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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.
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