Kevin Knudson: Welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined by your other co-host person.

Evelyn Lamb: Hi, I am Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. And I was actually thinking we should have a quiz at the end of this one.

KK: We really should.

EL: It’s just so jam-packed. There's gonna be so many different things floating around. So, like, be prepared…actually don't because we haven't prepared a quiz for you, so we don’t want you to be disappointed.

KK: I’ll start writing the quiz now. Yeah, today we have an interesting new experiment that we're going to try. So Mike Krebs from Cal State University in Los Angeles reached out to us with an idea. Mike, why don't you just introduce yourself and explain?

Mike Krebs: Hi, my name is Mike Krebs. I'm a professor of mathematics at California State University Los Angeles. Graduation is tomorrow, and I think our students have had enough of quizzes, so thank you for passing on the quiz. Yeah, I listen to a lot of podcasts, and my origin story of finding your podcast is sometimes to find a new one, I will go to Wikipedia and click the “random article” button, and then whatever comes up, search to find a podcast on that.

KK: Okay.

MK: I found various things that way like the story of Sylvia Weiner, an octogenarian marathon runner, and so on and so forth. And then one time, I clicked “random article,” and up came a page on differential geometry of surfaces.

KK: Okay.

MK: And one Google Search later, I started screaming at my laptop, “There’s a podcast called My Favorite Theorem!” So, yeah, I discovered that at the time I was teaching, this past semester, a capstone course for our math majors, in which students select a topic and then have to write about it and present about it. And I said, “Oh, I wonder if the good folks at my favorite theorem would be interested in doing something like that with students.” So I recruited some students from that class, as well as a bunch of other students from our university. And here we are now.

KK: All right.

EL: That’s amazing. And so you're mostly graduating seniors about to graduate and you're spending the morning before your graduation with us? I feel so honored.

KK: I really do. This is something else. Yeah.

EL: Well, let's get to it.

KK: We have nine students. And so as Mike pointed out, there are nine factorial or 362,880 possibilities here. And we have chosen one of those orders.

EL: Yes. You know, if you so choose, you can always divide this into tracks and listen to them in every possible order and then get back to us and tell us what the optimal order would have been. But for now, it's the order in which they appeared on my Zoom screen. So our first guest today is Pablo Martinez Gutierrez. Great to have you. Would you like to say a little bit about yourself and let us know your favorite theorem?

Pablo Martinez Gutierrez: Hi, thank you for having me on the show. Yes, I'm Pablo. I'm currently a math undergraduate at Cal State LA, hoping to complete my Bachelor's, not this semester, but hopefully next fall next semester. And my favorite theorem that I'm covering for today is Euler’s formula and Euler’s identity. It's something that I got exposed to back in Professor Krebs’s class when I took his class for differential equations. He was teaching us about second order linear homogeneous differential equations. And in one class session, he introduced the topic of Euler’s formula and identity as a side gem. And I was like, “Oh my goodness, this thing is so incredibly beautiful.” The way that I learned it in his class was he introduced the mathematical expression ex as a Taylor series, and he expanded it out as a series. And then when plugging in eix, then you spat out that series and because of the i, something interesting happens where it starts to be, you could split it up into two individual, or two smaller series, so to speak, of cosine, and i sine x. So you would have the expression eix equal to cosine x plus i sine x. And that to me just seemed that for me, it was like I was gobsmacked. It was just baffling. It was incredible.

EL: Yeah, everything just falls out after that, right?

PMG: Yeah, you're seeing all these terms that come from math, you have e, that comes from compounded interest back when you're learning about it in algebra, you have sine and cosine, that are coming in from the unit circle and trig. And then you have i from complex numbers. So all those just coming in together is is like mind-boggling, right? And then if that wasn't amazing in and of itself, something interesting and amazing, even more amazing, happens when you plug in π for x, right? So you have eiπ is equal to cosine π plus i sine π. And so the cosine π just becomes negative one. And the i sine π becomes zero, which just goes away. So then you have eiπ equal to negative one. And then if you add one to both sides, you get eiπ+1 is equal to zero. And that's just — when I saw it, I was in awe. And I was just like, how do these things align and assemble so beautifully and neatly and concisely? It doesn't seem like, it seems crazy that it would happen that way.

EL: I have an unpopular, or possibly controversial opinion about this, which is, it's cooler to leave it with the minus one on the other side, instead of doing the plus one equals zero. Don't cancel me for my controversial Euler formula takes, but I’ve just got to put that out there.

KK: My favorite part about complex exponentials like this is that you can forget all of those sum formulas, right? Like, if you want to know the cosine of three theta, you just use the complex exponential. It makes your life so much simpler. So that's my fun thing. Okay, this is a really beautiful fact. So what have you chosen to pair with this fact?

PMG: So my pairing for this formula and identity is this. I don't know if anyone's seen the Stephen Hawking movie Theory of Everything. The ending scene of that movie has this musical score that I like to listen to, that evokes a similar feeling of elegance and beauty, and awe about the universe, which is the same feeling I get from this identity and formula. It's called the Arrival of the Birds by the Cinematic Orchestra and the London Metropolitan Orchestra. You can give it a listen on YouTube. And any you as you listen to it, it elicits that feeling of awe.

EL: Yeah, listen to it while you do some complex integrals, maybe. I like it. Yeah. Thank you.

KK: All right.

PMG: Thank you.

EL: Yes, well, the bar has been set high. But yeah, we will see — no, I won’t pit anyone against each other. Our next guest is Holly Kim. So yes, Holly, if you'd like to tell us about yourself, and tell us your favorite theorem.

Holly Kim: Hi. So my name is Holly. And I'm currently a grad student at Cal State Los Angeles. And I'm not graduating this semester, so I still have about, like a year or year and a half before I graduate. But I'm happy to be here. So thank you for having me on the show as well.

KK: Absolutely.

HK: My favorite theorem is currently Ore’s theorem from graph theory, which states that for a given graph that’s simple and finite, and for two vertices that are distinct and non-adjacent, if the sum of the degrees of those two vertices is greater than or equal to the total number of vertices of your graph, then the graph is Hamiltonian, meaning you can find a Hamiltonian cycle, meaning you can find a spanning cycle that reaches every vertex once it's in the graph. So that one is my favorite. And it's interesting because I was not a math person when I got my bachelor's. So when I took my first proof-based course, the professor quickly mentioned Hamiltonian graphs. And I had not seen graph theory, I think in that form, at least, ever. So it was really interesting at that time. And he had made a joke about like, “It's not the Hamilton that they made the musical about.” And around that time, I thought that was so funny because I was also listening to Hamilton the musical, or had started listening to it, even though it had been out for a while by the time I'd taken that course. But it just sort of stuck with me, and I thought Hamiltonian graphs, and Hamilton the musical, they’re just sort of like, every time I thought about it, I thought, oh, how fun and how interesting and how funny Hamiltonian graphs are. And then what makes them even more interesting is that unlike Eulerian graphs, where you can tell a graph is Eulerian quickly by looking at the — you know, it's if and only if every vertex has an even degree. So then you know that graph is definitely learning. The Hamiltonian graphs don't have sort of a defining characteristic, like Eulerian graphs. So Hamiltonian graphs are sort of elusive, like there are some theorems that will work for certain families, or types of graphs, but nothing that quite, I think, captures, yes, for sure every graph — or this graph is Hamiltonian if and only if these conditions are satisfied. So it's not been discovered or found out yet. So that's my current favorite theorem.

KK: I don't know this theorem. So this is sort of interesting, right? So it basically says that if you have two vertices in the graph that have enough edges out of them, basically, you're guaranteed a Hamiltonian cycle. That's just, that's pretty remarkable, actually.

HK: Yeah, and the proof has like a funny — it's like a proof by contradiction, but there are certain edges, like you cannot have, as you construct this proof, otherwise, you will end up having a Hamiltonian cycle. So it's like, you’ve got to have just enough, but not too many, or where you actually end up with another, like a Hamiltonian cycle kind of embedded in, in your graph. And so it's very fun. The converse is of course not true. You can see, the other direction would not.

EL: I'm sitting here trying to doodle myself a graph and see, but I think I think I need to do it a little bigger graph, because there weren't enough vertices in this one. And I can't doodle and talk at the same time.

KK: I can’t either. What’s that about?

EL: So yeah, no multitasking for me.

KK: So what pairs well, with this theorem?

HK: Well, it might be on the nose, but I'm going to do it anyway. I paired it with Hamilton the musical. And I've mentioned it before, but beyond just, it being a good soundtrack to listen to with just about everything, I thought, well, certainly, there must be a deeper connection I can draw between the musical and Hamiltonian graphs. And if you listen to the musical, a motif of it is that oh, Alexander Hamilton is just like never satisfied in terms of his goals and ambitions always wants to do more. And he's never at a point where he's like, Oh, I'm, I'm good. And I don't need to keep going. Just based on the musical. And I kind of thought that Hamiltonian graphs, they aren’t personifications of Alexander Hamilton, but that you know, there is nothing that quite satisfies them at this time, or at least as a whole, like Hamiltonian graphs as a whole. So, there is nothing that that would satisfy like, oh yeah, for sure, I am a Hamiltonian graph if these conditions are met. And so that was my connection. And if you want to go further, that symbol, like the iconic logo of Hamilton, or the Hamiltonian graph of Hamilton the musical, there is a star, which is isomorphic to a C5, a five-cycle, so my pairing was Hamilton the musical.

EL: I like that. Well, just in case you have not seen it, there is an excellent parody of it. (singing) William Rowan Hamilton (end singing) of the the Alexander Hamilton song, done by, I'm forgetting the name of the YouTube channel [Editor’s note: it’s A Cappella Science]. I think it's it's run by a guy called Tim Blais, B-L-A-I-S. So, yeah, check that out. I was actually, I was trying to write a parody, I just would always get in my head that (singing) William Rowan Hamilton (end singing).

HK: That’s so funny.

EL: Then I discovered some other person did it already. But they've got a bunch of people coming in. They've got, like, someone playing Emmy Noether, and a few other math contemporaries. So yes, definitely check that out. We'll include a link to that in the show notes. Yes, excellent. We're doing great here.

KK: Yeah. Who’s next?

EL: So thank you, Holly. Bryce Van Ross, welcome to the podcast!

BVR: Hi, thanks for having me. Excited to be here. I guess I'll introduce myself.

EL: Yes, please do.

BVR: My name is Bryce. Come tomorrow, I'll be a master's graduate in math from Cal State LA. So I'm pretty excited about that.

KK: Congratulations!

BVR: Thank you. Yeah, so the theorem today, I like learning a lot and learning new things. So in the process of choosing a theorem was like, find something new. I found the Hales–Jewett theorem, I believe that's how it's pronounced. The general idea is it's a combinatorial theorem, usually applied to game theory, and you start off with two positive integers, N and C. N would correspond to, for example, like a grid, right, like how many columns or how many rows you have in a grid, and C would correspond to the amount of players alternating in some game. So, the Hales–Jewett Theorem states that for any two positive integers N and C, there will exist some H-dimensional cube corresponding to that N and C. So by cube I mean, like an N by N by N by N H-many times, grid. Now, more than that, that theorem says that corresponding to that cube, there has to exist some, at least one, row, column, or diagonal that is all of the same color, which is pretty impressive. Yeah. And a great way to conceptualize that is, for example, like generalizing the notion of Tic-Tac-Toe, where you have a bunch of like Tic-Tac-Toe, grids as faces of some very big cube. This theorem tells us that, at some point, you're going to find a very long line that's either a row, a column or a diagonal where someone wins, guaranteeing that also someone loses. So that's the theorem I picked.

KK: Wow, okay.

EL: Yeah, so what what kind of — so you gave the example of Tic-Tac-Toe. I must admit I'm fairly ignorant in game theory. It's like, I sort of get, I like the part where you're like, Oh, you've got an N-dimensional, or H-dimensional cube. It’s the part where it's actually the games, that’s why I don't know that much about game theory. Because like, I'm a little bored by my that kind of thing. So what other kinds of games can show up in this?

BVR: Yeah, I'm also unfamiliar with game theory, never even like read a book on game theory or taking a course, but from my understanding, you could extend it towards notions are familiar games like Connect 4, any turn-based game such that it requires somebody satisfying a line of something of the same color.

EL: I know Game Theorists always have these super tricksy ways of like, oh, yeah, a chess game you could just make into this or, you know, Nim you can represent by this or, you know, any game you want, whether it's like a real game, like chess, or a math game like Nim. Yeah. Sorry, hashtag not all math games. But yeah, that's interesting. So you just kind of stumbled on this theorem in a some sort of curiosity rabbit hole?

BVR: Lots of googling at 2am. And Wiki-ing at 2 am.

KK: That's what happens. This sort of reminds me of Sperner’s lemma, right, where you try to, if you're coloring the vertices, you start coloring vertices, then you have to get a triangle with all the same color vertices. I wonder if it's sort of related to that in some way. Or maybe — I'm a topologist, so I’m always trying to think of ways to just turn this into a topology question. It feels like it should be one. But anyway, yeah, all right, so this is a good theorem. What pairs with it well?

BVR: Yeah, so I was trying to think of something where audiences can very much relate to. I'm a big fan of games and media. And I think several people in the world last year, watched the show Squid Game. And season two is upcoming. So Squid Game that's my pairing. The reason why it's my pairing is not just because game is in the title there, but also because for those familiar having seen season one, at certain points in the game, I think it was with marbles or something, they were like, oh, make your own game using these marbles. And I was like, if I were to like change up and impose my own ideas of season two, what if a player gets to choose a game, any game and make it and compete against any number of players. I think that the Hales–Jewett theorem is ideal because everyone would be intimidated by a giant cube of Tic-Tac-Toe. But if you know the theorem, then technically you have an advantage because you know, hey, someone's got to win and ideally, it's you.

KK: Okay.

EL: I must admit, I am too much of a weenie to watch Squid Game.

KK: Yeah, I never watched it either.

EL: From just — from even the SNL sketch that was based on Squid Game, I was like, Nope, not gonna watch that one. It looks too scary for me.

KK: Yeah, my wife really doesn't like anything violent. So we're watching a couple of things right now that there is some violence and she has to you look away. It's just really pretty rough. But I know it’s very popular.

EL: Our very brave listeners, I think, will enjoy that pairing.

KK: Okay, excellent.

EL: Well thank you. Oh, and Bryce, I know that you wanted to share an exhibit that you've got up right now.

KK: That’s right.

BVR: Yeah. Yeah. So I really value, for example, like the themes of this podcast, to give an opportunity for people of any background to get a different perspective on math. And I like that for a variety of topics within STEM. So I work as a library archivist at Cal State LA, and I'm developing a STEM exhibit for all faculty, students, etc, to just visit and change their minds. So there are going to be interesting math related artifacts, as well as just unconventional things. You're going to see, like, dinosaur bones and whatnot. And it will be debuting in the fall. So just wanted to give a hype for anyone interested, they could reach out to Cal State LA Special Collections and Archives to find out more, and it'll be open for everyone.

EL: Awesome. Yeah. I hope our LA-based listeners will check that out.

KK: Sounds very cool.

EL: Okay, thank you, Alvin. Alvin Lew? Would you like to join us? Sorry, I don't want to introduce people differently. Yes, please introduce yourself and let us know about your favorite theorem.

Alvin Lew: Hello, everyone. I'm very glad to be here. I'm Alvin I'm currently a third-year computer science major and math minor. Not directly just in math, but I've been doing machine learning research and kind of being at the intersection of two different subjects there. And so today, I'd like to share not so much a theorem, but I guess the actual theorem is that the cardinality of the real numbers is greater than the cardinality of natural numbers. And specifically, I really, really enjoyed the proof of Cantor's diagonalization.

KK: Class.

AL: And so I don't know, has anyone played the game in elementary school where each person is trying to come up with a bigger number?

KK: Sure.

EL: Yeah.

AL: And then someone would play the trump card of infinity, you know.

EL: And then infinity plus one.

AL: But I think what people don't realize is that there are actually different levels of infinity. And I remember watching a TED Talk video back in middle school, and my mind was so blown by the fact that there could be an infinity greater than a different one. And so having taken a more formal class last semester, talking about Cantor's diagonalization, I think it's such a beautiful proof, because it's so easy for even non-math majors to understand. And it just goes something like: suppose you can list everything out so that you can match each natural number to a real number. And what you'll find is that the real number you can write out as an infinite decimal, so a number with infinite decimal places. And so you just write all of them out, you pair each of them up. And what you ended up doing is you take one of the digits from every single real number, and then you create a new number based on that by changing up all the digits, and you'll find a contradiction that you actually didn't list out that number. And so just that very simple idea there leads to actually quite a few paradoxes. I think there's like a Hilbert hotel paradox if anyone's heard of that, and some other ones, but something so simple like that, just one idea, and it opens the kind of, you know, mathematics world where there's different levels of infinity. And for a normal person who's not too involved in math, I think it's actually very interesting that it's accessible. It's a simple idea that anyone can go ahead and check out.

EL: Yeah, yeah, that such a great one. And I remember that is the one when I was in undergrad, when I saw that and finally understood it, it was it was one of those like, mind blown kind of moments for me, a very special place in my heart.

KK: Yeah, yeah. And then you want all these other weird things like, you know, like the cantor set is also uncountable. It's kind of the same argument. Yeah, it definitely blows blows everyone's mind the first time. And but the second time, sometimes too.

EL: I’ll just think about it every once in a while be like, is that really true? Do I just need to add that number to the list? And then I'll fix everything.

KK: Well, yeah, it doesn't. You can you can prove that too. But you know, I saw some Twitter thread the other day, that was always trying to argue that there's some models out there where the or the reals are countable, you know, like, you can change your rules and get different answers. But I don't like that. I like my real numbers to be what I think they are. Right? Although, who knows what they are really. Anyway.

EL: So what what is your pairing for?

AL: So I was really actually thinking quite hard about how to match some concept of infinity with real life. And so I think some people might have heard the analogy of, like, you put a box in a box in a box or whatever. And so to me, that's actually very hard to actually even imagine, right? Because in real life, you could never actually make an infinitely small box. It's not practical. Theoretically, it's possible. So what I was thinking is I pulled the old mathematician trick and pair it with alphabets. Because mathematicians, every time we struggle to match numbers to something, we just slap a variable. We take the Greek alphabet, we take everything. And in terms of uncountable and countable infinities, we actually take from the Hebrew alphabet, right, the Aleph and the Bet characters to represent the two different sets. And so I decided, you know, by pairing it with different languages, I figure, if we ever do come up with infinitely, or more more discoveries like that, we can just slap another letter on it and hope it pairs up. And we'll have a new method of explaining something without without the numbers with the more general letter from from a random alphabet.

EL: I like that. Yeah, I've seen, you know, the Latin alphabet, the Greek, Hebrew, and Cyrillic. But yeah, we need to just the next time when one of you comes up with with some new concept that just needs you know, Greek letters are not sufficient for the amazement of this, we need to start using other alphabet.

KK: Oh, yeah. How about like those Southeast Asian ones? Like the like the Thai alphabet? That’s really beautiful and completely different. Right?

EL: Yeah, we’ve got a lot of options.

KK: We do.

EL: New goal. I like it. Thank you, Alvin. Very good. Okay, next up, we have Judith Landau. Judith, please tell us about yourself and share your favorite theorem.

Judith Landau: Well, thank you for having me. My name is Judith Landau, thank you for introducing me. I'm also graduating as an undergrad in math tomorrow.

KK: Congratulations!

JL: Thank you. And I'll be moving on to an interdisciplinary biology program, as a Ph. D. program. And I've been doing some research modeling biological systems. So my favorite theorem is actually the fundamental theorem of Markov chains, which is from the field of probability theory and statistics. And I'm very interested in Markov chains, because they are a way of modeling systems based off of probability instead of deterministically. So instead of saying we know what's going to happen next, we're going to say it's based off of probability. Markov chains can be represented by a directed graph. So a set of vertices with edges that are directed pointing to a specific vertex in one direction or the other. The outgoing edges of each vertex are actually, the that's the probability — sorry, the random variable associated with that vertex. So each vertex has its own random variable, its own probability distribution that tells you how likely you are to go to any other vertex, and so in a very simple weather model for down here in Southern California where we could only might only consider sunny, rainy and cloudy days, no snowy days, those could be our three vertices or our three states. And we could talk about the probability of moving between them from day to day. And so the fundamental theorem of Markov chains can be stated in many different ways. There are long ways and short ways to state theorem. But the simple way to state it is that an aperiodic irreducible Markov chain has a stationary distribution. So an aperiodic Markov chain for our sunny rainy cloudy model would basically mean that the chain is set up so that there is no regular period for any of those states. So there is no regular period between the sunny days, cloudy days, or rainy days. And the irreducibility just means that there is some path, directed path, obviously, between every vertex in that directed graph for this Markov chain, so you're able to reach every vertex from any other in some number of steps. And the stationary distribution: basically, for the Markov chain, if I were to give you a longer version of the theorem, the longer version of a theorem says that the stationary distribution for the Markov chain is actually the long-term probabilities of ending up on any given state. So in my example, sunny, rainy and cloudy. My pairing, if I can move on to that, is actually in biology. It's kind of a pun, because my pairing is proteins. Proteins are actually chains themselves, they’re chains of amino acids. And so since there's this dependency going on in Markov chains, where the next day's weather is dependent on today's weather, and that's a really characteristic idea in Markov chains is that there's this dependency from one state to the next, proteins, the order of those amino acids, is dependent on a gene, because genes are what code for proteins.

KK: Okay.

EL: I love that pairing. That's very nice. It it kind of makes me wonder, like if — I do not know anything about biology, the last biology class I took was in, probably you weren’t born yet.

KK: Ninth grade.

EL: But it makes me wonder like, are there things where like, it's a little more likely that you'll have a tryptophan after a lysine, than something else?

JL: Yes, we actually study that in bioinformatics heavily, what amino acids are more likely to be next to each other and things?

EL: Oh, cool! I’d never thought about that. But it's kind of like, of course there's going to be some sort of tendency, because it would be extremely improbable that it would all be exactly equal all the time.

KK: Right.

JL: And according to our theories, in biology, it's all related to the structure of that protein and how that structure will affect the function of the protein. And that's why proteins evolve as they do.

EL: That is so cool. Well, thank you for sharing that theorem and that biology connection, and I'm so glad that you are going on to study this more.

KK: Yeah, that's very cool. Yeah. Good luck.

JL: You too.

EL: All right. Next up, we've got Kevin Alfaro. So yes, welcome. Please let us know about yourself. And I see that you've got the Golden Gate Bridge behind you in your Zoom background. Are you from the Bay Area?

Kevin Alfaro: No, it’s just my favorite bridge.

EL: It’s a good bridge.

KK: I mean, I've stood in that spot and taken that picture. I think I think a lot of us have, right?

KA: Yeah. Thank you for having me. It's a pleasure to be here. I'm Kevin and I'm majoring in math at Cal State Los Angeles. And for my theorem, it's actually Archimedes’ theorem on his approximation of pi. So I'm going to be taking us back a couple thousand years.

EL: Yeah, I love it. I love the variety.

KA: Yeah, I'm a big daydreamer. So this is all something I could daydream about all day. And so what he was trying to do, he was trying to approximate pi. And what he did was since it’s included using a circle, he placed the hexagon inside the circle. And this hexagon had already included the radius of the circle as one of its sides. And this is because the hexagon is made of six equilateral triangles. So each side ends up being the same, and each side also corresponds to the radius. So he uses this to calculate the circumference of the hexagon. And since it’s inside the circle, he knows it has to be smaller than the circumference of the circle. So he, he knows that the circumference of the hexagon is 6R since it's six radiuses each, each part of the triangles. So already has a bound on that. He knows that it has to be greater than 6R. So now what he does is a series of calculations to try to get a bound. He tries to squeeze pi in two hexagons, a hexagon outside of the circle and a hexagon inside of the circle. So he knows that that circle has to be between these two hexagons. But the way he does it is so so laborious. Yeah, so for the for each of these. So he starts with the first hexagon, right? And it's a six-piece hexagon. So he has to build that up each time, he has to create a longer hexagon that corresponds to the actual circle. So he has to cut up each arc in half. And he does that. So he cuts every one in half, and he creates a midpoint. So he has to do the Pythagorean theorem for each one. And yeah, so he's just doing it like every day. So if you guys have like, a couple of days of free time, you guys could do this yourself.

KK: And you didn't have algebra either, right? I mean, he didn't actually have algebra. So he was — and these numbers didn't even exist to him like square root of two or whatever.

EL: Yeah, it’s amazing. I’m trying to remember, is the last level, was it, like, a 96-gon, or a polygon with 96 sides or 100-something sides?

KK: I think that's how far he got. Yeah, yeah.

EL: And it's just like, I immediately give up on hearing that. And I've got a little calculator in my phone that I carry in my pocket all the time. And like, I've just like, I've noped right out of doing that. But yeah, it's amazing. Yeah. So is that called the method of exhaustion, which I think is perfectly named?

KA: Yeah.

EL: Yeah, it’s so amazing to think about 1000s of years ago, the people like like Archimedes, and other people like having, having the wherewithal, the persistence, to go through and be like, Well, I don't have a calculator, and I don't know what pi is yet, so I guess I'm just going to, you know, figure out the length of this 96-sided polygon. I don't have anything to do for the next week. So did you first encounter this theorem? Or this idea?

KA: I actually got this from a really great book. It's called Infinite Powers by Steven Strogatz. So I recommend reading that book to anyone listening. It is good. Yeah.

EL: And so yes, what is your pairing with Archimedes exhaustive proof of — or exhaustive approximation of pi.

KA: My pairing is actually more of an abstract pairing. And it's more of an idea, because I think around the context of math around this time, too, and the context is always changing. And around this time, it was, it was more like a spiritual level of math, like people were more into it. And it was more spiritual. So then if we connect it to now, and how advanced math is, and we know what Archimedes was doing, since he could never find out the numerical value of pi, since it's an irrational number. And he couldn't do that back then. But he still knew that it was between two fractions, between 3+10/71, and 3+10/72, I believe. And that's all he needed to know. So he knew that it was between two numbers, but he can never really find out what exactly this number was. And I think that really speaks a lot about what math is, or what people do, in terms of knowing what we don't know. So in this time, he knew that he didn't know, which is a lot what, what's happening on today. And I think that's just a cycle of what we do for maths or for anything, really. We try to get to the point where we know that we don't know, and then we're done.

EL: That can be such a difficult part of basically anything in life. This whole pandemic thing we've been living through, like knowing what we do and don't know about what's happening with that has been such a challenge. It's like, oh, that really affects my life in a way that maybe knowing the exact value of pi doesn’t.

KK: Right.

EL: But yeah, it's so hard. And sometimes you think you know what, you know, what you don't know, and you don't realize what you don't know. So yeah, I like that kind of, from the very concrete polygons to this sort of abstract. Thank you, Kevin.

KA: Thank you.

KK: That’s a good one.

EL: All right. Next up, welcome to the show, Francisco Leon. Would you like to tell us a little bit about yourself and share your favorite theorem? Yeah,

FL: Thank you for the welcome. Tomorrow, along with Bryce, I'm graduating from Cal State LA through the master's program, so I'm really excited about that.

KK: Congratulations.

FL: Yeah, thank you. Yeah. Today, I want to mention, when I was asked, what's my favorite theorem, at the time, I really had one on mine from my topology class. I’ll state the theorem now. It says a topological space is regular if and only if for every element in the set and open set containing the element, that there exists another open set that also contains elements whose closure is in the first mentioned set. So maybe to label some of this, to get a better visualization, so we have some element, say P. And then say, some open set that contains it, U. Now in between P and U, this alternate characterization of regularity says there's going to have to exist another open set V, that also contains P inside of U whose closure is also inside of U. So it goes, maybe nested: P, V-closure, V, and then U.

KK: Yup.

FL: And that's equivalent for regularity. And you know, at first, this was a homework assignment from the class. And one of the key tools to do this problem is to understand that if you illustrate an element in open set, so you draw it ,right, you kind of draw a little point P, and a dashed circle around an open set, that's going to be equivalent to having P outside that complement. So you can draw P outside of a box and a box, just kind of know that, since of the complement of open sets are closed, that's why it's represented as a box, to be able to comfortably go between those two perspectives of the same situation, I found difficult to get at the time. And so, you know, I fostered an appreciation for this theorem because it made me resolve that difficulty. So it led me to some trains of thoughts that were really interesting for me at the time. And it's been a while since a theorem held my attention for so long. So just to share some lines of thoughts I was having when working through this exercise: okay, I was like, how do I go from being an element in this open set to being outside its complement? I started imagining myself inside of a room, just as the element would be inside the open set. And I'm like, how do I push the walls of this room so that suddenly the outside is contained inside these walls, right? Because you want to go from being inside the open set to being outside the complement. So how can I move this boundary, so that the outside is contained? I felt like I couldn't do it. I look outside, and I know the universe is huge. How can I push the walls of this room to contain everything else outside and suddenly be the one outside? So I had great difficulty. And then as I would think about this problem and look outside, I would see the window. And then I realized that the illustration of the element inside the open set, you know, when you have that point P and those dashed lines around it, those dashed lines aren't necessarily representative of a border, it's really representative of a window, because what you do at the illustration, is you look at what's within those dashed lines to see the elements. You are really looking into the set to see what's inside of it. So what I really needed was a window in this room for me to be able to understand the outside. So now if I reimagine the situation, I'm in a room, which is representative of the open set, and there's some element in here. What I need is a window so that I can see what's on the outside. And then if I change my perspective, if I pass through the window and then turn back around, then I see through the window, the element inside the room. And now I have the outside perspective that I want it. So now I see the element inside the room. So now I'm seeing the complement of the open set. And that really helped me change that perspective from being inside the open set to being outside the complement, which is a closed set. And then I was able to do that exercise pretty straightforwardly, but I was really struggling before that. And I never really think about theorems as visually as I just described, so that's why I really appreciate this theorem and that's why it's one of my current favorites.

KK: You know, when I was an undergrad, I was always going to be a math major. That was always what I was going to do. But that class, point-set topology, is when I really fell in love. I mean, I was already planning to go to graduate school, but my professor for that, Peter Fletcher, who passed away a couple of years ago, really put me on this path to being a topologist. I didn't know what I was going to do until I actually hit that. And of course, you know, that flavor of topology is pretty well settled and has been for quite a while. But it's still really beautiful to think about these basic properties of topological spaces. And I love your visual description of this sort of thing, like regular spaces. You know, I haven't thought about them in a long time, but I think you really nailed it right there. So nice. Yeah.

EL: And I like, I like seeing that little glimpse into how you were thinking this through, even if that isn't what you wrote down in your proof in the end. But it seeing — so often, when we read math, we don't, we don't see those little glimpses of how the person actually thought about it. We see the cleaned up version that, you know, is presentable for professional company or something, but I really liked seeing that. So what what is your pairing for this theorem?

FL: Oh, yeah, so I'm thinking maybe a hot apple pie, just because we typically do place it on a windowsill to cool down and I just figured that this theorem is cooler with a window.

KK: That’s good.

EL: And yeah, who can argue with a nice warm slice of apple? It's delicious. Yeah. Excellent. Well, thank you so much.

FL: Thank you guys.

EL: All right. Next up, we have Marlene Enriquez. Marlene, would you like to introduce yourself and tell us about your favorite theorem?

Marlene Enriquez: Yes. So hi, my name is Marlene, and my favorite theorem has to be the ham sandwich theorem.

KK: Yes!

EL: You've got company there with Kevin.

KK: So in episode zero, Evelyn and I had our favorite theorems. That was mine. Here we go. All right.

EL: So please tell us what you like about it.

ME: What I like about it is just, it's the first time that I encountered a theorem that really didn't have an exact answer to how to solve it. So the idea of basically having a line cut something and split it into equal volumes, per se. So when I first read it, I was like, oh, that sounds cool. When I read it some more. I was like, oh, no, this is really interesting. And it's actually the the theorem that I used to write the my research paper for my senior seminar class with Dr. Krebs. So the theorem, when I read it, I read it like this is like, you have a sandwich made out of bread, ham, and another piece of bread and like, they don't have to be exactly aligned. Or even touching each other. But there exists, a line, or a cut, a straight cut that will split the bread, ham, and the other piece of bread in equal volumes. And then when I saw that, like, wait, you don't really have to be next to each other, or like on top of each other, or even in the same room as each other? And you can still be cut and split into equal volumes? I was like, Oh, wow. I can read more about this, it’s pretty remarkable.

EL: Yeah, it can be the sloppiest made ham sandwich ever, and it’ll still work.

KK: Like you can have one slice of bread in Los Angeles. I have one here in Gainesville and a piece of ham could be in Salt Lake with Evelyn, and there is — it will take a big knife — there is a big knife that can cut them in half. That's right.

EL: Where did you first see this theorem?

ME: I was actually searching for a topic for my paper and I was online Google searching, every time at 2am, it always happens. And I kind of stumbled upon it. And I think I saw a YouTube video and I was like, Oh, this is interesting. Then I clicked to another one. I'm like, okay, this is very interesting. And then I just started searching and searching and reading and yeah

EL: Excellent. Yeah, I guess this podcast brought to you by the time 2am: the perfect time to be just finding weird math stuff to learn about.

KK: So what's what pairs with a ham sandwich there?

ME: Well, it's not on the nose, not a ham sandwich. But actually, when I was typing my paper, I was in my living room, and I have siblings. And they were fighting over a chocolate bar. They were saying, oh, like, let's split it equally. And they're like, No, you got the bigger piece or you got the bigger piece. And just the idea of sharing. Even when you go out with friends, you split the bill and stuff like that. That kind of just kind of brought it to, like, that's kind of like this ham sandwich. But like, in like, money, or a chocolate bar, who got the biggest piece, right?

KK: That’s a hard problem, actually, the whole equal division problem is extremely difficult. As I'm sure you found out with your siblings already over a chocolate bar.

EL: The answer is probably just giving up all earthly desires.

KK: Yeah. Right. The Buddha, the Buddhists understand.

EL: Well, thank you so much Marlene. And our final guest is Daniel Argueta. Thank you for joining us. And please tell us a little bit about yourself and your favorite theorem.

Daniel Argueta: Well, thank you, Kevin, and Evelyn, for having me and Evelyn, don't worry, you said my last name correctly. So I'm also graduating tomorrow. I'm getting my bachelor's in math.

KK: Congratulations.

DA: It’s a big thing. I’m first gen. So when Dr. Krebs approached me about this, I was like, What am I going to talk about? What's my favorite theorem? And when I was doing research for our capstone class, I actually stumbled across you guys’ podcast. And the one, I think it was episode 17 with Dr. Naomi Joshi or something like that.

EL: Nalini.

KK: Nalini Joshi.

DA: And it was the Mittag Leffler theorem. And that theorem was just way too complicated for me. Dr. Krebs helped me bring it down a little bit. And I stumbled across the infinite product convergence theorem. And I was like, okay, maybe that’s what I’m going to talk about. I wrote a whole paper on that. And then I thought about it and I was like, you know what, that's not my favorite theorem. So I started doing some googling, and I came across the headline “The theorem shook math to its core.” So maybe that might give you guys some ideas as to where I'm going to. And I'm actually doing I'm going to talk about Kurt Gödel’s, incompleteness theorems.

KK: Oh, yeah.

EL: Okay, yes, is a great way. I’d just like to say, of all the however many thousands of possibilities, I like that, at least that this one was at the end, to sort of be like, okay, we talked about all this math that we can do. Now let's finish it off with: Oh, yeah, we can't do math. That's probably not the best summary of the Incompleteness Theorems. So why don't you tell us about the incompleteness theorem?

DA: Well, what Gödel pretty much set out to prove, everybody at the time was trying to prove that math is perfect. And they were trying to say, hey, math is perfect. They were trying to find like this theory of everything, and Gödel’s all like, hmm, I don't know about that. And so pretty much what his proof did, and not to get super technical or anything, was he showed that in any formal system, there are certain truth statements about the system that can not be proven by said system. So a way to think of this is our classic logical problems with the knights and the knaves. In this case, you only need the knight, where you have a knight in shining armor who can only tell you the truth. So so you ask yourself, what is a sentence that this knight cannot say if he can only tell the truth? And one sentence is: I cannot say that sentence. You know, because if he can say the sentence, then it's true. But if he can't say the sentence, then you know, it's not gonna work. So I just really like this theorem, because to me, you know, we usually tend to think of math as perfect. It's, a lot of people call it the universal language, because math across everywhere is perfect. But to me, it's super interesting that the fact that we don't know everything about it, and we will never know everything about it. So that's why I really like this theorem. And it actually brings me to my pairing, which is — I know usually you guys do food on the on the podcast, but in this case, my pairing is our Final Frontier, which is space. I think I think it super coincides with it. Because, you know, we're going to always keep discovering more and more about space as our technology advances. But at the same time, our universe is ever-expanding. So will we ever completely know everything about space? There's so much to learn and discover, and the same can be said for math. And that's my favorite theorem.

EL: Excellent. I love that. Yeah, it's almost like okay, yeah, math is like, perfect, and you can do everything. Just don't look too hard. This incompleteness theorem stuff is just saying, oh, yeah, you think you could do calculus? Just don't look too hard.

KK: Right, right. Yeah. Well that’s everybody.

EL: This has been so fun. I’m so optimistic about the future of math right now. And biology. And whatever else you all do.

KK: And machine learning, and whatever it is yes.

EL: Yeah. Good luck to all of you in your next steps. It’s so appropriate that we got to do this right before many of you are graduating and, you know, taking those next steps. So thank you so much for sharing part of your, Monday morning with us.

KK: And thanks to your professor, Mike, for reaching out to us and making this happen.

MK: Well thank you for hosting everybody. This is great.

EL: Yes. I don't know if you all want to unmute it and do a goodbye. I don't know. How do you end a podcast with 12 people? (I just did multiplication there.)

KK: Is there a CSULA kind of cheer? Like I'm at the University of Florida so it'd be like a Go Gators kind of thing. They’re looking at me like, no, we don’t do that.

EL: Math majors might not be the best choice for for you know, knowing all the sports things, not to invoke any stereotypes.

KK: I know every word of the Virginia Tech fight song.

EL: Well, it may not be universal.

KK: Maybe not. All right. Well, you’re all unmuted. Maybe say goodbye.

(General chaos of 12 unmuted people on Zoom.)

[outro]

In this Very Special Episode of My Favorite Theorem, we were excited to welcome nine students from California State University Los Angeles, along with their professor Mike Krebs. Each student shared their favorite theorem and a pairing with us. Below are some links to more information about their favorites.

Pablo Martinez Gutierrez talked about Euler's formula and identity, which connect trigonometric functions with the complex exponential.

Holly Kim told us about Ore's theorem about Hamiltonians in graph theory. Check out A Capella Science's Hamilton parody video!
Bryce Van Ross shared the Hales–Jewett theorem from game theory.

Alvin Lew chose Cantor's diagonalization proof of the uncountability of the reals, which was also a favorite of our previous guest Adriana Salerno.

Judith Landau shared the fundamental theorem of Markov chains, which relates to her work in bioinformatics.
Kevin Alfaro talked about Archimedes' approximation of pi.

Francisco Leon chose a theorem from point-set topology about regular spaces.
Marlene Enriquez chose Kevin's favorite, the ham sandwich theorem.
Daniel Argueta finished off the episode with Gödel's incompleteness theorems.