Kevin Knudson: Welcome to My Favorite Theorem! I’m your host Kevin Knudson, professor of mathematics at the University of Florida. And this is your other host.
Evelyn Lamb: Hi. I’m Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City. So how’s it going, Kevin?
KK: It’s okay. Classes are almost over. I’ve got grades for 600 students I still need to upload. But, you know, it’s an Excel nightmare, but once we get done with that, it’s okay. Then my son comes home for Christmas on Saturday.
EL: Oh great. I don’t miss grading. I miss some things about teaching, but I don’t miss grading.
EL: I don’t envy this time of the semester.
KK: Certainly not for a 600-student calculus class. But you know, I had a good time. It’s still fun. Anyway, today we are pleased to welcome Vidit Nanda. Vidit, why don’t you introduce yourself and tell everyone about you?
Vidit Nanda: Hello. My name is Vidit Nanda. I’m a research fellow at the University of Oxford and the amazing new Alan Turing Institute in London. This year I’m a member at the School of Mathematics at the Institute for Advanced Study in Princeton. I’m very happy to be here. Thank you both for doing this. This is a wonderful project, and I’m very happy to be a part of it today.
KK: Yeah, we’re having a good time.
EL: Can you tell us a little more about the Alan Turing Institute? I think I’ve heard a little bit about it, but I guess I didn’t even know it was that new. I thought I had just never heard of it before.
VN: Right. So about three years ago, and maybe longer because it takes time to set these things up, the UK decided they needed a national data science center, and what they did was they collected proposals from universities, and the ones who are now, well, the original five universities that got together and contributed funds and professors and students to the Turing Institute were Oxford, Cambridge, Warwick, UCL, and Edinburgh. Now we have a space on what they call the first floor of the British Library, and we would call the second floor of the British Library. Half of that floor is called the Alan Turing Institute, and it’s kind of crazy. You enter the British Library, and there’s this stack of books that kind of looks like wallpaper. It’s too beautiful, you know, but it is real. It’s behind glass. And then you turn to the right, and it’s Las Vegas, you know. There’s a startup-looking data science center with people dressed exactly the way you think they are with the hoodies, you know. It’s sort of nuts. But there are two things I should tell everyone about the Alan Turing Institute who’s listening. The first one is that if you walk down a flight of steps, there’s a room called Treasures of the British Library. Turn left, and the first thing you see is a table with Da Vinci’s sketches right next to Michaelangelo’s letters with the first printing of Shakespeare. Those are the first things you see. So if you’re ever thinking about cutting a corner in a paper you’re writing, you go down to that room, you feel bad about yourself for ten minutes, and you rush back up the stairs, inspired and ready to work hard.
KK: Yeah. This sounds very cool.
EL: Wow, that’s amazing.
VN: That’s the first table. There’s other stuff there.
KK: Yeah, I’m still waiting on my invitation to visit you, by the way.
VN: It’s coming. It would help if I’m there.
KK: Sure, once you’re back. So, Vidit, what’s your favorite theorem?
VN: Well, this will not be a surprise to the two of you since you cheated and you made me tell you this in advance. And this took some time. My favorite theorem is Banach’s fixed point theorem, also called the contraction mapping principle. And the reason it’s my favorite theorem is it’s about functions that take a space to itself, so for example, a polynomial in a single variable takes real numbers to real numbers. You can have functions in two dimensions taking values in two dimensions, and so on. And it gives you a criterion for when this function has a fixed point, which is a point that’s sent to itself by the function.
One of the reasons it’s my favorite theorem—well, there are several—but it’s the first theorem I ever discovered. For the kids in the audience, if there are any, we used to have calculators. I promise. They looked like your iPhone, but they were much stupider. And one of the most fun things you could do with them was mash the square root button like you were in a video game. This is what we had for entertainment.
KK: I used to do this too.
VN: Take a large number, and you mash the square root button, and you get 1. And it worked every time.
VN: And this is Banach’s fixed-point theorem. That’s my proof of Banach’s fixed-point theorem.
KK: That’s great. What’s the actual statement, though? Let’s be less loose.
VN: Right. The actual statement requires a little bit more work than having an old, beat-up calculator. The setup is kind of simple. You have a complete metric space, and by metric space you mean a space where points have a well-defined distance subject to natural axioms for what a distance is, and complete means if you have a sequence of points that are getting close to each other, they actually have a limit. They stop somewhere. If you have a function from such a complete metric space to itself so that when you apply the function to a pair of points, the images are closer together strictly than the original points were, so f(x) and f(y), the distance between them should be strictly less, some constant less than 1 times the distance between x and y. If this is true, then the function has a unique fixed point, and the amazing part about this theorem that I cannot stress highly enough is that the way to find this fixed point is you start anywhere you want, pick any initial point and keep hitting f, this is mashing the square root button, and very quickly, you converge to the actual fixed point. And when you hit the square root button, nothing changes, you just stay at 1.
KK: And it’s a unique fixed point?
VN: It’s a unique fixed point because wherever else you start, you reach that same place. So I’m an algebraic topologist by trade, and this is very much not an algebraic topology fixed-point theorem. The algebraic topology fixed-point theorem makes no assumptions on the function, like it should be bringing points closer together. It makes assumptions on the space where the function is taking its values. It says if the space is nice, maybe convex, maybe contractible, then there is a fixed point, no uniqueness and no recipe for converging to the fixed point.
KK: In fact, we recently had a guest who chose the Brouwer fixed-point theorem.
VN: Yes, the Brouwer fixed-point theorem is one of my favorites, it’s one of the tools I use in my work a lot, but I always have this sort of analyst envy where their fixed-point theorem comes with a recipe for finding the actual fixed point.
VN: Instead of an existence result.
KK: Yeah, we just wave our hands and say, “Yeah, yeah, yeah, if you didn’t have a fixed point there’d be some map on homology that couldn’t exist and blah blah blah.
VN: Right. And that’s sort of neat but sort of unsatisfying if what you actually care about are the fixed points.
EL: Yeah, so in some ways I kind of ended up more of an analyst because of this. I was really attracted to algebra and that kind of thing, and I felt like at some point I just couldn’t do anything. I felt like in analysis, at least I could get a bound on something, even if it was a really ugly bound, I could at least come in with my hands and play around in the dirt and eventually come up with something. This is probably showing that somehow my brain is more likely to succeed at analysis or something because I know there are people who get to algebra and they can do things, but I just felt like at some point it was this beautiful but untouchable thing, and analysis wasn’t so pretty, and I didn’t mind going and mucking it up.
KK: I had the opposite point of view. I never liked analysis. All those epsilons and deltas, and maybe it was a function of that first advanced calculus course, where you have to get at the end the thing you’re looking for is less than epsilon, not 14epsilon+3epsilon^2. It had to be less than epsilon. I was like, man, come on, this thing is small! Who cares? So I liked the squishiness of topology. I think that’s why I went there.
VN: I think with those epsilon arguments, I don’t know about you guys, but I always ended up doing it twice. You do it the first time and get some hideous function of epsilon, and then you feed back whatever you got to the beginning of the argument, dividing by whatever is necessary, and then it looks like, when you submit your solution, it looks like you were a genius the whole time, and you knew to choose this very awkward thing initially, and you change the argument.
KK: That’s mathematics, right, when you read a paper, it’s lovely. You don’t see all the ugly, horrifying ream of paper you used for the calculations to get it right, you know. I think that’s part of our problem as mathematicians from a PR point of view. We make it look so slick at the end, and people think, wait a minute, how did you do that? Like it’s magic.
VN: We’re very much writing for people next door in our buildings as opposed to people on the street. It helps sometimes, and it also bites us.
KK: This is where Evelyn’s so great, because she is writing for people on the street, and doing it very well.
EL: Well thank you. I didn’t intend this to come back around here, but I’ll take it. Anyway, getting back to our guest, so when did you first encounter this theorem, and was it something you were immediately really into, or did it take some more time?
VN: Actually, the first time I encountered this theorem in a semiformal setting, it just blazed by. I think this is where most people see it for the first time, is in a differential equations course. One of the things that’s so neat about this theorem is that it’s one of the things that guarantees you take f’(x), which equals some hideous expression of x, why should this have a solution, how long should it have a solution for, when is a solution unique? And this requires the hideous thing on the right side to satisfy the contraction mapping property. The existence and uniqueness of ordinary differential equations is the slickest, most famous application of the Banach fixed-point theorem.
KK: I’d never thought about it.
VN: And the analyst nods while Kevin stares off into space, wondering why this should be the case.
KK: No, no, you had a better differential equations course than I did. In our first diffeq’s course, we wouldn’t bring this up. This is too high-powered, right?
VN: It was sort of mentioned, this was at Georgia Tech. It was mentioned that this property holds, there was no proof, even though the proof is not difficult. It’s not so bad if you understand the Cauchy sequence, which not everyone in differential equations does. So we were not shown the proof, but there’s a contraction mapping principle. And then Wikipedia was in its infancy, so now I’m dating myself badly, but I did look it up then and then forgot about it. And then of course I saw it in graduate school all over the place.
KK: Hey, when I was in college, the internet didn’t exist.
VN: How did you get anything done?
KK: You went to the library.
EL: Did you use a card catalog?
KK: I’m a master of the card catalog.
EL: We had one at my elementary school library.
KK: Geez. So growing up in high school, we used to go to the main public library downtown where they had bound periodicals and so if you needed to do your report about, say, the assassination of John Kennedy, for example, you had to go and pull the old Newsweeks off the shelf from 1963. I don’t know, there’s something to that. There’s something to having to actually dig instead of just having it on your phone. But I don’t want to sound like an old curmudgeon either. The internet is great. Well, although wait a minute, the net neutrality vote is happening right now.
VN: It’s great while we speak. We don’t know what’s going to happen in 20 minutes.
KK: Maybe in the middle of this conversation we’re going to get throttled. So Vidit, part of the fun here is that we ask our guest to pair their theorem with something. So what have you chosen to pair the contraction theorem with?
VN: I’m certainly not going to suggest Plato like one of the recent guests. I have something very simple in mind. The reason I have something simple in mind is there’s an inevitability to this theorem, right? You will find the fixed point. So I wanted something inevitable and irresistible in some sense, so I want to pair it with pizza.
EL: Pizza is the best food. Hands down.
VN: Right. It is the best food, hands down. I’m imagining the sort of heathens’ way of eating pizza, right, you eat the edges and move in. I’ve seen people do this, and it’s sort of very disturbing to me. The edge is how you hold the damn thing in the first place. But if you imagine a pizza being eaten from the outside, that’s how I think of the contraction mapping, converging to the middle, the most delicious part of the pizza. I refuse to tell you what fraction of the last two weeks it took me to come up with this pairing. It’s disturbingly difficult.
KK: So you argue that the middle of the pizza is the most delicious part?
EL: Oh yeah.
KK: See, my dog would argue with you. She is obsessed with the crust. If we ever get a pizza, she’s just sitting there: “Wait, can I have the crust?”
EL: But the reason she gets the crust is because humans don’t find it the most delicious.
VN: If I want to eat bread, I’ll eat bread.
KK: I make my own pizza dough, so I make really good pizza crust. It’s worth eating. It’s not this vehicle. But you’re right. Yeah, sure.
EL: We’re going to press you now. What pizza toppings are we talking here? We really need specifics. It’s 9 am where I am, so I can’t have pizza now unless I made my own.
KK: You could. You can have it any time of the day.
EL: But I don’t think there’s a store open. I guess I could get a frozen pizza at the grocery store.
VN: Kevin would suggest having a quick-rise dough set up that, if you pour your yeast in it, it’ll be done in 20 minutes. I think, I’m not big into toppings, but it’s important to have good toppings. Maybe bufala mozzarella and a bit of basil, keep it simple. There’s going to be tomatoes in it, of course, some pizza sauce. But I don’t want to overload it with olives and peppers and sausage and all that.
EL: Okay. So you’re going simple. That’s what we do. We make our own pizza a lot, and a couple years ago we decided to just for fun buy the fancy canned tomatoes from Italy, the San Marzanos.
VN: The San Marzanos, yeah.
EL: Buy the good mozzarella. And since then, that’s all we do. We used to put a bunch of toppings on it all the time, and now it’s just, we don’t even make a sauce, we just squish the tomatoes onto the pizza. Then put the cheese on it, and then the basil, and it’s so good.
KK: I like to make, I assume you’ve both been to the Cheese Board in Berkeley?
EL: No, I haven’t. I hear about it all the time.
KK: It’s on Shattuck Ave in Berkeley, and they have the bakery. It’s a co-op. The bakery is scones—delicious scones, amazing scones—and bread and coffee and all that. And right next door is a pizza place, and they make one kind of pizza for the day, and that’s what you’re going to have. You’re going to have it because it’s delicious. Even the ones where you’re like “eh,” it’s amazing. The line goes down the block, and everybody’s in a good mood, there’s a jazz trio. Anyway, I got the cookbook, and that’s how I make my crust. There’s a sourdough crust, and then our favorite one is the zucchini-corn pizza.
KK: It’s zucchinis, onions, and cheese, and then corn, and a little feta on top. And then you sprinkle some cilantro and a squeeze of lime juice.
VN: God, I’m so hungry right now.
KK: This is amazing. Yeah, it’s almost lunchtime. My wife and I are going to meet for lunch after this, so can we wrap this up?
EL: Hopefully you’re going to have pizza.
KK: We’re going to a new breakfast place, actually. I’ve got huevos rancheros on my mind.
EL: That’s good too.
KK: Well this has been great fun, Vidit. Thanks for joining us.
VN: Thanks so much again for having me and for doing this. I’m looking forward to seeing who else you’ve managed to rope in to describe their favorite theorems.
KK: There are some good ones.
EL: We’re enjoying it.
KK: We’re having a good time.
VN: Wonderful. Thank you so much, and have fun.
EL: Nice talking to you.
KK: See you. Bye.