Kevin Knudson: Welcome to My Favorite Theorem, a special Valentine’s Day edition this year.

Evelyn Lamb: Yes.

KK: I’m one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida. This is your other host.

EL: Hi. I’m Evelyn Lamb. I’m a math and science writer in Salt Lake City, Utah.

KK: How’s it going, Evelyn.

EL: It’s going okay. We got, we probably had 15 inches of snow in the past day and a half or so.

KK: Oh. It’s sunny and 80 in Florida, so I’m not going to rub it in. This is a Valentine’s Day edition. Are you and your spouse doing anything special?

EL: We’re not big Valentine’s Day people.

KK: So here’s the nerdy thing I did. So Ellen, my wife, is an artist, and she loves pens and pencils. There’s this great website called CW Pencil Enterprise, and they have a little kit where you can make a bouquet of pencils for your significant other, so this is what I did. Because we’ve been married for almost 27 years. I mean, we don’t have to have the big show anymore. So that’s our Valentine’s Day.

EL: Yeah, we’re not big Valentine’s Day people, but I got very excited about doing a Valentine’s Day episode of My Favorite Theorem because of the guests that we have. Will you introduce them?

KK: I know! We’re pleased to have Nikita Nikolaev and Beatriz Navarro, and they had some popular press. So why don’t you guys introduce yourselves and tell everybody about you?

Nikita Nikolaev: Hi. My name is Nikita. I’m a postdoctoral fellow at the University of Geneva in Switzerland. I study algebraic geometry of singular differential equations.

Beatriz Navarro Lameda: And I’m Beatriz. I’m currently a Ph.D. student at the University of Toronto, but I’m doing an exchange program at the University of Geneva, so that’s why we’re here together. And I’m studying probability, in particular directed polymers in random environments.

EL: Okay, cool! So that is actually applicable in some way.

BNL: Yes, it is.

EL: Oh, great!

KK: So why don’t we talk about this whole thing with the wedding? So we had this conversation before we started recording, but I’m sure our listeners would love to hear this. So what exactly happened at your wedding?

NN: Both of us being mathematicians, of course, we had almost everybody was either a mathematician or somehow mathematically related, most of our guests. So we decided to have a little bit of fun at the wedding, sprinkle a little bit of maths here and there. And one of the ideas was to, when the guests arrive at the dinner, in order for them to find which table they’re sitting at, they would have to solve a small mathematical problem. They would arrive at the venue there, and they would open their name card, and the name card would contain a first coordinate.

BNL: And a question.

NN: And a question. And the questions were very bespoke. It really depended on what we know their mathematical background to be. We had many people in my former research group, so I pulled questions from their papers or some of the talks they’ve given.

EL: This is so great! Yeah.

NN: And there were some people who are, maybe they’re not mathematicians, they’re engineers or chemists or something, and we would have questions which are more mathematically flavored rather than actual mathematical questions just to make everyone feel like they’re at a wedding of mathematicians.

EL: Right.

NN: So right. They had to find out two coordinates. All the tables were named after regular polyhedra, and they had to find out what their polyhedron of the night was.

EL: Okay.

NN: In order to do that, there was a matrix of polyhedra. Each one had two coordinates, and once you find out what the two coordinates are, you look at that matrix, and it gives you what polyhedron you’ve got. So as a guest, you would open the name card, and it would contain your first coordinate and a question and a multiple choice answer. And the answers were,

BNL: Usually it was one right answer and two crazy answers that had nothing to do with the question. Most of them were 2019 because that’s the year we got married, and then some other options. And then once you choose your answer, you would be directed to some other card that had a name of some mathematical term or some theorem, and that one would give you the second coordinate.

NN: So we made this cool what we called maths tree. We had several of these manzanita trees, and we put little cards on them with names, with these mathematical terms, with the answers, with people’s questions, and we just had this tree with cards hanging down, more than 100 cards hanging down. What I liked, in mathematicians it induced this kind of hunting instinct. You somehow look at this tree, and there are all these terms that you recognize, and you’ve seen before in your mathematical career, and you’re searching for that one that you know is the correct one.

BNL: And of course we wanted to make sure everyone found their table, so if they for any reason chose the wrong answer, they would also be directed to some card with a mathematical term. And when they opened it, it would say, “Oops, try again.” So that way you knew, okay, I just have to go and try again and find what the correct coordinate would be.

KK: This is amazing.

NN: And then to foolproof the whole thing, during the cocktail hour, they would do this kind of hunting for mathematical terminology, but then the doors would open into the dinner room, and just like most textbooks, when you open the back of the textbook, there’s the answer key, in the dinner room we had the answer key, which was a poster with everyone’s names

BNL: And their polyhedra

NN: And their polyhedron of the night.

EL: Yeah.

NN: So it was foolproof. I think some commentators on the internet were very concerned that some guests wouldn’t find their seat and starve to death.

EL: Leave hungry.

NN: No, it was all thought through completely.

KK: Some of the internet comments, people were just incredulous. Like, “I can’t believe these people forced their guests to do this!” They don’t understand, we would think this was incredible. This is amazing!

EL: Yeah! So delightfully nerdy and thoughtful. So, yeah, we’ve mentioned, this did end up on the internet in some way, which is how I heard about it, because I sadly was not invited to your wedding. (Since this is the first time I have looked at you at all.) So yeah, how did it end up making the rounds on some weird corner of the internet?

NN: So basically a couple of weeks before the wedding, I made a post on Facebook. It was a private Facebook post, just to my Facebook friends. You know, a Facebook friend is a very general notion.

EL: Right.

NN: I kind of briefly explained that all our guests are mathematicians, so we’re going to do this cool thing, we’re going to come up with mathematical questions, and one of my Facebook “friends,” a Facebook acquaintance, I later found out who it was, they didn’t like it so much, and they did a screengrab, and then they posted, with our names redacted and everything redacted, they made a post on Reddit which was like, “Maths shaming, look, these people are forcing their guests to solve a mathematical question to find their seat, maths shaming.”

BNL: It was in the “bridezilla” thread. “This crazy bride is forcing their guests to solve mathematical problems,” and how evil she is. Which, funny because Nikita was the one who wrote that Facebook post.

NN: I actually was the one.

BNL: So it was not a bridezilla, it’s what I like to call a Groom Kong.

NN: That’s right. So then this Reddit thread kind of got very popular, and later some newspaper in Australia picked it up, and then it just snowballed from there. Fox News, Daily Mail, yeah.

KK: Well, this is great. This is good, now you’ve had your 15 minutes of fame, and now life can get back to normal.

EL: Yeah.

KK: This is a great story. Okay, but this is a podcast about theorems, so what is your favorite theorem?

NN: Right, yeah. So we kind of actually thought long and hard about what theorem to choose. Like, what is our favorite theorem is such a difficult question, actually.

KK: Sure.

NN: It's kind of like, you know, what is your favorite music piece? And it's, I mean, it's so many variables. Depends on the day, right?

EL: Yeah.

NN: But we ended up deciding that we were going to choose the intermediate value theorem.

KK: Oh, nice.

NN: As our favorite theorem.

KK: Good.

BNL: So, yes, the intermediate value theorem is probably one of the first theorems that you learn when you go to university, right? Like calculus you start learning basic calculus, and it's one of the first theorems that you see. Well, what it says is that well, suppose you start with a continuous function f, and you look at some interval (a, b), so the function f sends a to f(a), b to f(b), and then you pick any value y that is between f(a) and f(b). And then you know that you will find a point c that is between a and b, such that f(c) equals y.

So it looks like an incredibly simple statement. Obvious.

KK: Sure.

BNL: Right, but it is a quite powerful statement. Most students believe it without proof. They don't need it. It's, yes, absolutely obvious. But, well, we have lots of things that we like about the theorem.

NN: Yeah, I mean, it feels incredibly simple and completely obvious. You look at it, and, you know, it's the only thing that could possibly be true. And the cool thing about it, of course, is that it represents kind of the essence of what we mean by continuous function.

KK: Sure.

EL: Yeah.

NN: In fact, actually, if you look at the history, before our modern formal definition of continuity, that was part of the property that was a required property that people used to use as part of the definition of continuity. In fact, actually, if you look at the history, before we formalize the definition of continuity, people were very confused about what a continuous function actually should mean. And many thought, erroneously, that this intermediate value property was equivalent to continuity. In some sense, it's what you would want to believe because it really is the property that more or less formalizes what we normally tell our students, that heuristically, a continuous function is one that if you want to draw it, you can draw it without taking off your pencil off of a piece of paper, and that's what the Intermediate Value Theorem, this intermediate value property, it represents that really.

But for all its simplicity and triviality, if you actually look at it properly, if you look at our modern definition of continuity using epsilon-delta, then it becomes not obvious at all.

BNL: Yes. So if you see epsilon, you look at the definition of continuity, and have epsilons, deltas, how is it possible that from this thing, you get such an obvious statement, like the intermediate value theorem? So what the intermediate value theorem is telling us is that, well, continuous functions do exactly what we want them to do. They are what we intuitively think of a continuous function. So in a sense, what the intermediate value theorem is doing for us is serving as a bridge between this formal definition that we encounter in university. So we start first year calculus, and then our professor gives us this epsilon-delta definition of continuity. And it's like, oh, but in high school, I learned that a continuous function is one that I can draw without lifting my pencil. Well, the intermediate value theorem is precisely that. It's connecting the two ideas, just in a very powerful way.

NN: Yeah. And also, you know, it cannot be overstated how useful it is. I mean, we use it all the time. As a geometer, of course, you know, you use some generalization of it, that continuous functions send connected sets to connected, and we use it all the time, absolutely, without thinking, we take it absolutely for granted.

BNL: So even if you do analysis, you are using it all the time, because you can see that the intermediate value theorem is also equivalent to the least upper bound property, so the completeness axiom of the real numbers. Which is quite incredible, to see that just having the intermediate value theorem could be equivalent to such a fundamental axiom for the real numbers, right? So it appears everywhere. It's surprising. We know, it's very easy, when you see the proof of the intermediate value theorem, you see that it is a consequence of this least upper bound property, but the converse is also true. So in a sense, we have that very powerful notion there.

KK: I don't think I knew the converse. So I'm a topologist, right? So to me, this is just a statement that the continuous image of a connected set is connected. But then, of course, the hard part is showing that the connected subsets of the real line are precisely the intervals, which I guess is where the least upper bound property comes in.

BNL: Yes, indeed, yes. Exactly yes.

KK: Okay. I haven't thought about analysis in a while. As it turns out, we're hiring several people this year. And, for some of them, we've asked them to do a teaching demonstration. And we want them all to do the same thing. And as it so happens, it's a calculus one demonstration about continuity and the intermediate value theorem.

BNL: Oh.

EL: Nice.

KK: So in the last month, I've seen, you know, 10 presentations about the intermediate value theorem. And I've come to really appreciate it as a result. My favorite application is, though, that you can use it to prove that you can always make any table level, or not level, but all four legs touch the ground at the same time.

BNL: Yes.

KK: Yeah, that's, that's great fun. The table won't be level necessarily, but all four feet will be on the ground, so it won't wobble.

BNL: Yes.

EL: Right.

NN: If only it were actually applied in classrooms, right?

KK: Right.

EL: Yeah.

NN: The first thing you always do when you come to, you sit at a desk somewhere, is to pull out a piece of paper to actually level it.

EL: Yeah. So was this a theorem that you immediately really appreciated? Or do you feel like your appreciation has grown as you have matured mathematically?

BNL: In my case, I definitely learned to appreciate it more and more as I matured mathematically. The first time I saw the theorem, it's like, "Okay, yes, interesting, very cool theorem." But I didn't realize at the moment how powerful that theorem was. Then as my mathematical learning continued, then I realized, "Oh, this is happening because of the intermediate value theorem. And this is also a consequence of it." So there's so many important things that are a consequence, of the intermediate value theorem. That really makes me appreciate it.

NN: Well, there's also somehow, I think this also comes with maturity, when you realize that some very, what appear to be very hard theorems, if you strip away all the complexity, you realize that they may be really just some clever application of the intermediate value theorem.

BNL: Like Sharkovskii's theorem, for example, is a theorem about periodic points of continuous functions. And it just introduces some new ordering in the natural numbers. And it tells you that if you have a periodic point of some period m, then you will have periodic points of any period that comes after m in that ordering. You can also look at the famous "period three implies chaos."

KK: Right.

BNL: A big component of it is period three implies all other periods. And the proof of it is really just a clever use of the intermediate value theorem. It's so interesting, that such an important and famous theorem is just a very kind of immediate--though, you know, it takes some work to get it--but you can definitely do it with just the Intermediate Value Theorem. And I actually like to present that theorem to students in high school because they can believe the Intermediate Value Theorem.

EL: Yeah.

BNL: That's something that if you tell someone, "This is true," no one is going to question it is definitely true.

KK: Sure.

BNL: And then you tell them, "Oh, using this thing that is obvious, we can also prove these other things." And I've actually done with high school students to, you know, prove Sharkovskii's theorem just starting from the fact that they believe the intermediate value theorem. So they can get to higher-level theorems just from something very simple. I think that's beautiful.

NN: Yeah, that's kind of a very astonishing thing, that from something so simple, and what looks obvious, you can get statements which really are not obvious at all, like what she just explained, Sharkovskii's theorem, that's kind of a mind blowing thing.

EL: Yeah, you're making a pretty good case, I must say.

KK: That's right.

EL: So when we started this podcast, our very first episode was Kevin and I just talking about our own favorite theorems. And I have already since re-, you know, one of our other guests has taken my loyalty elsewhere. And I think you're kind of dragging me. So I think, I think my theorem love is quite fickle, it turns out. I can be persuaded.

KK: You know, in the beginning of our conversation, you pointed out, you know, how does one choose a favorite theorem, right? And, and it's sort of like, your favorite theorem du jour. It has to be.

BNL: Exactly, yes.

EL: Yeah.

KK: All right, so what does one pair with the intermediate value theorem?

BNL: So we thought about it. And to continue with the Valentine's Day theme, we want to pair the intermediate value theorem with love in a relationship.

KK: Ah, okay, good.

BNL: The reason why we want to pair it with love is because when you love someone, it's completely obvious to you. You just know it's true, you know you love someone.

KK: That's true.

BNL: You just feel like there's no proof required. It's just, you know it, you love this person.

NN: It's the only thing that can possibly be true, there's no reason to prove it.

BNL: But also, just like any good theorem, you can also prove, you can provide a proof of love, right? You can show someone that you love them.

NN: Any good mathematical theorem can always be supplied with a very rigorous, detailed to whatever required level proof. And if you truly, really truly love someone, you can prove it. And if someone questions, any part of that proof, you can always supply more details and a more detailed explanation for why why you love that person. And that's why there's a similarity between the intermediate value theorem and love in a relationship.

EL: Yeah, well, I'm thinking of the poem now, "How do I love thee? Let me count the ways," This is a slightly mathematically-flavored poem.

KK: But I think there must be at least, you know, the, the continuity of the continuum ways, right? Or the cardinality of the continuum ways.

NN: Absolutely.

KK: That's an excellent pairing.

EL: Yeah.

BNL: We also thought that love is something that we feel, we take it as an obvious statement, and then from love, we can build so many other things, right? Like in the intermediate value theorem case, we start from a theorem that looks obvious, and using it, we can prove so many other theorems. So it's the same, right, in a relationship. You start from love, and then you can build so many other great things.

EL: Yeah, a marriage for example.

BNL and NN: For example, yes.

EL: Yeah. And a ridiculously amazing wedding game as part of that.

NN: There were some other mathematical tidbits in the wedding. So one of them I'll mention is our rings. Our wedding bands are actually Möbius bands.

KK: Oh, I see.

EL: Okay, very nice.

NN: We had to work with a jeweler. And there's a bit of a trick, because if you just take a wedding band, and you do the twist to make it a Möbius band, than the place where it twists would stick out too much.

EL: Yeah.

NN: So the idea is to try to squish it. And that, of course, is a bit challenging if you want to make a good-looking ring, so that was part of the problem to be solved.

EL: Yeah. Well, my wedding ring is also--it's not a Möbius band. But it's one that I helped design with a particular somewhat math-ish design.

KK: My wife and I are on our second set of wedding bands. The first ones, because we were, I was a graduate student and poor, we got silver ones. And silver doesn't last as long, so we're on our second ones. But the first ones were handmade, and they were, they had sort of like a similar to Evelyn's sort of little crossing thing. So they were a little bit mathy, too. I guess that's a thing that we do, right?

EL: Yeah.

NN: It's inevitable.

KK: Yeah, excellent.

KK: So we like to give our guests a chance to plug anything. Do you have any websites, books, wedding registries that you want to plug?

NN: Actually, in terms of the wedding registry, lots of our guests, of course, were asking. We didn't have a wedding registry because given the career of a postdoc, where you travel from place to place every few years, a wedding registry isn't the most practical thing. Yeah, difficult.

BNL: Yes. So we said, well, you can just give us anything you like, we'll have a box where you can leave envelopes. And some of our guests were very creative. They gave us, some of them decided to give us money. But the amounts they chose were very interesting, because they were, like, some integer times e or times π, or some combination. They wrote the number and then they explained how they came up with that number. And that was very interesting and sweet.

NN: Some of them didn't explain it. But we kind of understood. We cracked the code, essentially, except one. So one of our friends wrote us a check with a very strange number. And to this day,

BNL: We still don't know what the number is.

NN: We kept trying to guess what it could be. But no, I don't know. Maybe eventually I'll just have to ask. I'd like to know.

KK: Maybe it was just random.

NN: Maybe it was just random.

BNL: Yeah, I think one of the best gifts people gave us was their reaction right after seeing their card. In particular, there is a very nice story of a guest who really, really loved the way we set up everything and maybe you can tell us about that.

NN: Yeah, so we, at the dinner we would approach tables to say hi to some guests, and so this particular, he's actually Bea's teaching mentor.

BNL: So I'm very much into teaching. And he's the one who taught me most of the things I know.

NN: So we approached him, and, and he looked at us, and he pulled out the name card out of his breast pocket like, "This. This is the most beautiful thing I've ever seen. This is incredible. It's from my last paper, isn't it?" Yes. Yeah, that's right. He's like, "I have to send it to my collaborator. He's going to love it." And just seeing that reaction, him telling us how much he loved the card, just made all those hours that I and Bea spent reading through papers and trying to come up with some kind of, you know, short sounding question to be put into multiple choice, made all of that worthwhile.

EL: Yeah, I'm just imagining it. Like, usually you don't have to, like, cram for your wedding. But yeah, you've got all these papers you've got to read.

BNL: Yeah, we spent days going through everyone's papers and trying to find questions that were short enough to put in a small card and also easy to answer as a multiple choice question.

NN: Yeah, some were easy. So for example, my former PhD advisor came to our wedding, and I basically gave him a question from my thesis, you know, just to make sure he'd read it.

EL: Yeah.

NN: So when we approached him at the dinner and I said, "Oh, did you like the question?" and he just looked at me like, "Yeah, well I gave you that question two years ago!"

EL: Yeah

NN: So, yes, some questions were easy to come up with. Some questions were a bit more difficult. So we had a number of people from set theory, and neither of us are in set theory. I'd never, ever before opened a paper in set theory. It was all very, very new to me.

EL: Nice.

KK: Well, this has been great fun. Thanks for being such good sports on short notice.

EL: Yeah.

KK: Thank you for joining us.

BNL: Yeah.

EL: Yeah, really fun to talk to you about this. It's so much better than even the Reddit post and weird news stories led me to believe.

KK: Well, congratulations. it's fun meeting you guys. And let me tell you, it's fun being married for 27 years, too.

NN: We're looking forward to that.

KK: All right, take care.

NN: Thank you. Bye bye.

BNL: Bye.