Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the math theorem with no test at the end. I think I decided I liked that tagline. [Editor’s note: Nope, she really didn’t notice that slip of the tongue!]

Kevin Knudson: Okay.

EL: So we’re going to go with that. Yeah. I'm one of your hosts, Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is the other host.

KK: I’m Kevin Knudson, a professor of mathematics at the University of Florida. How are you doing?

EL: I’m doing well. Yeah, not not anything too exciting going on here. My mother-in-law is coming to visit later today. So the fact that I have to record this podcast means my husband has to do the cleaning up to get ready.

KK: Wouldn’t he do that anyway? Since it’s his mom?

EL: Yeah, probably most of it. But now I've got a really good excuse.

KK: Yeah, sure. Well, Ellen and I had our 27th anniversary yesterday.

EL: Oh, congratulations.

KK: Yeah, we had a nice night out on the town. Got a hotel room just to sit around and watch hockey, as it turns out.

EL: Okay.

KK: But there's a pool at the hotel. And you know, it's hot in Florida, and we don't have a pool. And this is absurd—which Ellen reminds me of every day, that we need a pool—and I just keep telling her that we can either send the kid to college or have a pool. Okay.

EL: Yeah.

KK: I mean, I don't know. Anyway, we're not here talking about that, we're talking about math..

EL: Yes. And we're very excited today to have Carina Curto on the show. Hi, Carina, can you tell us a little bit about yourself?

Carina Curto: Hi, I'm Carina, and I'm a professor of mathematics at Penn State.

EL: Yeah, and I think I first—I don't think we've actually met. But I think the first time I saw you was at the Joint Meetings a few years ago. You gave a really interesting talk about, like, the topology of neural networks, and how your brain has these, like, basically kind of mental maps of spaces that you interact with. It was really cool. So is that the kind of research you do?

CC: Yeah, so that was—I remember that talk, actually, at the Joint Meetings in Seattle. So that was a talk about the uses of typology for understanding neural codes. And a lot of my research has been about that. And basically, everything I do is motivated in some way by questions in neuroscience. And so that was an example of work that's been motivated by neuroscience questions about how your brain encodes geometry and topology of space.

KK: Now, there's been a lot of a lot of TDA [topological data analysis] moving in that direction these last few years. People have been finding interesting uses of topology in neuroscience and studying the brain and imaging stuff like that, very cool stuff.

CC: Yeah.

EL: And did you come from more of a neuroscience background? Or have you been kind of picking that up as you go, coming from a math background?

CC: So I originally came from a mathematical physics background.

EL: Okay.

CC: I was actually a physics major as an undergrad. But I did a lot of math, so I was effectively a double major. And then I wanted to be a string theorist.

KK: Sure, yeah.

CC: I started grad school in 2000. So this is, like, right after Brian Greene’s The Elegant Universe came out.

EL: Right. Yeah.

CC: You know, I was young and impressionable. And so I kind of went that route because I loved physics, and I loved math. And it was kind of an area of physics that was using a lot of deep math. And so I went to grad school to do mathematical string theory in the math department at Duke. And I worked on Calabi-Yaus and, you know, extra dimensions and this kind of stuff. And it was, the math was mainly algebraic geometry, is what right HD thesis was in So this had nothing to do with neuroscience.

EL: Right.

CC: Nothing. And so basically about halfway through grad school—I don't know how better to put it, then I got a little disillusioned with string theory. People laugh now when I say that because everybody is.

KK: Sure.

CC: But I started kind of looking for other—I always wanted to do applied things, interdisciplinary things. And so neuroscience just seemed really exciting. I kind of discovered it randomly and started learning a lot about it and became fascinated. And so then when I finished my PhD, I actually took a postdoc in a neuroscience lab that had rats and, you know, was reporting from the cortex and all this stuff, because I just wanted to to learn as much neuroscience as possible. So I spent three years working in a lab. I didn't actually do experiments. I did mostly computational work and data analysis. But it was kind of a total cultural immersion sort of experience, coming from more of a pure math and physics background.

EL: Right. Yeah, I bet that was a really different experience

CC: It was really different. So I kind of left math in a sense for my first postdoc, and then I came back. So I did a second postdoc at Courant at NYU, and then started getting ideas of how I could tackle some questions in neuroscience using mathematics. And so ever since then, I've basically become a mathematical neuroscientist. I guess I would call myself.

KK: So 2/3 of this podcast is Duke alums. That's good.

CC: Oh yeah? Are you a Duke alum?

KK: I did my degree there too. I finished in ’96.

CC: Oh, yeah.

KK: Okay. Yeah.

CC: Cool.

EL: Nice. Well, so what is your favorite theorem?

CC: So I have many, but the one I chose for today is the Perron-Frobenius theorem.

KK: Nice.

EL: All right.

CC: And so you want to know about it, I guess?

KK: We do. So do our listeners.

CC: So it's actually really old. I mean, there are older theorems, but Perron proved it, I think in 1907 and Frobenius in 1912, so it carries both of their names. So it's over 100 years old. And it's a theorem and linear algebra. So it has to do with eigenvectors and eigenvalues of matrices.

KK: Okay.

CC: And so I'll just tell you quickly what it is. So, if you have a square matrix, so like an n×n square matrix with all positive—so there are many variations of that theorem. I'm going to tell you the simplest one—So if all the entries of your matrix are positive, then you are guaranteed that your largest eigenvalue is unique and real and is positive, so a positive real part. So eigenvalues can be complex. They can come in complex conjugate pairs, for example, but when we talk about the largest one, we mean the one that has the largest real part.

EL: Okay.

KK: All right.

CC: And so one part of the theorem is that that eigenvalue is unique and real and positive. And the other part is that you can pick the corresponding eigenvector for it to be all positive as well.

EL: Okay. And we were talking before we started taping that I'm not actually remembering for sure whether we've used the words eigenvector and eigenvalue yet on the podcast, which, I feel like we must have because we've done so many episodes, but yeah, can we maybe just say what those are for anyone who isn't familiar?

CC: Yeah. So when you have a matrix, like a square matrix, you have these special vectors. So the matrix operates on vectors. And so a lot of people have learned how to multiply a matrix by a vector. And so when you have a vector, so say your matrix is A and your vector is x, if A times x gives you a multiple of x back—so you basically keep the same vector, but maybe scale it—then x is called an eigenvector of A. And the scaling factor, which is often denoted λ, is called the eigenvalue associated to that eigenvector.

KK: Right. And you want x to be a nonzero vector in this situation.

CC: Yes, you want x to be nonzero, yes, otherwise it's trivial. And so I like to think about eigenvectors geometrically because if you think of your matrix operating on vectors in some Euclidean space, for example, then what it does, what the matrix will do, is it will pick up a vector and then move it to some other vector, right? So there's an operation that takes vectors to vectors, called linear transformations, that are manifested by the matrix multiplication. And so when you have an eigenvector, the matrix keeps the eigenvector on its own line and just scales, or it can flip the sign. If the eigenvalue is negative, it can flip it to point the other direction, but it basically preserves that line, which is called the eigenspace associated. So it has a nice geometric interpretation.

EL: Yeah. So the Perron-Frobeius theorem, then, says that if your matrix only has positive entries, then there's some eigenvector that's stretched by a positive amount.

CC: So yeah, so it says there's some eigenvector where the entries of the vector itself are all positive, right, so it lies in the positive orthant of your space, and also that the the corresponding eigenvalue is actually the largest in terms of absolute value. And the reason this is relevant is because there are many kind of dynamic processes that you can model by iterating a matrix multiplication. So, you know, one simple example is things like Markov chains. So if you have, say, different populations of something, whether it be, say, animals in an ecosystem or something, then you can have these transition matrices that will update the population. And so, if you have a situation where if your matrix that's updating your population has—whatever the leading eigenvalue is of that matrix is going to control somehow the long-term behavior of the population. So that top eigenvalue, that one with the largest absolute value, is really controlling the long-term behavior of your dynamic process.

EL: Right, it kind of dominates.

CC: It is dominating, right. And you can even see that just by hand when you sort of multiply, if you take a matrix times a vector, and then do it again, and then do it again. So instead of having A times x, you have A squared times x or A cubed times x. So it's like doing multiple iterations of this dynamic process. And you can see how, then, what’s going to happen to the to the vector if it's the eigenvector. Well, if it's an eigenvector, well, what's going to happen is when you apply the matrix once, A times x, you're going to get λ times x. Now apply A again. So now you're applying A to the quantity λx, but the λ comes out, by the linearity of the of the matrix multiplication, and then you have Ax again, so you get another factor of λ, so you get λ^2 times x. And so if you keep doing this, you see that if I do A^k times x, I get λ^k times x. And so if that λ is something, you know, bigger than 1, right, my process is going to blow up on me. And if it's less than 1, it's going to converge to zero as I keep taking powers. And so anyway, the point is that that top eigenvector is really going to dominate the dynamics and the behavior. And so it's really important if it's positive, and also if it's bigger or less than 1, and the Perron-Frobenius theorem basically tells you that you have, it gives you control over what that top eigenvalue looks like and moreover, associates it to an all-positive eigenvector, which is then a reflection of maybe the distribution of population. So it's important that that be positive too because lots of things we want to model our positive, like populations of things.

KK: Negative populations aren't good. Yeah,

CC: Yes, exactly. And so this is one of the reasons it's so, useful is because a lot of the things we want to model are—that vector that we apply the matrix to is reflecting something like populations, right?

KK: So already this is a very non-obvious statement, right? Because if I hand you an arbitrary matrix, I mean, even like a 2×2 rotation matrix, it doesn't have any eigenvalues, any real eigenvalues. But the entries aren't all positive, so you’re okay.

CC: Right. Exactly.

KK: But yeah, so a priori, it's not obvious that if I just hand you an n×n matrix with all real entries that it even has a real eigenvalue, period.

CC: Yeah. It's not obvious at all, and let alone that it's positive, and let alone that it has an eigenvector that's all positive. That's right. And the positivity of that eigenvector is really important, too.

EL: Yeah. So it seems like if you're doing some population model, just make sure your matrix has all positive entries. It’ll make your life a lot easier.

CC: So there's an interesting, so do you do you know what the most famous application of the Perron-Frobenius theorem is?

EL: I don't think I do.

KK: I might, but go ahead.

CC: You might, but I’ll go ahead?

KK: Can I guess?

CC: Sure.

KK: Is it Google?

CC: Yes. Good. Did you Google it ahead of time?

KK: No, this is sort of in the dark recesses of my memory that essentially they computed this eigenvector of the web graph.

CC: Right. Exactly. So back in the day, in the late ‘90s, when Larry Page and Sergey Brin came up with their original strategy for ranking web pages, they used this theorem. This is like, the original PageRank algorithm is based on this theorem, because they're, they have again the Markov process where they imagine some web—some animal or some person—crawling across the web. And so you have this graph of websites and edges between them. And you can model the random walk across the web as one of these Markov processes where there's some matrix that that reflects the connections between web pages that you apply over and over again to update the position of the of the web crawler. And and so now if you imagine a distribution of web crawlers, and you want to find out in the long run what pages do they end up on, or what fraction of web crawlers end up on which pages, it turns out that the Perron-Frobenius theorem gives you precisely the existence of this all-positive eigenvector, which is a positive probability that you have on every website for ending up there. And so if you look at the eigenvector itself, that you get from your web matrix, that will give you a ranking of web pages. So the biggest value will correspond to the most, you know, trafficked website. And smaller values will correspond to less popular websites, as predicted by this random walk model.

EL: Huh.

CC: And so it really is the basis of the original PageRank. I mean, they do fancier things now, and I'm sure they don't reveal it. But the original PageRank algorithm was really based on this. And this is the key theorem. So I think it's a it's kind of a fun thing. When I teach linear algebra, I always tell students about this.

KK: Linear Algebra can make you billions of dollars.

CC: Yes.

KK: That’ll catch students’ attention.

CC: Yes, it gets students’ attention.

EL: Yes. So where did you first encounter the Perron-Frobenius theorem?

CC: Probably in an undergrad linear algebra class, to be honest. But I also encountered it many more times. So I remember seeing it in more advanced math classes as a linear algebra fact that becomes useful a lot. And now that I'm a math biologist, I see it all the time because it's used in so many biological applications. And so I told you about a population biology application before, but it also comes up a lot in neural network theory that I do. So in my own research, I study these competitive neural networks. And here I have matrices of interactions that are actually all negative. But I can still apply the theorem. I can just flip the sign.

EL: Oh, right.

CC: And apply the theorem, and I still get this, you know, dominant eigenvalue and eigenvector. But in that case, the eigenvalue is actually negative, and I still have this all-positive eigenvector that I can choose. And that's actually important for proving certain results about the behavior of the neural networks that I study. So it's a theorem I actually use in my research.

EL: Yeah. So would you say that your appreciation of it has grown since you first saw it?

CC: Oh for sure. Because now I see it everywhere.

EL: Right.

CC: It was one of those fun facts, and now it’s in, you know, so many math things that I encounter. It's like, oh, they're using the Perron-Frobenius theorem. And it makes me happy.

EL: Yeah, well, when I first read the statement of the theorem, it's not like it bowled me over, like, “Oh, this is clearly going to be so useful everywhere.” So probably, as you see how many places it shows up, your appreciation grows.

CC: Yeah, I mean, that's one of the things that I think is really interesting about the theorem, because, I mean, many things in math are like this. But you know, surely when Perron and Frobenius proved it over 100 years ago, they never imagined what kinds of applications it would have. You know, they didn't imagine Google ranking web pages, or the neural network theory, or anything like this. And so it's one of these things where it's like, it's so basic. Maybe it could look initially like a boring fact of linear algebra, right? If you're just a student in a class and you're like, “Okay, there's going to be some eigenvector, eigenvalue, and it's positive, whatever.” And you can imagine just sort of brushing it off as another boring fact about matrices that you have to memorize for the test, right? And yet, it's surprisingly useful. I mean, it has applications in so many fields of applied math and in pure math, and so it's just one of those things that gives you respect for even seemingly simple and not obviously, it doesn't bowl you over, right, you can see the statement and you're not like, “Wow, that's so powerful!” But it ends up that it's actually the key thing you need in so many applications. And so, you know, it's earned its place over time. It's aged nicely.

EL: And do you have a favorite proof of this theorem?

CC: I mean, I like the elementary proofs. I mean, there are lots of proofs. So I think there's an interesting proof by Birkhoff. There are some proofs that involve the Brouwer fixed point theorem, which is something maybe somebody has chosen already.

EL: Yes, actually. Two people have chosen it!

CC: Two people have chosen the Brouwer fixed point theorem. Yeah, I would imagine that's a popular choice. So, yeah, there are some proofs that rely on that, which I think is kind of cool. So those are more modern proofs of it. That's the other thing I like about it, is that it has kind of old-school elementary proofs that an undergrad in a linear algebra class could understand. And then it also has these more modern proofs. And so it's kind of an interesting theorem in terms of the variety of proofs that it admits.

KK: So one of the things we like to do on this podcast is we like to invite our guests to pair their theorem with something. So I'm curious, I have to know what pairs well with the Perron-Frobenius theorem?

CC: I was so stressed out about this pairing thing!

KK: This is not unusual. Everybody says this. Yeah.

CC: What is this?

KK: It’s the fun part of the show!

CC: I know, I know. And so don't know if this is a good pairing, but I came up with this. So I went to play tennis yesterday. And I was playing doubles with some friends of mine. And I told them, I was like, I have to come up with a pairing for my favorite theorem. So we chatted about it for a while. And as I was playing, I decided that I will pair it with my favorite tennis shot.

EL: Okay.

CC: So, my favorite shot in tennis is a backhand down the line.

KK: Yes.

CC: Yeah?

KK: I never could master that!

CC: Yeah. The backhand down the line is one of the basic ground strokes. But it's maybe the hardest one for amateur players to master. I mean, the pros all do it well. But, you know, for amateurs, it's kind of hard. So usually people hit their backhand cross court. But if you can hit that backhand down the line, especially when someone's at the net, like in doubles, and you pass them, it's just very satisfying, kind of like, win the point. And for my tennis game, when my backhand down the line is on, that's when I'm playing really well.

EL: Nice.

CC: And I like the linearity of it.

EL: Right, it does seem like, you know, you're pushing it down.

CC: Like I'm pushing that eigenvector.

KK: It’s very positive, everything's positive about it.

CC: Everything’s positive. The vector with the tennis ball, just exploding down the line. It's sort of maybe it's a stretch, but that's kind of what I decided.

EL: A…stretch? Like with an eigenvalue and eigenvector?

CC: Right, exactly. I needed to find a pairing that was a stretch.

EL: I think this is a really great pairing. And you know, something I love about the pairing thing that we do—other than the fact that I came up with it, so of course, I'm absurdly proud of it—is that I think, for me at least it's built all these bizarre connections with math and other things. It's like, now when I see the mean value theorem, I'm like, “Oh, I could eat a mango.” Or like, all these weird things. So now when I see people playing tennis, I'll be like, “Oh, the Perron-Frobenius theorem.”

CC: Of course.

EL: So are you a pretty serious tennis player?

CC: I mean, not anymore. I played in college for a little bit. So when I was a junior, I was pretty serious.

EL: Nice. Yeah, I’m not really a tennis person I've never played or really followed it. But I guess there's like some tennis going on right now that's important?

CC: The French Open?

EL: That’s the one!

KK: Nadal really stuck it to Federer this morning. I played obsessively in high school, and I was never really any good, and then I kind of gave it up for a long time, and I picked up again in my 30s and did league tennis when I lived in Mississippi. And my team at our level—we were just sort of very intermediate players, you know—we won the state championship two years in a row.

CC: Wow.

KK: And then and then I gave it up again when I moved to Florida. My shoulder can't take it anymore. I was one of these guys with a big booming serve and a pretty good forehand and then nothing else, right?

CC: Yeah.

KK: So you know, if you work my backhand enough you're going to destroy me.

EL: Nice. Oh, yeah, that's a that's a lot of fun. And I hope other our tennis appreciator listeners will now have have an extra reason to enjoy this theorem too. So yeah, we also like to give our guests a chance, like if they have a website or book or anything they want to mention—you know, if people want to find them online and chat about tennis or linear algebra— is there anything you want to mention?

CC: I mean, I don't have a book or anything that I can plug, but I guess I wanted to just plug linear algebra as a subject.

KK: Sure.

CC: I feel like linear algebra is one of the grand achievements of humanity in some ways. And it should really shine in the public consciousness at the same level as calculus, I think.

EL: Yeah.

KK: Maybe even more.

CC: Yeah, maybe even more. And now, everybody knows about calculus. Every little kid knows about calculus. Everyone is like, “Oh, when when are you going to get to calculus?” You know, calculus, calculus. And linear algebra—it also has kind of a weird name, right, so it sounds very elementary somehow, linear and algebra—but it's such a powerful subject. And it's very basic, like calculus, and it's used widely and so I just want to plug linear algebra.

EL: Right. I sometimes feel like there are basically—so math can boil down to like, doing integration by parts really well or doing linear algebra really. Like, I joked with somebody, like, I didn't end up doing a PhD in a field that used a lot of linear algebra, but I sort of got my PhD in applied integration by parts, it's just like, “Oh, yeah. Figure out an estimate based on doing this.” And I think linear algebra, especially now with how important social media and the internet are, it is really an important field that, I agree, more people should know about. It is one of the classes that when I took it in college, it's one of the reasons I—at that time, I was trying to get enough credits to finish my math minor. And I was like, “Oh, yeah, actually, this is pretty cool. Maybe I should learn a little more of this math stuff.” So, yeah, great class.

CC: And you know, it's everywhere. And you know, there are all these people, almost more people have heard of algebraic topology than linear algebra, outside, you know, because it's this fancy topology or whatever. But when it comes down to it, it's all linear algebra tricks. With some vision of how to package them together, of course, I’m not trying to diminish the field, but somehow linear algebra doesn't get it’s—it’s the workhorse behind so much cool math and yeah, doesn't get its due.

EL: Yes, definitely agree.

KK: Yeah. All right. Well, here's to linear algebra.

EL: Thanks a lot for joining us.

CC: Thank you.

KK: It was fun.

[outro]

Our guest on this episode, Carina Curto, is a mathematician at Penn State University who specializes in applications in biology and neuroscience. She talked about the Perron-Frobenius theorem. Here are some links you may find useful as you listen to this episode.

Curto’s website
A short video of Curto talking about how her background in math and physics is useful in neuroscience and a longer interview in Quanta Magazine
An article version of Curto’s talk a few years ago about topology and the neural code
Curto ended the episode with a plug for linear algebra as a whole. If you’re looking for an engaging video introduction to the subject, check out this playlist from 3blue1brown.