KK: Welcome to My Favorite Theorem. I’m Kevin Knudson, and I’m joined by my cohost. EL: I’m Evelyn Lamb. KK: This is Episode 0, in which we’ll lay out our ground rules for what we’re going to do. The idea is every week we’ll have a guest, and that guest will tell us what his or her favorite theorem is, and they’ll tell us some fun things about themselves, and Evelyn had good ideas here. What else are we going to do? EL: Yeah, well, with any great thing in life, pairings are important. So we’ll find the perfect wine, or ice cream, or work of 19th century German romanticism to include with the theorem. We’ll ask our guests to help us with that. KK: Since this is episode 0, we thought we should probably set the tone and let you know what our favorite theorems are. I’m going to defer. I’m going to let Evelyn go first here. What’s your favorite theorem? EL: OK, so we’re recording this on March 23rd, which is Emmy Noether’s birthday, her 135th, to be precise. I feel like I should say Noether’s theorem. It’s a theorem in physics that relates, that says basically conserved quantities in physics come from symmetries in nature. So time translation symmetry yields conservation of energy and things like that. But I’m not going to say that one. I’m sorry, physics, I just like math more. So I’m going to pick the uniformization theorem as my favorite theorem. KK: I don’t think I know that theorem. Which one is this? EL: It’s a great theorem. When I was doing math research, I was working in Teichmüller theory, which is related to hyperbolic geometry. This is a theorem about two-dimensional surfaces. The upshot of this theorem is that every two-dimensional surface can be given geometry that is either spherical, flat — so, Euclidean, like the flat plane — or hyperbolic. The uniformization itself is related to simply connected Riemann surfaces, the ones with no holes, but using this theorem you can show that 2-d surfaces with any number of holes have one of these kinds of 2-d geometry. This is a great theorem. I just love that part of topology where you’re classifying surfaces and everything. I think it’s nice A little of the history is that it was conjectured by Poincaré in 1882 and Klein in 1883. I think the first proof was by Poincaré in the early 1900s. There are a lot of proofs of it that come from different approaches. KK: Now that you tell me what the theorem is, of course I know what it is. Being a topologist, I know how to classify surfaces, I think. That is a great theorem. There’s so much going on there. You can think about Riemann surfaces as quotients of hyperbolic space, and you have all this fun geometry going. I love that theorem. In fact, I’m teaching our graduate topology course this year, and I didn’t do this. I’m sorry. I had to get through homology and cohomology. So yeah, surfaces are classified. We know surfaces. So what are you going to pair this with? EL: So my pairing is Neapolitan ice cream. I’m going a bit literal with this. Neapolitan ice cream is the ice cream that has part of it vanilla, part of it chocolate, and part of it strawberry. So this theorem says that surfaces come in three flavors. KK: Nice. EL: When I was a little kid, when we had our birthday parties at home, my mom always let us pick what ice cream we wanted to have, and I always picked Neapolitan so that if my friends liked one of the flavors but not the others, they could have whichever flavor they wanted. KK: You’re too kind. EL: Really, I’m just such a good-hearted person. KK: Clearly. EL: Yeah, Neapolitan. Three flavors of surfaces, three flavors of ice cream. KK: Nice. Although nobody ever eats the strawberry, right? EL: Yeah, I love strawberry ice cream now, but yeah, when I was a little kid chocolate and vanilla were a little more my thing. KK: I remember my mother would sometimes buy the Neapolitan, and I remember the strawberry would just sit there, uneaten, until it got freezer burn, and we just threw it away at that point. EL: I guess the question is, which of the kinds of geometry is strawberry? KK: Well, vanilla is clearly flat, right? EL: Yeah, that’s good. I guess that means strawberry must be spherical. KK: That seems right. It’s pretty unique, right? Spherical geometry is kind of dull, right? There’s just the sphere. There’s a lot more variation in hyperbolic geometry, right? EL: Yeah, I guess so. I feel like there are more different kinds of chocolate-flavored ice cream, and hyperbolic, there are so many different hyperbolic surfaces. KK: Right. Here in Gainesville, we have a really wonderful local ice cream place, and twice a year they have chocolate night, and they have 32 different varieties of chocolate. EL: Oh my gosh. KK: So you can go and you can get a ginormous bowl of all 36 flavors if you want, but we usually get a little sample of eight different flavors and try them out. It’s really wonderful. I think that’s the right classification. EL: OK. So Kevin? KK: Yes? EL: What is your favorite theorem? KK: Well, yeah, I thought about this for a long time, and what I came up with was that my favorite theorem is the ham sandwich theorem. I think it’s largely because it’s got a fun name, right? EL: Yeah. KK: And I remember hearing about this theorem as an undergrad for the first time. This was a general topology course, and you don’t prove it in that, I think. You need some algebraic topology to prove this well. I thought, wow, what a cool thing! There’s something called the ham sandwich theorem. So what is the ham sandwich theorem? It says: say you have a ham sandwich, which consists of two pieces of bread and a chunk of ham. And maybe you got a little nuts and you put one piece of bread on top of the fridge, and one on the floor, and your ham is sitting on the counter, and the theorem is that if you have a long enough knife, you can make one cut and cut all of those things in half. Mathematically what that means is that you have three blobs in space, and there is a single plane that cuts each of those blobs in half exactly. I just thought that was a pretty remarkable theorem, and I still think it’s kind of remarkable theorem because it’s kind of hard to picture, right? Your blobs could be anywhere. They could be really far apart, as long as they have positive measure, so as long as they’re not some flat thing, they actually have some 3-d-ness to them, then you can actually find a plane that does this. What’s even more fun, I think, is that this is a consequence of the Borsuk-Ulam theorem, which in this case would say that if you have a continuous function from the 2-sphere to the plane, then two antipodal points have to go to the same place. And that’s always a fun theorem to explain to people who don’t know any mathematics, because you can say, somewhere, right now, there are two opposite points on the surface of the earth where the temperature and the humidity are the same, for example. EL: Yeah. KK: I love that kind of theorem, where there’s a good physical interpretation for it. And of course there are higher-dimensional analogues, but the idea of the ham sandwich theorem is great. Everybody’s had a ham sandwich, probably, or some kind of sandwich. It doesn’t have to be ham. Maybe we should be more politically correct. What’s a good sandwich? EL: A peanut butter sandwich is a great sandwich. KK: A peanut butter sandwich. But the peanut butter is kind of hard to get going, right? You don’t really want that anywhere except in the middle of the sandwich. You don’t want to imagine this blob of peanut butter. The ham you can kind of imagine. EL: It’s really saying that you don’t even have to remove the peanut butter from the jar. You can leave the peanut butter in the jar. KK: There you go. EL: You can cut this sandwich in half. KK: Your knife’s going to have to cut through the whole jar. It’s gotta be a pretty strong knife. EL: Yeah. We’re already asking for an arbitrarily long knife. KK: Yes. EL: You don’t think our arbitrarily long knife can cut through glass? Come on. KK: It probably can, you’re right. How silly of me. If we’re being so silly and hyperbolic, we might as well. EL: We’re mathematicians, after all. KK: You’re right, we are. So I thought about the pairing, too. Basically, I’ve got a croque monsieur, right? EL: Right. KK: You’re in France. You probably eat these all the time. So what does one have with a croque monsieur? It’s not really fancy food. So I think you’ve got to go with a beer for this, and if I’m getting to choose any beer, we have a wonderful local brewery here, First Magnitude brewery, it’s owned by a good friend of mine. They have a really nice pale ale. It’s called 72 Pale Ale. I invite everyone to look up First Magnitude Brewing on the internet there and check them out. It’s a good beer. Not too hoppy. EL: OK. KK: It’s hoppy enough, but it’s not one of those West Coast IPA’s that makes your mouth shrivel up. EL: Yeah, socks you in the face with the hops. KK: Yeah, you don’t need all of that. EL: So actually, if you think of the two pieces of bread as one mass of bread and the ham as its own thing, then you could also bisect the bread, the ham, and the beer with one knife. KK: That’s right, we could do that. EL: Yeah, if you really wanted to make sure to eat your meal in two identical halves. KK: Right. So you have vanilla donuts and balls of chocolate, no, no, the donuts, wait a minute. The hyperbolic spaces were chocolate. This is starting to break down. But the flat geometry is the plane. But there’s a flat torus too, right? So you could have a flat donut, or a flat plane. Very cool. This is fun. I think we’re going to have a good time doing this. EL: I think so too. And I think we’re going to end each episode hungry. KK: It sounds that way, yeah. In the weeks to come, we have a pretty good lineup of interesting people from all areas of mathematics and all parts of the world, hopefully. I’m excited about this project. So thanks, Evelyn, for coming along with me on this. EL: Yeah. Thank you for inviting me. I’m looking forward to this. KK: Until next time, this has been My Favorite Theorem. KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Dell Mitchell, and Baochau Nguyen. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at firstname.lastname@example.org. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics.