, for any energy function, you get a symmetry. But if I have some random geometric space, it’s not clear how to do that. A symplectic structure is the ingredient which allows you to make that translation.
So is it about constructing worlds in which classical mechanics is allowed to behave differently from what we’re used to?
Yeah, it’s abstracting into a very foreign world. One way in which symplectic geometry is more general is, it works over any notion of energy.
With Einstein and relativity, a big insight is that space and time don’t really exist as separate entities so much as there’s this one thing called the space-time continuum. In classical mechanics, you see something similar, in that the equations can’t tell the difference between position and momentum. So when we’re building these abstract symplectic manifolds, a lot of these spaces don’t have separate notions of what position and momentum are.
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