My Favorite Theorem

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I am one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida, and your other host is…

Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where we sadly are past our beautiful, not too hot spring and fully into summer. So we enjoyed it while it lasted. I didn’t have to turn on any air conditioning until after the start of July.

KK: I think we started air conditioning in March.

EL: Slightly different.

KK: Little different vibe down here in Florida, but that’s where we are. So anyway, it’s summertime here, which means that there are tumbleweeds rolling through my department and I’m answering a few emails a day and trying to work, trying to do math. And boy, sometimes it’s hard, you know, but sometimes it isn’t. So. Anyway, so today, though, we are — this is great — we are very pleased to welcome Tatiana Toto, who will introduce herself and let us know what she’s all about.

Tatiana Toro: Thank you very much for the invitation. I’m very glad to be here. And in fact, I’m very glad to see Evelyn’s cloud that I had heard about in other podcasts. So I’m Tatiana Toro. I’m a mathematician at the University of Washington, where I have been a faculty member since 1996. And currently I am the director of the Simon’s Lab for Mathematical Sciences Institute, formerly known as MSRI. And I’m in Berkeley, California, and summer hasn’t arrived yet.

KK: It never will.

EL: Yeah, that’ll be November, right?

KK: I had actually forgotten that the name of MSRI had changed to the Simon’s business. That’ll take some getting used to. I think I mentioned before we started talking, I spent a semester there, way back in 2006, and my son came with me, and my wife did too, and he was seven at the time. And now he’s an adult living in Vancouver. It’s weird how things change. I love that building, though. And the panoramic view you have the bay, and you can watch the fog roll in through the gate at tea time. It’s just a really wonderful place. So congratulations. How long have you been director? Has it been a year yet?

TT: It’s almost a year, a year August first.

KK: Yeah. That’s fantastic. What a terrific position. And I’m glad that you’re willing to take it on. Do you split your time between Berkeley and Seattle? Or are you mostly in Berkeley these days?

TT: I am mostly in Berkeley. My students are still in Seattle, so I see them mostly on Zoom. But once in a while on a Friday, in Seattle.

KK: Oh, so you go there. You don’t fly them down?

TT: Some of them have come, actually one of them this year to the summer school.

KK: All right. So what is this podcast about? Favorite theorems. And you told us yours ahead of time, but we’ll let you share. What is your favorite theorem?

TT: Okay, so my favorite theorem is the Pythagorean theorem, and I know that everybody’s gonna say what on earth are you talking about?

EL: No, I really, really love this choice. And, you know, I’ve said this on many other iterations of this podcast, but I love that, you know, we’ll get things that span the gamut from Pythagoras theorem, or the infinitude of primes, or something like that, all the way up to something that you, you know, you need to have been researching for 20 years in some very ultra-specific field to even understand, and so, you know, it just like shows how math connects with us in different ways at different times in our lives, and how we can appreciate some maybe things that seem very simple about math, even when we have had math careers for for many years. So yeah, tell us about how did you end up settling on the Pythagoras theorem?

TT: So, actually, it has played a very important role in my career. Like, when I describe it to my students, when I’m teaching a graduate class and I talk about the some of the theorems I’ll describe in a minute, I tell them, you know, one of the key ideas in my thesis was the Pythagorean theorem. So let me explain. It appears in many other results in this area of geometric analysis. So for example — let me give you two examples. So what was my thesis about? You have a surface, a blob in space, and you’re trying to — two dimensions in R3 — and you’re trying to understand if you can find a parameterization, which means a good way to describe it in terms of the plane. So can you deform the plane in a nice way so that it covers the surface? And a nice way means that distances are not changed too much. So I had some specific conditions for this surface, and the answer, the key, is in the situation I was looking at, yes, you could do it. And when you go and deeply look at what makes this possible, it is the Pythagorean theorem because the basic point is that if you can control how distances are distorted, you can control how the whole shape is mapped from the plane. And at the time, it looked like a curiosity. You know, I graduated many years ago. At the time, a few years earlier, Peter Jones had solved the analyst’s traveling salesman problem, which I’m gonna — just in general terms, let’s imagine you have a lot of points in a square, and you’re trying to understand whether you can pass a curve of finite length to all of these points. You’re going to tell me, “If they’re finite, of course you can.” But you want to do it in an efficient way, in a way that doesn’t depend on the number of points. And so he had found the condition that told you if this condition is satisfied, then yes. And there’s not an algorithm, that doesn’t exist yet, that tells you what’s the best curve, but there’s a curve, and he tells you that the length is no more than something. And what’s behind that is the fact that if you have a straight triangle that has sides, A and B, and the other one is B, A squared plus B squared equals C squared. And it really is understanding that. And there’s another important thing, the fact that the square root also plays an important role in these, but really, really, if you ask me, “What are the tools you need in this area?” I’ll tell you how the square root behaves in the Pythagorean theorem, and then a couple of good ideas and you’re able to reconstruct the whole thing.

KK: I’m now curious about this traveling salesman problem. So there’s no algorithm though?

TT: No, there’s no algorithm. I used the word analyst’s traveling salesman problem because the analyst wants to know whether you can pass a curve of finite length. Maybe you can say you’re not ambitious enough. You don’t want the shortest possible curve. To build the shortest curve, there’s no algorithm. And the construction of Peter Jones builds a curve, but it’s not necessarily the best one.

KK: Sure. Yeah.

TT: It doesn’t tell you it tells you the length is no more than D. But it’s not. Yeah, no.

EL: Yeah. I’m trying to remember if, like, I think there probably are some algorithms or some, like results that say like, you can get within a certain percentage of something. But yeah, the algorithm for the actual fastest path doesn’t exist yet. Which is, you know, it’s one of those things, it’s like, huh, that’s kind of surprising that we don’t have a way to do that yet. Just means that there’s still work to be done. Still jobs out there for mathematicians.

KK: Well, because the combinatorial on the graph theory one is, is NP complete, right? I mean, yeah. So that that are NP-hard, or whatever. NP-something. I’ve never been clear about the differences. But is this one known to be that too?

TT: I believe.

KK: Okay. All right.

TT: But you can construct — you know, so this was what was interesting about the problem, the result of Peter Jones, is that — the result of Peter Jones, and I have to say, I was very ignorant of that result, which had just happened a few years prior to my thesis. I have to remind the young audience that at the time, there was no internet the same way, and there was no arXiv, and you know, there was no Zoom. And then Peter Jones had a couple of postdocs at Yale, Stephen Semmes and Guy David, who started working on this. And the truth is, may I tell story about my thesis?

EL: Yeah.

KK: Please do.

TT: So my thesis came out of misunderstanding. I went to my advisor, and I showed that these surfaces that I was looking at, which were some that he had looked at, that there was this property about distances over the surfaces, like if an ant traveled on the surface between two points, you know, taking the shortest path, it was comparable to the Euclidean distance. And so I went to my advisor, Leon Simon, and I told him, you know, I’ve been able to do this about these surfaces. And then he told me, oh, then I guess they have about they admitted bilipschitz parameterization, which is this good description. So okay, so I went to the library, and I looked through every possible book that I could find, and I couldn’t find that. So I went back two weeks later and asked if he’d mind giving me a reference for this results, and he said, oh, I don’t have a reference. That must be true.

KK: It must be true.

TT: And that became my thesis problem. And then, oh, there were many iterations of attempts. And I could do specific cases, but I could not do the general case. And on May of my fourth year, finally, somebody gives a colloquium where he talks about good parameterizations. And he talks about things like what I was thinking. I was thrilled. I mean, I thought, oh, I’m going go read everything this guy has written and my answer will be there. And then I told my advisor afterwards, I think I’m going to go read this guy’s work. And this guy was Stephen Semmes, and he comes from harmonic analysis. And my advisor says, no, stop reading, I don’t want you reading anymore. You just prove that theorem and that’s it. I don’t want you reading. But one good thing, you know, harmonic analysts use squares, rather than balls. That’s the most useful comment my advisor had.

EL: Huh!

TT: And what’s interesting is that Stephen Semmes was talking about a broader class of surfaces than mine. And for those, he was asking, “Do bilipschitz parameterizations exist?” And for those the answer still is not known. And if I had gone and read everything that he had written, I mean, he was the big shot, I was the student, I might not have gotten my result. And I remember when I told Stephen at some point in the fall, oh, you know, I proved this, his first question, his first reaction, was, “I don’t believe you.” And he said, “How did you do this?” And I said, “Using the Pythagorean theorem.” And so that’s why the Pythagorean Theorem really is very dear to my heart.

EL: Yeah. So I imagine that you saw the Pythagorean Theorem many years before you were in grad school. Do you remember, did it make a big impact on you when you saw it in school for the first time? I don’t know what what year that would have been, elementary or middle school or whatever it was?

TT: So I remember, I think I remember when I saw it because I remember the book. I had a beautiful — I went through the French system. I’m Colombian, but I went through the French system, and in the French system at the time, they tracked us very early on. And so we had these beautiful math book that, you know, I still remember how it smelled, and it was in there. But I remember the book, not especially the theorem. I never thought much about it until I got to graduate school. I used it other times.

EL: Right. I mean, I think maybe the beauty of that kind of thing isn’t necessarily what you’re looking at, when you’re a kid and first seeing math. You’re more like, okay, how can I use this to do the problems on the homework or something like that? So you were tracked into math pretty early on? You knew very early on that you were interested in math?

TT: Yeah.

KK: It’s nice they let you just do math. I think in the US what happens, I think, is students who are good at math are told they should be engineers. As if they’re kind of the same thing, and they’re not.

TT: But that, you see, now, you feel free to remove this if you want. That’s what the boys were told. The girls — since math was roughly like philosophy, and I come from a South American country, it was okay.

KK: That’s fascinating. Okay, interesting.

EL: Yeah. Well, I mean, there’s a lot of different, you know, philosophies about whether tracking that early, you know, kind of deciding on what direction you want to go that early, is good or not. You know, it works for some people and not others, definitely.

TT: Absolutely. I think it worked for me very well. And it didn’t work on any of my classmates who were in the same class. I mean, I thought everybody loved it the same way I did and had as much fun. And then, it’s interesting. Later on, I’ve learned that that wasn’t the case. And then some of them suffered through it, you know. But to me, it was great.

KK: So this is a French system in Colombia? Okay, this is a bit — okay, let’s get there. How did that actually happen? Why were there French schools in Colombia?

TT: Well, I’ll explain why there were French schools in Colombia and how I got into a French school. So there’s something that’s called a cooperation agreement between France and developing countries, where they have schools. The primary reason to have them is so the kids of their diplomats can continue their studies, but then they also offer them to the general population at a very reasonable price. They are private schools, but they are not as expensive. They’re a fraction, or they used to be a fraction, of what the other private schools were. And at the time, so Colombia for a long time was what was called a Sacred Heart country. And so the ties with the Catholic Church were very strong. And so in terms of education for the girls, it was most girls went to nun school. But I am not Catholic, and therefore I couldn’t, that was not an option for me. And so we needed a coed school. I mean, my parents wanted a coed school. The girls schools were all nuns. They wanted a coed school, and we needed an affordable coed school, and public schools were not good, and still unfortunately are not good. That’s how I landed in the French school.

KK: Fascinating.

EL: Wow. Okay. Yeah.

KK: Our listeners are learning all kinds of stuff, right?

EL: Yeah, yeah, we’ve wandered a little away. But luckily, we know, thanks to the Pythagorean theorem, that we can walk back in a certain amount of time. So yeah, the other things that we like to do on this podcast is have you pair your theorem with, you know, some food, beverage, sport, you know, whatever, delight in life you would like.

TT: So I actually will pair it with walking. So I’m going to give myself the title of urban hiker. I do walk long distances around town and in cities on a regular basis. I mean, I walk about two hours a day, at least. And so I pair it with that, because most often when I walk, I’m actually doing exactly the opposite of the Pythagorean theorem. I want to go the longest possible way, not the shortest possible way. But once in a while, I take the diagonal. And now that I’m living here in Berkeley, there’s a beautiful diagonal that I take. And so I think about that here often.

EL: Yeah. Do you like the hills?

KK: Yeah, I was about to say, do you actually hike all the way up to the building there? Because that is quite a hike.

TT: Not when I’m coming to work. But sometimes on weekends I do. You know, I want to crease and it depends. It depends what I’m doing while I walk. I use walking as a way — if I am listening to a book, then I can go up the hill. But if I want to talk on the phone, I need to go down the hill, because the reception here is terrible! I know exactly at what point on the hill, you lose AT&T.

KK: That’s true. Yeah, like I said, I was there some time ago and cell phones weren’t quite as good as they are now. But yeah, my reception was terrible at the institute.

TT: Well, your cell phone might have improved, but the reception hasn’t.

EL: Yeah. I love this pairing I love walking and biking as like, ways to, you know, see the city on a human scale instead of when you’re in a car or something and you just almost teleport from point A to point B, you don’t like see the — you kind of don’t get the same environment around you, that kind of effect. So I like that. Even though walking is also a great time to sort of, like, let your mind wander and not think about what’s around you, listen to your book, or talk on the phone with someone, or think about proving that next theorem, or anything like that. So it’s kind of that, it has both of those things.

KK: You live in two great cities for walking.

TT: Ye.s. With respect to, you know, seeing things differently, one thing I find amazing is that depending on what side of the street you walk, you see things differently.

KK: Absolutely. All right, this has been terrific.

EL: Gotta be some metaphor in here.

KK: I’m sure, I’m sure. Yeah. So it’s always nice to get another perspective on the Pythagorean theorem. So we didn’t even — it’s one of those things that everyone knows so much, we didn’t even tell them what it was. I think it was embedded in there somewhere.

EL: Yeah.

KK: But the idea that it is still vital, like still important in modern research mathematics, you know, is a really interesting thing to know about. We all just sort of take it for granted. Right?

EL: Yeah, this theorem that has been known by humans for millennia. And, you know, still is important.

TT: One of the things that these ways of building parameterizations, so they developed into a whole field, and then they moved to other areas. So there are some recent results by Naber and Valtorta trying to look at a singular set of minimizing surfaces, varifolds, you know, that minimize some sort of energy. And they have been able to give a very good description of the singular set by using these type of parameterizations. And they’re all basically, the basis is always the Pythagorean theorem. It’s really, that’s how distances change.

KK: That’s right. It’s completely fundamental.

EL: Thank you so much. This was really fun.

TT: Thanks for the invitation.

[outro]

In this episode, we were happy to talk with Tatiana Toro, mathematician at the University of Washington and director of the Simons Laufer Math Foundation (formerly known as MSRI), about the Pythagorean theorem. Here are some links that you may find interesting.
Toro’s website and the SLMath website
Our episodes with Henry Fowler and Fawn Nguyen, who also love the Pythagorean theorem
The analyst’s traveling salesman problem on Wikipedia
Naber and Valtorta’s work on singular sets of minimizing varifolds