Episode 54 - Steve Strogatz

My Favorite Theorem

Kevin Knudson: Welcome to My Favorite Theorem, a math podcast and so much more. I'm Kevin Knudson, professor of mathematics at the University of Florida, and here is your other host.
Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer based in Salt Lake City, where it is snowy, but I understand not as snowy as it is for our guest.
KK: I know, and we've been trying to make this one happen for a long time. So I'm super excited that this is finally going to happen. So today we are pleased to welcome Professor Steve Strogatz. Steve, why don't you introduce yourself?
Steve Strogatz: Well, wow, thank you. Hi, Kevin. Hey, Evelyn. Thanks for having me on. Yeah, I've wanted to be on the show for a very long time. And I think it's true what Evelyn just said, we have a very big snowstorm here today in not-so-sunny Ithaca, New York, upstate. I just took my dog out for a walk, and the snow was over my boots and going into them and making my feet wet.
KK: See, I have a Florida dog. She wouldn't know what to do. Actually, we were in North Carolina a few years ago at Christmas, and it snowed, and she was just alarmed. She had no idea what to do. And she's small, too, she just couldn't take it.
SS: Yeah, well, it would be more like tunneling than running.
KK: Right.
EL: Yeah. So we actually met quite a few years ago at this point — actually, I know the exact date because it was, like, two days before my brother's wedding the first time we met because you were on the thesis committee for my sister in law, who is a physicist, many years ago, and so we have this weird, it was when I had just moved to New York to work at Scientific American for the first time. So it was at the very beginning of my life as a math writer. And I remember just being floored by how generous you were with being willing to meet with a nobody like me.
SS: Well that’s nice.
EL: At this time when I was first starting.
SS: But actually, I had a crystal ball, and I knew you were going to become the voice of mathematics for the country, practically. I mean, so I let me brag on Evelyn’s behalf a little bit. If you go on Twitter, you—I wonder if you know this, Kevin, do you know this little factoid I'm going to unreel?
KK: I bet I do.
SS: You know where I'm going. On Twitter, if you ask “What mathematician do other mathematicians follow?” I think Evelyn is the number one person the last time I checked.
KK: She is indeed number one. That's right.
SS: Yeah.
EL: I like to say I'm the queen of math, Twitter, although I don't actually like to say this because it feels really weird.
SS: Well that’s okay. You didn't say it. But yeah, I do remember our meeting that day in my office. And right, it was on this happy occasion of a family, of a wedding. Okay, sorry, I interrupted you, Kevin.
KK: Oh, I don't know. I was going to say with the Twitter thing. I think you're not far behind, right? Like, aren't you number two, probably?
SS: I think the last time I looked I was number two.
KK: Yeah.
SS: So look at that. Okay, so look at that, the two tweet monsters here.
KK: And now the funny thing is I'm not even on that list. So here we go.
SS: Okay. Yeah, well you could catch up. I'm sure you'll be coming right on our heels.
KK: Maybe. I have over 1000 followers now, but apparently not that many mathematicians. So this is how this goes. Anyway, what weird times we live in, right?
SS: It's very weird. I mean, I don't know what this can get us, a cup of coffee or what.
KK: Maybe, maybe. Okay. Let's talk theorems. So Steve, you must have a favorite theorem. What is it?
SS: Yeah, I have a very sentimental attachment to a theorem and complex analysis called Cauchy’s theorem, or sometimes called Cauchy’s integral theorem.
KK: Oh, I love that theorem.
SS: It’s a fantastic theorem. And so I don't know. I mean, I feel like I want to say what I like about it mathematically and what I like about it personally. Does that work?
EL: Yeah, that’s exactly what we want.
SS: Well, okay. So then, the scene is, it's my sophomore year of college. Maybe I'll start with the emotional.
KK: Okay.
SS: It’s my sophomore year of college. I've just gotten very demoralized in my freshman year, taking the the honors linear algebra course that a lot of universities offer as a kind of first introduction to what college math is really going to be like. You know, a lot of kids in high school have done perfectly well in their precalculus and calculus courses, and then they get to college and suddenly it's all about proofs and abstraction. And it can be—I mean, we sometimes call it a transition course, right? It's a transition into the rigorous world of pure math. And so it was a shock for me. I had a lot of trouble with that course. I couldn't read the book very well, it didn't have pictures. And I'm kind of visual. And so I was always at a loss to figure out what was going on. And being a freshman I didn't have any sense about, why don't I look at a different book, you know, or maybe, maybe I should switch sections. Or I could ask my teaching assistant, or I could go to office hours. I didn't know to do any of that stuff.
So anyway, this is not my favorite theorem. I was very demoralized after this experience in linear algebra. And then when I took a second semester, also an honors course, that was a rigorous calculus course with the Heine-Borel theorem, and, you know, like, all kinds of—again, no formulas, it was all about, I remember hearing this stuff about “every open cover has a finite subcover,” and I thought, “I want to take a derivative! I can't do anything here. I don't know what to do!” So anyway, after that first year, I thought, “I don't have the right stuff to be a mathematician. And so maybe I'll try physics,” which I also always loved. I say all that as preamble to this complex analysis course that I was taking in sophomore year, which, you know, I still wanted to take math, I heard complex variables might be useful for physics, I thought it would be an interesting course. I don't know. Turned out it was a really great course for me because it really looked a lot like calculus, except it was f(z) instead of f(x).
KK: Right.
SS: You know, but everything else was kind of what I wanted. And so I was really happy. I had a great teacher, a famous person actually named Elias Stein.
KK: Oh.
SS: So Stein is a well-known mathematician, but I didn't know that. To me, he was a guy who wore Hush Puppies and, you know, had always kind of a rumpled appearance, came in with his notes. And he seemed nice, and I really liked his lectures. But so one day, he starts proving this thing, Cauchy’s theorem, and he draws a big triangle on the board. And he's going to prove that the integral of an analytic function f around this triangle is zero no matter what f is. All he needs is that it's analytic, meaning that it has a derivative in the sense of a function of a complex variable. It's a little more stringent condition—actually a lot more stringent than to say a function of a real variable is differentiable, but I didn't appreciate that at the time. I mean, that's sort of the big reveal of the whole subject.
KK: Right.
SS: That this is an unbelievably stringent condition. You can’t imagine how much stuff follows from this innocuous-looking assumption that you could take a derivative, but okay, so I'm kind of naive. Anyway, he says he's going to prove this thing, only assuming that f is analytic on this triangle and inside it. And that's enough. And then, you know, I feel like you don't have enough information, there's nothing to do! So then he starts drawing a little triangle inside the big triangle, and then little triangles inside the little triangle. And it starts making a pattern that today I would call a fractal, though I didn't know it at the time, and he didn't say the word fractal. And actually, nobody ever says that when they're doing this proof. But it’s—right, they don’t—but it's triangles inside of triangles in a self-similar way that doesn't actually play any particular role in the proof, other than it's just this bizarre move, like, What is going on? Why is he drawing these triangles inside of triangles? And by the end, I mean, I won't go into the details of the proof, but he got the whole thing to work out, and it was so magnificent that I started clapping.
And at that point, every kid in the room whipped their head around to look at me, and the professor looked at me, like what is wrong with you? You know, and yet, I thought, “Wow, why are you guys looking at me?” This was the most amazing theorem and the most amazing proof.” You know, so anyway, to me, it was a very significant moment emotionally because it made me feel that math was, first of all, something I could do again, something I could appreciate and love, after having really been turned off for a year and having a kind of crisis of confidence. But also, you know, aside from any of that, it's just, I think people who know would regard this proof —this is actually by a mathematician named Goursat, a French mathematician who improved on Cauchy’s original proof. Goursat’s proof of Cauchy’s theorem is just one of the great— you know, it's from “The Book” in the words of Paul Erdős, right? If God had a proof of this theorem, it would be this proof. Do you guys have any thoughts about that? I mean, I'm assuming you know what I'm talking about with this theorem and this proof.
KK: Well, this is one of my favorite classes to teach because everything works out so well. Right? Every answer is zero because of Cauchy’s theorem, or it's 2πi because you have a pole in the middle, right?
SS: Yeah.
KK: And so I sort of joke with my students that this is true. But then the things you can do with this one theorem, which does—you’re right, it's very innocuous-looking, you know, you integrate an analytic function on a closed curve, and you get zero. And then you can do all these wonderful calculations and these contour integrals and, like, the real indefinite integrals and all this stuff. I just love blowing students’ minds with that, and just how clean everything is.
EL: Yeah, I kind of—I feel like I go back and forth a little bit. I mean, like, in my Twitter bio, it does have “complex analysis fangirl.” And I think that's accurate. But sometimes, like you said, it's so many of these, you know, you're you're like teaching it or reading it and you're like, “Oh, this is complex analysis is so powerful,” but in another way, it's like our definition of derivative in the complex plane is so restrictive that like, we're just plucking the very nicest, most well-behaved things to look at and then saying, “Oh, look what we can do when we only look at the very most well-behaved things!” So yeah, I kind of go back and forth, like is it really powerful or are we just, like, limiting ourselves so much in what we think about?
KK: And I guess the real dirty secret is that when you try to go to two complex variables, all hell breaks loose.
SS: Ah, see, I've never done that subject, so I don't appreciate that. Is that right?
KK: I don't, either. Yeah. But I mean, apparently, once you get into two variables, like none of this works.
SS: Ohhh. But that's a very interesting comment you make there, Evelyn, that—you know, in retrospect, it's true. We've assumed, when we make this assumption that a function is analytic, that we are living in the best of all possible worlds, we just didn't realize we were assuming that. It seems like we're not assuming much. And yet, it turns out, it's enormously restrictive, as you say. And so then it's a question of taste in math. Do you like your math really surprising and really beautiful and everything works out the way it should? Or do you like it thorny and full of rich counterexamples and struggles and paradoxes? And I feel like that's sort of the essential difference between real analysis and complex analysis.
EL: Yeah.
SS: In complex analysis, everything you had dreamed to be true is true, and the proofs are relatively easy. Whereas in real analysis, sort of the opposite. Everything you thought was true is actually false. There are some nasty counterexamples, and the proofs of the theorems are really hard.
EL: Yeah, you kind of have to MacGyver things together. “Yeah, I got this terrible epsilon and like, you know, it's got coefficients and exponents and stuff, but okay, here you go. I stuck it together.
KK: But but that's interesting, Steve, that this is your favorite theorem because, you know, you're very famous for studying kind of difficult, thorny mathematics, right? I mean, dynamics is not easy.
SS: Huh, I wouldn't have thought that, that's interesting that you think that. I don't think of myself as doing anything thorny.
KK: Okay.
SS: So that's interesting. I mean, yes, dynamical systems in the hands of some practitioners can be very subtle. I mean, those are people who have a taste for those those kinds of issues. I've never been very sophisticated and haven't really understood a lot of the subtleties. So I like my math very intuitive. I’m on the very applied end of the applied-pure spectrum, so that sometimes people will think I'm not really a mathematician at all. I look more like a physicist to them, or maybe even, God forbid, a biologist or something. So yeah, I don't really have much taste for the difficult and the subtle. I like my math very cooperative and surprising. I like—well, not surprising for mathematical reasons, but more surprising for its power to mirror things in the real world. I like math that is somehow tapping into the order in the world around us.
EL: Yeah, so this it's interesting to me, also that you picked this because, yeah, as you say, you are a very applied mathematician. And I think of complex analysis as a very pure—I actually, I'm trying to not say “pure” math, because I think it's this weird, like, purity test or something. But you know, that like a very theoretical thing. So does it play into your field of research at all?
SS: Well, uh, not particularly. Yeah. So that's a good question. I mean, I have to say I was a little intimidated by the title of the podcast. If you ask me what's my favorite theorem, the truth is for me, theorems are not my favorite things.
KK: Okay.
SS: My favorite things are examples or mathematical models. Like there’s a model in my field called the Kuramoto model after a Japanese physicist Yoshiki Kuramoto. And if you asked me what's my favorite mathematical object, I would say the Kuramoto model, which is a set of differential equations that mirrors how fireflies can get their flashes in sync, or how crickets can chirp in sync, or how other things in nature can self-organize into cooperative, collective oscillation. So that's my favorite object. I've been studying that thing for 30 years. And I suppose there are theorems attached to it, but it's the set of equations themselves and what they do that is my favorite of all. So I don't know, maybe that's my real answer.
KK: Well, that’s fine. So yeah, it's true. We've had people who've done that in the past, they didn't have a favorite theorem, but they had a favorite thing.
SS: But still, I mean, I am still a mathematician, part of me is, and I do have theorems that I love, and one of the things I love about Cauchy’s theorem is that in the proof, with this drawing of all the nested triangles inside the big triangle, you end up using a kind of internal cancellation. The triangles touch other triangles except on their common edge, sometimes you're going one way, and sometimes you're going in the opposite direction on that same edge. And so those contributions end up cancelling. And you end up, the only thing that doesn't cancel is what's going on around the boundary. And then that can be sort of pulled all the way into a tiny triangle in the interior, which is where you end up using the local property that is the derivative condition to get everything that you need to prove the result about the big triangle on the outside.
But the reason I'm going into all that is that this is a principle, this internal cancellation, that is at the heart of another theorem that's been featured on your show, the fundamental theorem of calculus, which uses a telescoping sum to convert what's happening on the boundary to what's happening when you integrate over the interior. This idea of telescoping I think, is really deep. I mean, it's what we use to prove Stokes’ theorem. It's what we would use to prove all the theorems about line integrals. It comes up in topology when you're doing chains and cochains. So this is a principle that goes beyond any one part of math, this idea of telescoping. And I've been thinking I want to write an article, someday (I haven't written it yet) called “Calculus Through the Telescope” or “A Telescopic View of Calculus” or something like that, that brings out this one principle and shows its ramifications for many parts of math and analysis and topology. I think some people get it, people who really understand differential forms and topology know what I'm talking about. But no one ever really told me this, and I feel like maybe it should be mentioned, even though it is well-known to the people who know it.
KK: Right, it's the air we breathe, right? So we don't we don't think about it.
SS: I guess, but like, I think there are probably high school teachers, or others who are teaching calculus—like for instance, when I learned about telescoping series in my first calculus course, that's just seems like a trick to find an exact sum of a certain infinite series of numbers. You know, they show you, “Okay, you could do this one because it's a telescoping series.” And it seems like it's an isolated trick, but it's not isolated. This one idea—you can see the two- dimensional version of it in Cauchy’s theorem, and you can see the three-dimensional version of it in the divergence theorem, and so on. Anyway, so I like that. I feel like this idea has tentacles spreading in all directions.
EL: Yeah. Well, this makes me want to go back and think about that idea more because, yeah, I wouldn't say that I would necessarily have thought to connect it to this many other things. I mean, you did preface your statement with “those who really understand differential forms,” and my dark secret is that the word “form” really scares me. It's a tough one. It's somehow, that was one of those really hard things, when I started doing more, like, hard real analysis. It's like, I feel like I always had to just kind of hold on to it and pray. And you get to the end of it. You're like, “Well, I guess I did it.” But I feel like I never really got that full deep understanding of forms.
SS: Huh. I don't I don't claim that I have either. I'm reminded of a time I was a teaching assistant for a freshman course for the the whiz kids that—you know, every university has this where you throw outrageous stuff at these freshmen, and then they rise to the occasion because they don't know what you're asking them to do is impossible. But so I remember being in a course, like I say, as a teaching assistant, where it was called A Course in Mathematics for Students of Physics, based on a book by Shlomo Sternberg, at Harvard, and Paul Bamberg, who's a physicist there too, and a very good teacher. And that book tried to teach Maxwell's equations and other parts of physics with the machinery of differential forms and homology and cohomology theory to freshmen. But what was amazing is it sort of worked, and the students could do it. And in the course of teaching it, I came to this appreciation of integrating forms, and how it really does amount to this telescoping sum trick. And, anyway, yeah, it's true, that maybe it's not super widely appreciated. I don't know. I don't know if it is, I don't want to insult people who already know what I'm talking about. But I I do feel like there's a story to tell here.
KK: Okay. Well, we'll be looking for that.
EL: Yeah.
SS: Someday.
KK: In the New York Times, right?
SS: Well.
KK: So another thing we do on this podcast is we ask our guests to pair their theorem with something. And we might have sprung this on you, but you seem to have thought of a solution here. So what have you chosen to paired with Cauchy’s theorem?
SS: Cubist painting.
KK: Oh, excellent. Okay. Explain.
EL: Yeah, tell us why.
SS: Well, I'm thinking of Cubism. I don’t—look, I don't know much about art. So it might be a dumb pairing. But what I'm thinking is there's a there's a painting. I think it's by Georges Braque of a guy, or maybe it's Picasso. Someone walking down stairs. And maybe it's called a Nude Descending a Staircase, or something like that. You're nodding, do you know what I mean?
EL: I'm a little nervous about saying, I think it is Picasso, but I'm looking it up on my phone surreptitiously.
SS: I could try too. For some reason, I'm thinking it's George Braque, but that may be wrong. But so I'll describe the painting I have in my head and it may be totally not—
EL: No, it’s Marcel Duchamp!
SS: Oh, it's Marcel Duchamp?
EL: Yeah.
SS: And what's the name of it?
EL: Nude descending a staircase, number two. I think.
SS: Yeah, that's the one. Would that be considered Cubism?
EL: Yeah.
SS: It says according to Wikipedia, it’s widely considered a modernist classic. Okay, I don't know if it's the best example of what I'm thinking. But it's, let me just blow it up and look at it here. So, what hits me about it is it's a lot of straight lines. It's very rectilinear. And you don't see anything that really looks curved like a human form. You know, people are made of curved surfaces, our faces, our cheeks are, you know. What I like is this idea that you can build up curved objects out of lots of things made of straight lines. You know what you can do? mesh refinement on it. For instance, there's an old proof of the area of a circle where you chop it up into lots of pizza-shaped slices, right, and then you add up the areas of all those. And they can be approximated by triangles, and if you make the triangles thin enough, then those slivers can fill out more and more of the area, the method of exhaustion proof for the area of a circle. So this idea that you can approximate curved things with triangles, reminds me of this idea in Cauchy’s theorem that you first you prove it for the triangle, and then later Professor Stein proved the result for any smooth curve by approximating it with triangles, you know, a polygonal approximation to the curve, and then he could chop up the interior into lots of triangles. So I sort of think it pairs with this vision of the human form and it's sinuous descent down. You know, this person is smooth and yet they're being built out of these strange Cubist facets, or other shapes. I mean, think of other Cubist paintings you you represent smooth things with gem-like faceted structures, it sort of reminds me of Cauchy’s theorem.
KK: Okay, good pairing. Yup.
EL: Yeah, glad we got to the bottom of that before we made false statements about art on this math podcast.
SS: Yeah, it may not be the best Cubist example. But what are you gonna do? You invited a mathematician.
KK: So we also like to let our guests make pitches for things that they're doing. So you have a lot going on. You have a new podcast.
EL: Yeah, tell us about it.
SS: Okay. Yeah, thank you for mentioning it. I have a podcast with the confusing name Joy of X. Confusing because I also wrote a book by that name. And before that I had written an article by that name.
KK: Yes.
SS: So I did not choose that name for the podcast. But my producer felt like it sort of works for this podcast because it's a show where I interview scientists and mathematicians—in spirit, very similar to what we're doing here. And I talk to them about their lives and their work. And it's sort of the inner life of a scientist, but it could be a neuroscientist, it could be a person who studies astrophysics, or a mathematician. It's anything that is covered by Quanta Magazine. So Quanta Magazine, some of your listeners will know, is an online magazine that covers fundamental parts of math and science and computer science. Really, it's quite terrific. If people haven't read it, they might want to look at it online. It's free. And anyway, so Quanta wanted to start a podcast. And they asked me to host it, which was really fun because I get to explore all these parts of science. I've always liked all of the different parts of science, as well as math. And so yeah, that's the show. It's called the Joy of X where here, X takes on this generalized meaning of the unknown, not just the unknown in algebra, but anything that's unknown, and the joy of doing science and the scientific question. We'll be sure to link to that.
EL: Yeah.
KK: Also, I think Infinite Powers came out last year, right? 2019?
SS: That’s true. Yes, I had a book, Infinite Powers, about calculus. And that was an attempt to try to explain to the general public what's so special about calculus, why is it such a famous part of math. I try to make the case that it really did change the world and that it underpins a lot of modern science and technology as well as being a gateway to modern math. I really do think of it as one of the greatest ideas that human beings have ever come up with. Of course, that raises the question, did we discover it or invent it? But that’s a good one.
EL: Put that on a philosophy podcast somewhere. We don’t need that on this math podcast.
SS: Yeah, I don't really know what to say about that. That's a good timeless question. But anyway, yes, Infinite Powers was a real challenge to write because I'm trying to tell some of the history, but I'm not a historian of math. I wanted to really teach some of the big ideas for people who either have math phobia or who took calculus but didn't see the point of it, or just thought it was a lot of, you know, doing one integral after another without really understanding why they're doing it. So it's my love song to calculus. It really is one of my favorite parts of math, and I wanted other people to see what's so lovable and important about it.
KK: Yeah.
SS: The book, as I say, was hard because I tried to combine history and applications and big ideas without really showing the math.
KK: Yeah, that's hard.
SS: And make it fun to read.
KK: Right. It is. It's a very good book, though. I did read it.
SS: Oh, thanks.
KK: And I enjoyed it quite a bit.
EL: Well, it is on my table here under a giant pile of books to read, because people need to just stop publishing.
SS: That’s right.
EL: There’s too much. We just need to have a year to catch up, and then we could start going again but what's what's
KK: What’s that Japanese word, sort of the joy of having unread books? [Editor’s note: Perhaps tsundoku, “aquiring reading materials but letting them pile up in one’s home without reading them.”] There's a Japanese concept of like these books that you’ll, well, maybe even never read. But that you should have stacks and stacks of books. Because, you know, maybe you'll read them. Maybe you won't. But the potential is there.
SS: Nice.
KK: So I have a nightstand, on the shelf of my nightstand there's probably 20 books there right now, and I haven't read them all. I've read half of them, maybe, but I'm going to read them. Maybe.
SS: Yeah, yeah.
KK: Actually, you know, when you were talking about your sort of emotional feelings about Cauchy’s theorem, it reminded me of your—I don't know if it was your first book, but The Calculus of Friendship, about your relationship with your high school teacher.
SS: Well, how nice of you to mention it.
KK: Yeah. That was interesting to it, because it reminded me a lot of me, in the sense of, I thought I knew everything too when I was 18. Like, I thought, “Calculus is easy.” And then I get to university and math wasn't necessarily so easy. You know. And so these same sort of challenges, you know?
SS: Well, I appreciate that, especially because that book is pretty obscure. As far as I know, not many people read it. And it's very meaningful to me because I love my old teacher, Mr. Joffrey, who is now, let’s see, he's 90 years old. And I stayed in touch with him for about 35 years after college, and we wrote math problems to each other, and solutions. And it was really a friendship based on calculus. But over the course of those 35 years, a lot happened to both of us in our lives. And yet, we didn't tend to talk about that. It was like math was a sanctuary for us, a refuge to get away from some of the ups and downs of real life. But of course, real life has a way of making itself, you know, insinuating itself whether you like it or not. And so it's it's that story. The subtitle of the book is “what a teacher and a student learned about life while corresponding about math.” And I sometimes think of it as, like, there's a Venn diagram where there's one circle is people who want to read math books with all the formulas, because I include all the formulas from our letters.
KK: Yeah.
SS: And then there's people who want to read books about emotional friendships between men. And if you intersect those two circles, there's a tiny sliver that apparently you're one of the people in it.
KK: And your book might be the unique book in that in that Venn diagram too.
SS: Maybe. I don't know. But yeah, so it was it was clear it would not be a big hit in any way. But I felt like I couldn't do any other work until I wrote that book. I really wanted to write it. It was the easiest book to write. It poured out of me, and I would sometimes cry while I was writing it. It was almost like a kind of psychoanalysis for myself, I think, because I did have a lot of guilty feelings about that relationship, which, you know, if you do read the book, anyone listening, you'll see what I felt guilty about, and I deserved to feel guilty. I needed to grow up, and you see some of that evolution in the course of the book.
KK: Yeah. All right. Anything else you want to pitch? I mean?
SS: Well, how about I pitch this show? I mean, I'm very delighted to be on here. Really, I think you guys are doing a great thing helping to get the word out about math, our wonderful subject. And so God bless you for doing that.
KK: Well, this has been a lot of fun, Steve, we really appreciate you taking time out of your snow day. And so now do you have to shovel your driveway?
SS: Oh, yeah, that may be the last act I ever commit.
KK: Don’t you still have a teenager at home? Isn't that what they're for?
SS: My kids, I do have—you know what, that's a good point. I have one daughter who is still in high school and has not left for college yet, so maybe I could deploy her. She's currently making oatmeal cookies with one of her friends.
KK: Well, that's a useful, I mean that that's helping out the family too, right? I mean,
SS: They’re both able bodied, strong young women. So I should get them out there and with me, and we could all shovel ourself out. Yeah.
KK: Good luck with that. Thank you. Thanks for joining us.
SS: My pleasure. Thanks for having me.
On this episode of My Favorite Theorem, we were happy to talk with Steve Strogatz, an applied mathematician at Cornell University, about the Cauchy integral theorem. Here are some links you might find helpful.

Strogatz’s website, which includes links to information about his books and article
The Joy of X, the podcast he hosts for Quanta Magazine
The Cauchy integral theorem on Wikipedia
The Kuramoto model
Nude Descending a Staircase no. 2 by Marcel Duchamp