My Favorite Theorem

Evelyn Lamb: Welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I’m a freelance math and science writer based in Salt Lake City. And today I am not joined by my cohost Kevin Knudson. Today I am solo for a very special episode of My Favorite Theorem because I am at MathFest, the annual summer meeting of the Mathematical Association of America. This year it’s in Chicago, a city I love. I lived here for a couple years, and it has been very fun to be back here with the big buildings and the lake and everything. There are about 2,000 other mathematicians here if I understand correctly. It’s a very busy few days with lots of talks to attend and friends to see, and I am very grateful that Ami Radunskaya has taken the time to record this podcast with me. So will you tell me a little bit about yourself?

Ami Radunskaya: Hi Evelyn. Thanks. I’m happy to be here at MathFest and talking to you. It’s a very fun conference for me. By way of introduction, I’m the current president for the Association for Women in Mathematics, and I’m a math professor at Pomona College in Claremont, which is a small liberal arts college in the Los Angeles County. My Ph.D. was in ergodic theory, something I am going to talk about a little bit. I went to Stanford for my doctorate, and before that I was an undergraduate at Berkeley. So I grew up in Berkeley, and it was very hard to leave.

EL: Yeah. You fall in love with the Bay Area if you go there.

AR: It’s a place dear to my heart, but I was actually born in Chicago.

EL: Oh really?

AR: So I used to visit my grandparents here, and it brings back memories of the Museum of Science and Industry and all those cool exhibits, so I’m loving being back here.

EL: Yeah, we lived in Hyde Park when we were here, so yeah, the Museum of Science and Industry.

AR: I think I was born there, Hyde Park.

EL: Oh? Good for you.

AR: My dad was one of the first Ph.D.s in statistics from the University of Chicago.

EL: Oh, nice.

AR: Although he later became an economist.

EL: Cool. So, what is your favorite theorem?

AR: I’m thinking today my favorite theorem is the Birkhoff ergodic theorem. I like it because it’s a very visual theorem. Can I kind of explain to you what it is?

EL: Yeah.

AR: So I’m not sure if you know what ergodic means. I actually first went into the area because I thought it was such a cool word, ergodic.

EL: Yeah, it is a cool word.

AR: I found out it comes from the Greek word ergod for path. So I’ve always loved the mathematics that describes change and structures evolving, so before I was a mathematician I was a professional cellist for about 10 years. Music and math are sort of as one in my mind, and that’s why I think I’m particularly attracted to the kinds of mathematics and the kinds of theory that describes how things change, what’s expected, what’s unexpected, what do we see coming out of a process, a dynamical process? So before I state the theorem, I need to tell you what ergodic means.

EL: Yeah.

AR: It’s an adjective. We’re talking about a function. We say a function is ergodic if it takes points: imagine you put a value into a function, you get out a new value. You put that value back in to the function, you get a new value. Repeat that over and over and over again, and now the function is ergodic if that set of points sort of visits everywhere in the space. So we say more technically a function is ergodic if the invariant sets, the sets it leaves alone, the sets that get mapped to themselves, are either the whole space or virtually nothing. A function is ergodic, a map is ergodic, if the invariant sets either have, we say, full measure or zero measure. So if you know anything about probability, it’s probability 1 or probability zero. I think that’s an easy way to think about measure.

EL: Yeah, and I think I’ve heard people describe ergodic as the time average is equal to the space average, so things are distributing very evenly when you look at long time scales. Is that right?

AR: Well that’s exactly the ergodic theorem. So that’s a theorem!

EL: Oh no!

AR: No, so that’s cool that you’ve heard of that. What I just said was that something is ergodic if the sets that it leaves unchanged are either everything or nothing, so these points, we call them the orbits, go everywhere around the set, but that doesn’t tell you how often they visit a particular piece of your space, whereas the ergodic theorem, so there are two versions of it. My favorite one is the one, they call it the pointwise ergodic theorem, because I think it’s easier to visualize. And it’s attributed to Birkhoff. So sometimes it’s called the Birkhoff ergodic theorem. And it’s exactly what you just said. So if you have an ergodic function, and then we start with a point and we sort of average it over many, many applications of the function, or iterations of the function, so that’s the time average. We think of applying this function once every time unit. The time average is the same as the integral of that function over the space. That’s the space average. So you can either take the function and see what it looks like over the entire space. And remember, that gives you, like, sizes of sets as well. So you might have your space, your function might be really big in the middle of the space, so when you integrate it over that piece, you’ll get a big hump. And it says that if I start iterating at any point, it’ll spend a lot more time in the space where the function is big. So the time average is equal to the space average. So that is the pointwise Birkhoff ergodic theorem. And I think it’s really cool because if you think about, say, if you’ve ever seen pictures of fractal attractors or something, so often these dynamical systems, these functions we’re looking at, are ergodic on their attractor. All the points get sucked into a certain subset, and then on that subset they stay on it forever and move around, so they’re ergodic on that attractor.

EL: Yeah.

AR: So if we just, say, take a computer and start with a number and plug it in our function and keep iterating, or maybe it’s a two-dimensional vector, or maybe it’s even a little shape, and you start iterating, you see a pattern appear because that point is visiting that set in exactly the right amount. Certain parts are darker, certain parts are lighter, and it’s as if, I don’t know in the old days, before digital cameras, we would actually develop photographs. Imagine you put that blank page in the developing fluid, and you sort of see it gradually appear. And it’s just like that. The ergodic theorem gives us that magical appearance of these shapes of these attractors.

EL: Yeah. That’s a fun image. I’m almost imagining a Polaroid picture, where it slowly, you know, you see that coming out.

AR: It’s the same idea. If you want to think about it another way, you’re sort of experiencing this process. You are the point, and you’re going around in your life. If your life is ergodic, and a lot of time it is, it says that you’ll keep bumping into certain things more often than others. What are those things you’ll bump into more often? Well the things that have higher measure for you, have higher meaning.

EL: Yeah. That’s a beautiful way to think about it. You kind of choose what you’re doing, but you’re guided.

AR: I call it, one measure I put on my life is the fun factor.

EL: That’s a good one.

AR: If your fun factor is higher, you’ll go there more often.

EL: Yeah. It also says something like, if you know what you value, you can choose to live your life so that you do visit those places more. That’s a good lesson. Let the ergodic theorem guide you in your life. OK, so what have you chosen to pair with this theorem?

AR: So the theorem has a lot of motion in it. A lot of motion, a lot of visualization. I think as far as music, it’s not so hard to think of an ergodic musical idea. Music is, after all, structures evolving through space.

EL: Exactly.

AR: I think I would pair Steve Reich’s Violin Phase. Do you know that piece?

EL: Yeah, yeah.

AR: So what it is, it’s a phrase on the violin, then you hear another copy of it playing at the same time. It’s a repetitive phrase, but one of them gets slightly out of phase with the other, and more and more and more and more. And what you hear are how those two combine in different ways as they get more and more and more and more out of phase. And if you think of that visually, you might think of rotating a circle bit by bit by bit, and in fact, we know irrational rotations of the circle are ergodic. You visit everywhere, so you hear all these different combinations of those patterns. So Steve Reich Violin Phase. He has a lot of pattern music. Some of it is less ergodic, I mean, you only hear certain things together. But I think that continuous phase thing is pretty cool.

EL: Yeah. And I think I’ve heard it as Piano Phase more often than Violin Phase.

AR: It’s a different piece. He wrote a bazillion of them.

EL: Yeah, but I guess the same idea. I really like your circle analogy. I almost imagine, maybe the notes are gears sticking out of the circle, and they line up sometimes. Because even when it’s not completely back in phase, sometimes the notes are playing at the same time but at a different part of the phrase. They almost lock in together for a little while, and then turn a little bit more and get out again and then lock in again at a different point in the phrase. Yeah, that’s a really neat visual. Have you performed much Steve Reich music?

AR: I’ve performed some, mostly his ensemble pieces, which are really fun because you have to focus. One of my favorites of his is called Clapping Music because you can do it with just two people. It’s the same idea as the Violin Phase, but it’s a discrete shift each time, so a shift by an eighth note. So the pattern is [clapping]. One person claps that over and over and over, and the other person claps that same rhythm but shifts it by one eighth note each time. So since that pattern is 12 beats long, you come back to it after 12 beats. So it’s discretized. You do each one twice, so it’s 24, so it’s long enough.

EL: So that’s a non-ergodic one, a periodic transformation.

AR: Exactly. So that one I do a lot when I give talks about how we can describe mathematics with its musical manifestations, but we can also describe music mathematically.

EL: Just like you, music is one of my loves too. I played viola for a long time. I’ve never performed any Steve Reich, and I’m glad you didn’t ask me to spontaneously perform Clapping Music with you. I think that would be tough to do on the spot.

AR: We can do that offline.

EL: Yeah, we’ll do that once we hang up.

AR: As far as foods, I think there are some great pairings of foods with the ergodic theorem. In fact, I think we apply the ergodic theorem often in cooking. You know, you mix stuff up. So one thing I like to do sometimes is make noodles, with a roller thing.

EL: Oh, from scratch?

AR: Yeah. You just get some flour, get some eggs, or if you’re vegan, you get some water. That’s the ingredients. You mix it up and you put it through this roller thing, so you can imagine things are getting quite mixed up. What’s really cool, I don’t know if you’ve ever eaten something they call in Italy paglia e fieno, straw and hay.

EL: No.

AR: And all it is is pasta colored green, so they put a little spinach in one of them. So you’ve got white and green noodles. So when you cook some spinach, you’ve got your dough. You put some blobs of spinach in. You start mushing it around and cranking it through, and you see the blobs make these cool streaks, and the patterns are amazing, until it’s uniformly, more or less, green.

EL: Yeah.

AR: So I’d say, paglia e fieno, we put on some Steve Reich, and there you go.

EL: That’s great. A double pairing. I like it.

AR: You can think of a lot of other things.

EL: Yeah, but in the pasta, you can really see it, almost like taffy. When you see pulling taffy. You can almost see how it’s getting transformed.

AR: It’s getting all mushed around.

EL: Thank you so much for talking to me about the Birkhoff ergodic theorem. And I hope you have a good rest of MathFest.

AR: You too, Evelyn. Thank you.

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Show notes (click here)