Episode 2 - Dave Richeson

08/03/17 • 23 min

This transcript is provided as a courtesy and may contain errors.
Evelyn Lamb: Welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I am a freelance math writer usually based in Salt Lake City but currently based in Paris. And this is your other host.
KK: I’m Kevin Knudson, professor of mathematics at the University of Florida.
EL: Every episode we invite a mathematician on to tell us about their favorite theorem. This week our guest is Dave Richeson. Can you tell us a little about yourself, Dave?
Dave Richeson: Sure. I’m a professor of mathematics at Dickinson College, which is in Carlisle, Pennsylvania. I’m also currently the editor of Math Horizons, which is the undergraduate magazine of the Mathematical Association of America.
EL: Great. And so how did you get from wherever you started to Carlisle, Pennsylvania?
DR: The way things usually work in academia. I applied to a bunch of schools. Actually, seriously, my wife knew someone in Carlisle, Pennsylvania. My girlfriend at the time, wife now, and she saw the list of schools that I was applying to and said, “You should get a job at Dickinson because I know someone there.” And I did.
KK: That never happens!
EL: Wow.
DR: That never happens.
KK: That never happens. Dave and I actually go back a long way. He was finishing his Ph.D. at Northwestern when I was a postdoc there.
DR: That’s right.
KK: That’s how old-timey we are. Hey, Dave, why don’t you plug your excellent book.
DR: A few years ago I wrote a book called Euler’s Gem: The Polyhedron Formula and the Birth of Topology. It’s at Princeton University Press. I could have chosen Euler’s Formula as my favorite theorem, but I decided to choose something different instead.
KK: That’s very cool. I really recommend Dave’s book. It’s great. I have it on my shelf. It’s a good read.
DR: Thank you.
EL: Yeah. So you’ve told us what your favorite theorem isn’t. So what is your favorite theorem?
DR: We have a family joke. My kids are always saying, “What’s your favorite ice cream? What’s your favorite color?” And I don’t really rank things that way. This was a really challenging assignment to come up with a theorem. I have recently been interested in π and Greek mathematics, so currently I’m fascinated by this theorem of Archimedes, so that is what I’m giving you as my favorite theorem. Favorite theorem of the moment.
The theorem says that if you take a circle, the area of that circle is the same as the area of a right triangle that has one leg equal to the radius and one leg equal to the circumference of the circle. Area equals 1/2 c x r, and hopefully we can spend the rest of the podcast talking about why I think this is such a fascinating theorem.
KK: I really like this theorem because I think in grade school you memorize this formula, that area is π r2, and if you translate what you said into modern terminology, or notation, that is what it would say. It’s always been a mystery, right? It just gets presented to you in grade school. Hey, this is the formula of a circle. Just take it.
DR: Really, we have these two circle formulas, right? The area equals π r2, and the circumference is 2πr, or the way it’s often presented is that π is the circumference divided by the diameter. As you said, you could convince yourself that Archimedes theorem is true by using those formulas. Really it’s sort of the reverse. We have those formulas because of what Archimedes did. Pi has a long and fascinating history. It was discovered and rediscovered in many, many cultures: the Babylonians, the Egyptians, Chinese, Indians, and so forth. But no one, until the Greeks, really looked at it in a rigorous way and started proving theorems about π and relationships between the circumference, the diameter, and the area of the circle.
EL: Right, and something you had said in one of your emails to us was about how it’s not even, if you ask a mathematician who proved that π was a constant, that’s a hard question.
DR: Yes, exactly. I mean, in a way, it seems easy. Pi is usually defined as the circumference divided by the diameter for any circle. And in a way, it seems kind of obvious. If you take a circle and you blow it up or shrink it down by some factor of k, let’s say, then the circumference is going to increase by a factor of k, the diameter is going to increase by a factor of k. When you do that division you would get the same number. That seems sort of obvious, and in a way it kind of is. What’s really tricky about this is that you have to have a way of talking about the length of the circumference. That is a curve, and it’s not obvious how to talk about lengths of curves. In fact, if you ask a mathematician who proved that the circumference over the diameter was the same value of π, most mathematicians don’t know the answer to that. I’d put money on it that most people would think it was in Euclid’s Elements, which is...

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Episode 1 - Amie Wilkinson

Kevin Knudson: Welcome to My Favorite Theorem. I’m Kevin Knudson, professor of mathematics at the University of Florida, and I’m joined by my cohost.
Evelyn Lamb: I’m Evelyn Lamb. I’m a freelance writer currently based in Paris.
KK: Yeah, Paris. Paris is better than Gainesville. I mean, Gainesville’s nice and everything.
EL: Depends on how much you like alligators.
KK: I don’t like alligators that much.
EL: OK.
KK: This episode, we’re thrilled to welcome Amie Wilkinson of the University of Chicago. Amie’s a fantastic mathematician. Say hi, Amie, and tell everyone about yourself.
AW: Hi, everyone. So Kevin and I go way back. I’m a professor at the University of Chicago. Kevin and I first met when we were pretty fresh out of graduate school. We were postdocs at Northwestern, and now we’ve kind of gone our separate ways but have stayed in touch over the years.
KK: And, let’s see, my son and your daughter were born the same very hot summer in Chicago.
AW: Yeah, that’s right.
KK: That’s a long time ago.
AW: Right. And they’re both pretty hot kids.
KK: They are, yes. So, Amie, you haven’t shared what your favorite theorem is with Evelyn and me, so this will be a complete surprise for us, and we’ll try to keep up. So what’s your favorite theorem?
AW: Fundamental theorem of calculus.
KK: Yes.
EL: It’s a good theorem.
KK: I like that theorem. I just taught calc one, so this is fresh in my mind. I can work with this.
AW: Excellent. Probably fresher than it is in my mind.
EL: Can you tell us, remind our listeners, or tell our listeners what the fundamental theorem of calculus is?
AW: The fundamental theorem of calculus is a magic theorem as far as I’m concerned, that relates two different concepts: differentiation and integration.
So integration roughly is the computation of area, like the area of a square, area of the inside of a triangle, and so on. But you can make much more general computations of area like Archimedes did a long time ago, the area inside of a curve, like the area inside of a circle. There’s long been built up, going back to the Greeks, this notion of area, and even ways to compute it. That’s called integration.
Differentiation, on the other hand, it has to do with motion. In its earliest forms, to differentiate a function means to compute its slope, or speed, velocity. It’s a computation of velocity. It’s a way of measuring instantaneous motion. Both of these notions go way back, to the Greeks in the case of area, back to the 15th century and the people at Oxford for the computation of speed, and it wasn’t until the 17th century that the two were connected. First by someone named James Gregory, and not long after, sort of concurrently, by Isaac Barrow, who was the advisor of Isaac Newton. Newton was the one who really formalized the connection between the two.
EL: Right, but this wasn’t just a lightning bolt that suddenly came from Newton, but it had been building up for a while.
AW: Building up, actually in some sense I think it was a lightning bolt, in the sense that all of the progress happened within maybe a 30-year period, so in the world of mathematics, that’s sort of, you could even say that’s a fad or a trend. Someone does something, and you’re like, oh my god, let’s see what we can do with this. It’s an amazing insight that the two are connected.
The most concrete illustration of this is actually one I read on Wikipedia, which says that suppose you’re in a car, and you’re not the driver because otherwise this would be a very scary application. You can’t see outside of the car, but you can see the odometer. Sorry, you can’t see the odometer either. Someone’s put tape over it. But you can see the speedometer. And that’s telling you your velocity at every second. Every instance there’s a number. And what the fundamental theorem of calculus says is that if you add up all of those numbers over a given interval of time, it’s going to tell you how far you’ve traveled.
KK: Right.
AW: You could just take the speed that you see on the odometer the minute you start driving the car and then multiply by the amount of time that you travel, and that’ll give you kind of an approximate idea, but you instead could break the time into two pieces and take the velocity that you see at the time and the velocity that you see at the midpoint, and take the average of those two velocities, multiplied by the amount of time, and that’ll give you a better sense. And basically it says to compute the average velocity multiplied by the time, and you’re going to get how far you’ve gone. That’s basically what the fundamental theorem of calculus means.
KK: So here’s my own hot take on the fundamental theorem: I think it’s actually named incorrectly. I think the mean value theorem is the real fundamental theorem of calculus.
AW: Ah-ha.
KK: If you think about the fundamenta...

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Episode 3 - Emille Davie Lawrence

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 ...