Episode 8 - Justin Curry

12/07/17 • 13 min

Kevin Knudson: Welcome to MFT. I'm Kevin Knudson, your host, professor of mathematics at the University of Florida. I am without my cohost Evelyn Lamb in this episode because I'm on location at the Banff International Research Station about a mile high in the Canadian Rockies, and this place is spectacular. If you ever get a chance to come here, for math or not, you should definitely make your way up here. I'm joined by my longtime friend Justin Curry. Justin.
Justin Curry: Hey Kevin.
KK: Can you tell us a little about yourself?
JC: I'm Justin Curry. I'm a mathematician working in the area of applied topology. I'm finishing up a postdoc at Duke University and on my way to a professorship at U Albany, and that's part of the SUNY system.
KK: Contratulations.
JC: Thank you.
KK: Landing that first tenure-track job is always
JC: No easy feat.
KK: Especially these days. I know the answer to this already because we talked about it a bit ahead of time, but tell us about your favorite theorem.
JC: So the theorem I decided to choose was the classification of regular polyhedra into the five Platonic solids.
KK: Very cool.
JC: I really like this theorem for a lot of reasons. There are some very natural things that show up in one proof of it. You use Euler's theorem, the Euler characteristic of things that look like the sphere, R=2.
There's duality between some of the shapes, and also it appears when you classify finite subgroups of SO(3). You get the symmetry groups of each of the solids.
KK: Oh right. Are those the only finite subgroups of SO(3)?
JC: Well you also have the cyclic and dihedral groups.
KK: Well sure.
JC: They embed in, but yes. The funny thing is they collapse too because dual solids have the same symmetry groups.
KK: Did the ancient Greeks know this, that these were the only five? I'm sure they suspected, but did they know?
JC: That's a good question. I don't know to what extent they had a proof that the only five regular polyhedra were the Platonic solids. But they definitely knew the list, and they knew they were special.
KK: Yes, because Archimedes had his solids. The Archimedean ones, you are allowed different polygons.
JC: That's right.
KK: But there's still this sort of regularity condition. I can never remember the actual definition, but there's like 13 of them, and then there's 5 Platonics. So you mentioned the proof involving the Euler characteristic, which is the one I had in mind. Can we maybe tell our listeners how that might go, at least roughly? We're not going to do a case analysis.
JC: Yeah. I mean, the proof is actually really simple. You know for a fact that vertices minus edges plus faces has to equal 2. Then when you take polyhedra constructed out of faces, those faces have a different number of edges. Think about a triangle, it has 3 edges, a square has 4 edges, a pentagon is at 5. You just ask how many edges or faces meet at a given vertex? And you end up creating these two equations. One is something like if your faces have p sides, then p times the number of faces equals 2 times the number of edges.
KK: Yeah.
JC: Then you want to look at this condition of faces meeting at a given vertex. You end up getting the equation q times the number of vertices equals 2 times the number of edges. Then you plug that into Euler's theorem, V-E+F=2, and you end up getting very rigid counting. Only a few solutions work.
KK: And of course you can't get anything bigger than pentagons because you end up in hyperbolic space.
JC: Oh yeah, that's right.
KK: You can certainly do this, you can make a torus. I've done this with origami, you sort of do this modular thing. You can make tori with decagons and octagons and things like that. But once you get to hexagons, you introduce negative curvature. Well, flat for hexagons.
JC: That's one of the reasons I love this theorem. It quickly introduces and intersects with so many higher branches of mathematics.
KK: Right. So are there other proofs, do you know?
JC: So I don't know of any other proofs.
KK: That's the one I thought of too, so I was wondering if there was some other slick proof.
JC: So I was initially thinking of the finite subgroups of SO(3). Again, this kind of fails to distinguish the dual ones. But you do pick out these special symmetry groups. You can ask what are these symmetries of, and you can start coming up with polyhedra.
KK: Sure, sure. Maybe we should remind our readers about-readers-I read too much on the internet-our listeners about duality. Can you explain how you get the dual of a polyhedral surface?
JC: Yeah, it's really simple and beautiful. Let's start with something, imagine you have a cube in your mind. Take the center of every face and put a vertex in. If you have the cube, you have six sides. So this dual, this thing we're constructing, has six ver...

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Episode 7 - Henry Fowler

Evelyn Lamb: Welcome to My Favorite Theorem, the show where we ask mathematicians what their favorite theorem is. I’m your host Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. And this is your other host.
Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. I had to wear a sweater yesterday.
EL: Oh my goodness! Yeah, I’ve had to wear a sweater for about a month and a half, so.
KK: Yeah, yeah, yeah.
EL: Maybe not quite that long.
KK: Well, it’ll be hot again tomorrow.
EL: Yeah. So today we’re very glad to have our guest Henry Fowler on. Henry, would you like to tell us a little bit about yourself?
Henry Fowler: I’m a Navajo Indian. I live on the Navajo reservation. I live by the Four Corners in a community, Tsaile, Arizona. It’s a small, rural area. We have a tribal college here on the Navajo Nation, and that’s what I work for, Diné College. I’m a math faculty. I’m also the chair for the math, physics, and technology. And my clan in Navajo is my maternal clan is Bitterwater and my paternal clan is Zuni Edge Water.
EL: Yeah, and we met at the SACNAS conference just a couple weeks ago in Salt Lake City, and you gave a really moving keynote address there. You talked a little bit about how you’re involved with the Navajo Math Circles.
HF: Yes. I’m passionate about promoting math education for my people, the Navajo people.
EL: Can you tell us a little bit about the Navajo Math Circles?
HF: The Navajo Math Circles started seven years ago with a mathematician from San Jose State University, and her name is Tatiana Shubin. She contacted me by email, and she wanted to introduce some projects that she was working on, and one of the projects was math circles, which is a collection of mathematicians that come together, and they integrate their way of mathematical thinking for grades K-12 working with students and teachers. Her and I, we got together, and we discussed one of the projects she was doing, which was math circles. And it was going to be here on the Navajo Nation, so we called it Navajo Math Circles. Through her project and myself here living on the Navajo Nation, we started the Navajo math circles.
KK: How many students are involved?
HF: We started first here at Diné College, we started first with a math summer camp, where we sent out applications, and these were for students who had a desire or engaged themselves to study mathematics, and it was overwhelming. Over 50 students applied for only 30 slots that were open because our grant could only sustain 30 students. So we screened the students and with the help of their regular teachers from junior high or high school, so they had recommendation letters that were also presented to us. So we selected the first 30 students. Following that we expanded our math circle to the Navajo Nation public school system, and there’s also contract schools and grant schools. Now we’re serving, I would say over 1,000 students now.
KK: Wow. That’s great. I assume these students have gone on to do pretty interesting things once they finish high school and the circle.
HF: Yes. We sort of strategized. We wanted to work with lower grades a little bit. We wanted to really promote a different way of thinking about math problems. We started off with the first summer math camp at the junior high or the middle school level, and also the students that were barely moving to high school, their freshman year or their 10th grade year. That cohort, the one that we started off with, they have a good rate of doing very well with their academic work, especially in math, at their high school and junior high school. We have four that have graduated recently from high school, and all four of them are now attending a university.
KK: That’s great.
EL: And some of our listeners may have seen there’s a documentary about Navajo math circles that has played on PBS some, and we’ll include a link to that for people to learn a little bit about that in the show notes for the episode. We invited you here to My Favorite Theorem, of course, because we like to hear about what theorems mathematicians enjoy. So what have you selected as your favorite theorem?
HF: I have quite a few of them, but something that is simple, something that has been an awe for mathematicians, the most famous theorem would be the Pythagorean theorem because it also relates to my cultural practices, to the Navajo.
KK: Really?
HF: The Pythagorean theorem is also how Navajo would construct their traditional home. We would call it a Navajo hogan. The Navajo would use the Pythagorean theorem charting how the sun travels in the sky, so they would open their hogan door, which is always constructed facing east. So once the sun comes out, it projects its energy, the light, into the hogan. The Navajo began to study that phenomenon, how that light travels in space in the h...

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Episode 9 - Ami Radunskaya

Evelyn Lamb: Welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I’m a freelance math and science writer based in Salt Lake City. And today I am not joined by my cohost Kevin Knudson. Today I am solo for a very special episode of My Favorite Theorem because I am at MathFest, the annual summer meeting of the Mathematical Association of America. This year it’s in Chicago, a city I love. I lived here for a couple years, and it has been very fun to be back here with the big buildings and the lake and everything. There are about 2,000 other mathematicians here if I understand correctly. It’s a very busy few days with lots of talks to attend and friends to see, and I am very grateful that Ami Radunskaya has taken the time to record this podcast with me. So will you tell me a little bit about yourself?
Ami Radunskaya: Hi Evelyn. Thanks. I’m happy to be here at MathFest and talking to you. It’s a very fun conference for me. By way of introduction, I’m the current president for the Association for Women in Mathematics, and I’m a math professor at Pomona College in Claremont, which is a small liberal arts college in the Los Angeles County. My Ph.D. was in ergodic theory, something I am going to talk about a little bit. I went to Stanford for my doctorate, and before that I was an undergraduate at Berkeley. So I grew up in Berkeley, and it was very hard to leave.
EL: Yeah. You fall in love with the Bay Area if you go there.
AR: It’s a place dear to my heart, but I was actually born in Chicago.
EL: Oh really?
AR: So I used to visit my grandparents here, and it brings back memories of the Museum of Science and Industry and all those cool exhibits, so I’m loving being back here.
EL: Yeah, we lived in Hyde Park when we were here, so yeah, the Museum of Science and Industry.
AR: I think I was born there, Hyde Park.
EL: Oh? Good for you.
AR: My dad was one of the first Ph.D.s in statistics from the University of Chicago.
EL: Oh, nice.
AR: Although he later became an economist.
EL: Cool. So, what is your favorite theorem?
AR: I’m thinking today my favorite theorem is the Birkhoff ergodic theorem. I like it because it’s a very visual theorem. Can I kind of explain to you what it is?
EL: Yeah.
AR: So I’m not sure if you know what ergodic means. I actually first went into the area because I thought it was such a cool word, ergodic.
EL: Yeah, it is a cool word.
AR: I found out it comes from the Greek word ergod for path. So I’ve always loved the mathematics that describes change and structures evolving, so before I was a mathematician I was a professional cellist for about 10 years. Music and math are sort of as one in my mind, and that’s why I think I’m particularly attracted to the kinds of mathematics and the kinds of theory that describes how things change, what’s expected, what’s unexpected, what do we see coming out of a process, a dynamical process? So before I state the theorem, I need to tell you what ergodic means.
EL: Yeah.
AR: It’s an adjective. We’re talking about a function. We say a function is ergodic if it takes points: imagine you put a value into a function, you get out a new value. You put that value back in to the function, you get a new value. Repeat that over and over and over again, and now the function is ergodic if that set of points sort of visits everywhere in the space. So we say more technically a function is ergodic if the invariant sets, the sets it leaves alone, the sets that get mapped to themselves, are either the whole space or virtually nothing. A function is ergodic, a map is ergodic, if the invariant sets either have, we say, full measure or zero measure. So if you know anything about probability, it’s probability 1 or probability zero. I think that’s an easy way to think about measure.
EL: Yeah, and I think I’ve heard people describe ergodic as the time average is equal to the space average, so things are distributing very evenly when you look at long time scales. Is that right?
AR: Well that’s exactly the ergodic theorem. So that’s a theorem!
EL: Oh no!
AR: No, so that’s cool that you’ve heard of that. What I just said was that something is ergodic if the sets that it leaves unchanged are either everything or nothing, so these points, we call them the orbits, go everywhere around the set, but that doesn’t tell you how often they visit a particular piece of your space, whereas the ergodic theorem, so there are two versions of it. My favorite one is the one, they call it the pointwise ergodic theorem, because I think it’s easier to visualize. And it’s attributed to Birkhoff. So sometimes it’s called the Birkhoff ergodic theorem. And it’s exactly what you just said. So if you have an ergodic function, and then we start with a point and we sort of average it over many, many applications of the function, or iterations of the function, so that’s the time average. We...