This transcript is provided as a courtesy and may contain errors.
EL: Welcome to My Favorite Theorem. I’m one of your hosts, Evelyn Lamb. I’m a freelance math and science writer currently based in Paris. And this is my cohost.
KK: Hi, I’m Kevin Knudson, professor of mathematics at the University of very, very hot Florida.
EL: Yeah. Not so bad in Paris yet.
KK: It’s going to be a 96-er tomorrow.
EL: Wow. So each episode, we invite a mathematician to come on and tell us about their favorite theorem. Today we’re delighted to welcome Emille Davie Lawrence to the show. Hi, Emille.
EDL: Hello, Evelyn.
EL: So can you tell us a little bit about yourself?
EDL: Sure! So I am a term assistant professor at the University of San Francisco. I’m in the mathematics and physics department. I’ve been here since 2011, so I guess that’s six years now. I love the city of San Francisco. I have two children, ages two and almost four.
EL: Who are adorable, if your Facebook is anything to go by.
EDL: Thank you so much. You’ll get no arguments from me. I’ve been doing math for quite a while now. I’m a topologist, and my mathematical interests have always been in topology, but they’ve evolved within topology. I started doing braid groups, and right now, I’m thinking about spatial graphs a lot. So lots of low-dimensional topology ideas.
EL: Cool. So what is your favorite theorem?
EDL: My favorite theorem is the classification theorem for compact surfaces. It basically says that no matter how weird the surface you think you have on your hands, if it’s a compact surface, it’s only one of a few things. It’s either a sphere, or the connected sum of a bunch of tori, or the connected sum of a bunch of projective planes. That’s it.
EL: Can you tell us a little bit more about what projective planes are?
EDL: Obviously a sphere, well, I don’t know how obvious, but a sphere is like the surface of a ball, and a torus looks like the surface of a donut, and a projective plane is a little bit stranger. I think anyone who would be listening may have run into a Möbius band at some point. Basically you take a strip of paper and glue the two ends of your strip together with a half-twist. This is a Möbius band. It’s a non-orientable half-surface. I think sometimes kids do this. They pop up in different contexts. One way to describe a projective plane is to take a Möbius band and add a disc to the Möbius band. It gives you a compact surface without boundary because you’ve identified the boundary circle of the Möbius band to the boundary of the disc.
EL: Right, OK.
EDL: Now you’ve got this non-orientable thing called a projective plane. Another way to think about a projective plane is to take a disc and glue one half of the boundary to the other half of the boundary in opposite directions. It’s a really weird little surface.
KK: One of those things we can’t visualize in three dimensions, unfortunately.
EDL: Right, right. It’s actually hard to explain. I don’t think I’ve ever tried to explain it without drawing a picture.
EL: Right. That’s where the blackboard comes in hand.
KK: Limitations of audio.
EL: Have you ever actually tried to make a projective plane with paper or cloth or anything?
EDL: Huh! I am going to disappoint you there. I have not. The Möbius bands are easy to make. All you need is a piece of paper and one little strip of tape. But I haven’t. Have you, Evelyn?
EL: I’ve seen these at the Joint Meetings, I think somebody brought this one that they had made. And I haven’t really tried. I’d imagine if you tried with paper, it would probably just be a crumpled mess.
EDL: Right, yeah.
EL: This one I think was with fabric and a bunch of zippers and stuff. It seemed pretty cool. I’m blanking now on who is was.
KK: That sounds like something sarah-marie belcastro would do.
EL: It might have been. It might have been someone else. There are lots of cool people doing cool things with that. I should get one for myself.
EDL: Yeah, yeah. I can see cloth and zippers working out a lot better than a piece of paper.
EL: So back to the theorem. Do you know what makes you love this theorem?
EDL: Yeah. I think just the fact that it is a complete classification of all compact surfaces. It’s really beautiful. Surfaces can get weird, right? And no matter what you have on your hands, you know that it’s somewhere on this list. That makes a person like me who likes order very happy. I also like teaching about it in a topology class. I’ve only taught undergraduate topology a few times, but the last time was last spring, a year ago, spring of 2016, and the students seemed to really love it. You can play these “What surface am I?” games. Part of the proof of the theorem is that you can triangulate any surface and cut it open and lay it flat. So basically any surface has a polygonal representation where you’re just some polygon in the plane with edges identified in pairs. I like to have this game in my class where I just draw a polygon and identify some of the edges in pairs and say, “What surface is this?” And they kind of get into it. They know what the answers, what the possibilities are for the answers. You can sort of just triangulate it and find the Euler characteristic, see if you can find a Möbius band, and you’re off to the races.
KK: That’s great. I taught the graduate topology course here at Florida last year. I’m ashamed to admit I didn’t actually prove the classification.
EDL: You should not be ashamed to admit that. It’s something at an undergraduate level you get to at the end, depending on how you structure things. We did get to it at the end of the course, so I don’t know how rigorously I proved it for them. The combinatorial step that goes from: you can take this polygonal representation, and you can put it in this polygonal form, always, that takes a lot of work and time.
EL: There are delicacies in there that you don’t really know about until you try to teach it. I taught it also in class a couple years ago, and when I got there, I was like, “This seemed a little easier when I saw it as a student.” Now that I was trying to teach it, it seemed a little harder. Oh, there are all of these t’s I have to cross and i’s I have to dot.
KK: That’s always the way, right?
KK: I assigned as a homework assignment that my students should just compute the homology of these surfaces, and even puncture them. Genus g, r punctures, just as a homework exercise. From there you can sort of see that homology tells you that genus classifies things, at least up to homotopy invariants, but this combinatorial business is tricky.
EDL: It is.
EL: Was this a love at first sight kind of theorem, or is this a theorem that’s grown on you?
EDL: I have to say it’s grown on me. I probably saw it my first year of graduate school, and like all of topology, I didn’t love it at first when I saw it as a first-year graduate student. I did not see any topology as an undergrad. I went to a small, liberal arts college that didn’t have it. So yeah, I have matured in my appreciation for the classification theorem of surfaces. It’s definitely something I love now.
KK: You’re talking to a couple of topologists, so you don’t have to convince us very much.
KK: I had a professor as an undergrad who always said, “Topology is analysis done right.”
EDL: I like that.
KK: I know I just infuriated all the analysts who are listening. I always took that to heart. I always took that to heart because I always felt that way too. All those epsilons and deltas, who wants all that?
EDL: Who needs it?
KK: Draw me a picture.
EL: I was so surprised in the first, I guess advanced calculus class I had, a broader approach to calculus, and I learned that all these open sets and closed sets and things actually had to do with topology not necessarily with epsilons and deltas. That was really a revelation.
KK: So you’re interested in braids, too, or you were? You moved on?
EDL: I would say I’m still interested in braids, although that is not the focus of my research right now.
KK: Those are hard questions too, so much interesting combinatorics there.
EDL: That’s right. I think that’s sort of what made me like braid groups in the first place. I thought it was really neat that a group could have that geometric representation. Groups, I don’t know, when you learn about groups, I guess the symmetric group is one of the first groups that you learn about, but then it starts to wander off into abstract land. Braid groups really appealed to me, maybe just the fact that I liked learning visually.
EL: It’s not quite as in the clouds as some abstract algebra.
KK: And they’re tied up with surfaces, right, because braid groups are just the mapping class group of the punctured disc.
EDL: There you go.
KK: And Evelyn being the local Teichmüller theorist can tell us all about the mapping class groups on surfaces.
EL: Oh no! We’re getting way too far from the classification of surfaces here.
KK: This is my fault. I like to go off on tangents.
EDL: Let’s reel it back in.
EL: You mentioned that you’ve matured into true appreciation of this lovely theorem, which kind of brings me to the next part of the show. The best things in life are better together. Can you recommend a pairing for your theorem? This could be a fine wine or a flavor of ice cream or a favorite piece of music or art that you think really enhances the beauty of this classification theorem.
EDL: I hate to do this, but I’m going to have to say coffee and donuts.
KK: Of course.
EDL: I really tried to say something else, but I couldn’t make myself do it. A donut and cup of coffee go great with the classification of compact surfaces theorem.
EL: That’s fair.
KK: San Francisco coffee, right? Really good dark, walk down to Blue Bottle and stand in line for a while?
EDL: That’s right. Vietnamese coffee.
KK: There you go. That’s good.
EL: Is there a particular flavor of donut that you recommend?
EDL: Well you know, the maple bacon. Who can say no to bacon on a donut?
KK: Or on anything for that matter.
EDL: Or on anything.
KK: That’s just a genus one surface. Can we get higher-genus donuts? Have we seen these anywhere, or is it just one?
EDL: There are some twisted little pastry type things. I’m wondering if there’s some higher genus donuts out there.
EL: If nothing else there’s a little bit of Dehn twisting going on with that.
EDL: There’s definitely some twisting.
EL: I guess we could move all the way over into pretzels, but that doesn’t go quite as well with a cup of coffee.
EDL: Or if you’re in San Francisco, you can get one of these cronuts that have been all the rage lately.
EL: What is a cronut? I have not quite understood this concept.
EDL: It is a cross between a croissant and a donut. And it’s flakier than your average donut. It is quite good. And if you want one, you’re probably going to have to stand on line for about an hour. Maybe the rage has died down by now, maybe. But that’s what was happening when they were first introduced.
EL: I’m a little scared of the cronut. That sounds intense but also intriguing.
EDL: You’ve got to try everything once, Evelyn. Live on the edge.
EL: The edge of the cronut.
KK: You’re in Paris. We’re not too concerned about your ability to get pastry.
EL: I have been putting away some butter.
KK: The French have it right. They understand that butter does the heavy lifting.
EDL: It’s probably a sin to have a cronut in Paris.
EL: Probably. But if they made one, it would be the best cronut that existed.
KK: Well I think this has been fun. Anything else you want to add about your favorite theorem?
EDL: It’s a theorem that everyone should dig into, even if you aren’t into topology. I think it’s one of those foundational theorems that everyone should see at least once, and look at the proof at least once, just for a well-rounded mathematical education.
KK: Maybe I should look at the proof sometime.
EL: Thanks so much for joining us, Emille. We really enjoyed having you. And this has been My Favorite Theorem.
EDL: Thank you so much.
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, Del Mitchell, and Bao-xian Lin. 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.