<input onclick="this.value=this.value=='Episode Transcript (click to expand)'?'Hide Transcript':'Episode Transcript (click to expand)';" type="button" class="spoilerbutton" value="Episode Transcript (click to expand)">
 
 
 
 
 
Kevin Knudson: Welcome to My Favorite Theorem. I’m Kevin Knudson, professor of mathematics at the university of Florida, and this is your other host. 

EL: Hi. I’m Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. 

KK: How’s it going, Evelyn? 

EL: How are you today? 

KK: I’m okay. I’m a little sleepy. So before I came to Florida, I was at Mississippi State University, and I still have a lot of good friends and colleagues there, and that basketball game last night. I don’t know if you guys saw it, but that last minute shot, last second shot that Notre Dame hit to win was just a crusher. I’m feeling bad for my old friends. But other than that, everything’s great. Nice sunny day. 

EL: Yeah, it’s gray here, so I’m a little, I always have trouble getting moving on gray mornings. 

KK: But you’ve got that nice big cup of tea, so you’re in good shape. 

EL: Yes. 

KK: That’s right. So today we are pleased to welcome Mike Lawler. Mike, why don’t you introduce yourself and tell everyone about yourself? 

1:24 ML: Hi. I’m Mike Lawler. I work in the reinsurance division for Berkshire Hathaway studying large reinsurance deals. And I also spend a lot of my spare time doing math activities for kids, actually mostly my own kids. 

KK: Yeah. 

EL: Yeah. 

KK: Yours is one of my favorite sites on the internet, actually. I love watching how you explain really complicated stuff to your kids. How old are they now? They’re not terribly old. 

ML: They’re in eighth grade and sixth grade. 

KK: But you’ve been doing this for quite a while. 

ML: We started. Boy, it could have been 2011, maybe before that. 

KK: Wow, right. 

ML: I think all three of us on the podcast today, and probably everybody listening, loves math. 

KK: One hopes. 

ML: And I think there’s a lot of really exciting math that kids are really interested in when they see. It’s fun finding things that are interesting to mathematicians and trying to figure out ways to share them with kids. 

EL: Yeah. Well I like, you always make videos of the things, so listening to your kids talking through what they’re thinking is really fun. Recently I watched one of the old ones, and I was like, “Oh my goodness! They’re just little babies there.” They’re so much bigger now. I don’t have kids of my own, so I don’t have that firsthand look at kids growing up the same way. They’re sweet kids, though. 

ML: I have to say, one of the first, it wasn’t actually the first one we did, but it’s called Family Math 1, where we do the famous “How many times can you fold a piece of paper?” And, you know, they’re probably 4 and 6 at the time, or maybe 5 and 7, and yeah, it’s always fun to go back and watch that one. 

EL: Yeah. 

KK: I see videos of my son, he’s now 18, he’s off in college. When I see videos of him, he’s a musician, so when he was 10, figuring out how to play this little toy accordion we got him, I kind of get a little weepy. You know. 

ML: It’s funny, I was picking him up somewhere the other day, and I confused him with a 20-year-old, my older son, and I just thought to myself: how did this happen? 

KK: So, all right. Enough talking about kids, I guess. So, Mike, we asked you on to talk about your favorite theorem. So what is it? 

ML: Well, it’s not quite a theorem, but it’s something that’s been very influential to me. Not in sharing math with kids, but in my own work. It comes from a paper from 1995 at a professor named Zvi Bodie at BU. And he was studying finance, and continues to study finance. And he published a paper showing that the cost of insurance for long holdings in the stock market actually increases with time. Specifically, if you want to buy insurance to guarantee your investments at least earn the risk-free rate, that cost of insurance goes up over time. And it just shocked me when I was just learning about finance, actually when I was just in grad school. And this paper has had a profound influence on me over the last 20 years. So that’s what I want to talk about today. 

KK: Okay. I know hardly any of those words. I have my retirement accounts and all that, but like most good quantitatively-minded people, I just ignore them. 

ML: Well, let’s take a simple example. Let’s just take actually the most simple example. Say you wanted to invest $100 in the stock market, and you thought, because you’ve read or you’ve heard that the stock market gives you good returns, you thought, “Well, in 10 years from now, I think I’ll probably have at least $150 in that account.” And you said, “Well, what I want to do is go out and buy some insurance that guarantees me that at least I’ll have that amount of money in the account. That’s the problem. That’s the math problem that Bodie studied. 

KK: Right. So how does one price that insurance policy, I guess? So right, on the insurance side, how do they price that correctly, and on the consumer side, how do you know you’re getting a worthwhile insurance policy, I guess. 

ML: Yeah, well this is kind of the fun of applied mathematics. So there’s a lot of theory behind this, and I think like a lot of good theories, it’s not named after the people who originally discovered that. So I think that’s important part of any theory. But then when you understand the theory, and you actually go into the financial markets, you have to start to ask yourself, “What parts of the theory apply here, and which ones don’t?” So the theory itself goes back to the early 1900s with a French mathematician and his Ph.D. thesis. His last name is Bachelier, and I’m probably butchering that. But then people began to study random processes, and Norbert Wiener studied those. And eventually all of that math came into economics, I think in the late 60s, early 1970s, and something called the Black-Scholes formula came to exist. The Black-Scholes formula is what people use to price this kind of insurance, sometimes called options. So that’s been around the financial markets since at least the early 1970s, so let’s call it 50 years now. And if you’re a consumer, I think you’d better be careful. 

EL: Well I find, I don’t know a lot about financial math, but I’ve tried to read a few books about the financial crash, actually one of which you suggested to me, I think, All the Devils Are Here. And I find, even with my math background, it’s very confusing what they’re pricing and how they’re calculating these, how they’re batching all of these things. It just really seems like a black box that you’re just kind of hoping what’s in the box isn’t going to eat you. 

ML: That’s a pretty good description. Yeah, Bethany MacLean’s book All the Devils Are Here is absolutely phenomenal, and Roger Lowenstein’s book, called When Genius Failed, is also an absolutely phenomenal book. You are absolutely right. The math is very heavy, and a lot of times, especially when you talk about the financial crisis, the math formulas get misused a little bit, and maybe are applied into situations where they might not necessarily apply. 

KK: Really? Wall Street does that? 

ML: So you really have to be careful. I think if you pull the original Black-Scholes paper, I think there are 7 or 8 assumptions that go into it. As long as these 7 or 8 things are true, then we can apply this theory. In theory we can apply the theory. 

KK: Right. 

ML: So when you go into the financial markets, a lot of times if you have that checklist of 7 things with you, you’re going to find maybe not all 7 are true. In fact, a lot of times, maybe you’re going to find not a single one of those things is true. And that is I think a problem that a lot of mathematicians have when they come into the markets, and they just think the theory applies directly, if you will. 

KK: Right, and we’ve all taught enough students to know they’re not very good at checking assumptions, right? So if you have to check off a list of 6 or 7 things, then after the first couple, you’re like, “Eh, I think it’s fine.” 

ML: Right. Maybe that seventh one really matters. 

KK: Right. 

EL: Yeah. 

ML: Or maybe you’re in a situation where the theory sort of applies 95% of the time, but now you’re in that 5% situation where it really doesn’t apply. 

KK: So should I buy investment insurance? I mean, I’ve never directly done such a thing. 

ML: Well… 

KK: I don’t know if it’s an option for me since I just have 401Ks, essentially. 

ML: Well, it’s probably not a great idea to give investment advice over a podcast. 

KK: Right, yeah, yeah. 

ML: But from a mathematical point of view, the really interesting thing about Bodie’s paper is Black-Scholes is indeed a very complicated mathematical idea, but the the thing that Bodie found was a really natural question to ask about pricing this kind of insurance, ensuring that your portfolio would grow at the risk-free rate. In that situation, and you can see it in Bodie’s paper, the math simplifies tremendously. And I think that is a common theme across mathematics. When someone finds exactly the right way to look at a problem, all of a sudden the problem simplifies. And I’m sure you can probably give me 3 or 4 examples in your own fields where that is then the case. 

KK: Sure. Well, I’m not going to, but yeah. 

EL: So something when you told us that this was the theorem or quasi-theorem you were going to talk about, it got me wondering how much the financial world—I’ve been trying to think about how to phrase this question—but how much your natural tendencies as mathematicians actually carry over into finance. How much are you able to think about your work in finance and insurance as math questions and how much you really have to shift how you’re thinking about things to this more realistic point of view. 

ML: I think it’s a great question because, you know, the assumptions and a lot of times the mathematical simplifications that allow you to solve these differential equations that stand behind the Black-Scholes theorem and generally stochastic processes, you know, you’re, that doesn’t translate perfectly to the real world. And you have to start asking questions like, “If this estimate is wrong, does it miss high? Does it miss low?” “In the 5% of the times it doesn’t work, do I lose all my money? 

EL: Right. 

ML: And so those, I can tell you as an undergrad I was also a physics major, and I spent a lot of time in the physics lab, and there’s not one single person who was ever in lab with me who misses me. I was a mathematician in the labs. But doing some of these physics experiments really teaches you that applying the theory directly, even in a lab situation, is very difficult. 

KK: Right. 

EL: Right. And your Ph.D. was in pure math, right? 

ML: Right, it was sort of mathematical physics. In the late 90s, people were really excited about the Yang-Mills equations. 

KK: Mirror symmetry. 

ML: Work that Seiberg and Witten were doing. So I was interested in that. 

EL: So your background is different from what you’re doing now. 

ML: Oh, totally. You know, I, it’s kind of a hard story for me to tell, but I really loved math from the time I was in fifth grade all the way up through about my third year of graduate school. 

EL: Yeah, I think that could be a painful story. 

ML: I don’t know why, I really don’t know why, I just kind of lost interest in math then. I finished my Ph.D., and I even took an appointment at the University of Minnesota, but I just lost interest, and it was an odd feeling because from about fifth grade until—what grade is your third year of graduate school? 

KK: Nineteenth. 

ML: Nineteenth grade. I really got out of bed every morning thinking about math, and I sort of drifted away from it. But my kids have brought me back into it, so I’m actually really happy about that. 

KK: Well that’s great. So, what have you chosen to pair with your quasi-theorem, we’re calling it? 

ML: Well, you know, so for me, this paper of Bodie’s goes back, and it sort of opened a new world for me, and for the last 20 years I’ve been studying more about it and learning more about it and all these different things, so I got to thinking about a journey. I have books on my table right now about this paper. So the journey I want to highlight is—and I think a lot of people can understand who are outside of math—is an athletic journey. I’m going to bring up a woman named Anna Nazarov, who represents the United States on the national ultimate frisbee team, which is a sport I’ve been around. And four years ago, she made it almost to being on the national team and got cut in the last minute and wrote this very powerful essay about her feelings about getting cut and then turned around and worked hard and improved and won three world and national championships in the last four years as a result of that work. 

KK: Wow. 

ML: Yeah, you know, it’s hard to compare world championships to just your plain old work. I think people in math understand that you kind of roll up your sleeves and over a long period of time you come to understand mathematics, or you come to understand in this case how certain mathematics applies, and so I want to pair this with that kind of athletic journey, which I think, to the general public, people understand a little bit better. 

EL: Yeah, so I played ultimate very recreationally in grad school. There was a math department pickup ultimate game every week, and playing with other math grad students is my speed in ultimate. I really miss it. When you, I can tell, follow ultimate, and I often read the links you post about ultimate frisbee, I’m like, oh, I kind of miss doing that. But a few years ago, I did get to, I happened to be in Vancouver at the same time that they were doing the world ultimate championships there and got to see a couple games there, and it’s really fun, and it’s been fun to follow the much-higher-level-than-math-grad-student ultimate playing thing through the things you’ve posted. 

ML: Yeah, it’s neat to follow an amateur sport, or not as well-known a sport because the players work so hard, and they spend so much of their own money to travel all over the world. You know, I think a lot of people do that with math. Despite the topic of today’s conversation, most people aren’t going into math because of the money. 

KK: Well this has been great fun. Thanks for joining us, Mike. Is there anything, we always want to give our guest a chance to plug anything. We already kind of plugged your website. 

EL: We’ll put links to your blog in the show notes there, and your Twitter. But yeah, if there’s anything else you want to plug here, this is the time for it. 

ML: No, that’s fine. If you want to follow Mike’s Math Page, it’s a lot of fun sharing math with kids. And like I said, I sort of lost interest in math in grad school, but sharing math with kids now is what gets me out of bed in the mornings. 

KK: Great. 

EL: Yeah. 

KK: All right. Well, thanks again, Mike. 

ML: Thank you. 

 

 




 <a href="https://blogs.scientificamerican.com/roots-of-unity/financial-mathematics-and-ultimate-frisbee/" target="_blank">Show notes</a>

<description>Mike Lawler is a mathematician working in finance. Join us to learn an interesting theorem about insurance pricing.</description>

In which we definitely do *not* give financial advice.

Mike Lawler is a mathematician working in finance. Join us to learn an interesting theorem about insurance pricing.

Episode 29 - Mike Lawler

<input onclick="this.value=this.value=='Episode Transcript (click to expand)'?'Hide Transcript':'Episode Transcript (click to expand)';" type="button" class="spoilerbutton" value="Episode Transcript (click to expand)">
 
 
 
 
 
 Kevin Knudson: Welcome to My Favorite Theorem. I’m your cohost Kevin Knudson, professor of mathematics at the University of Florida. I am joined by cohost number 2. 

Evelyn Lamb: I am Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City. So how are you? 

KK: I’m okay. And by the way, I did not mean to indicate that you are number 2 in this. 

EL: Only alphabetically. 

KK: That’s right. Yeah. Things are great. How are things in Salt Lake? 

EL: Pretty good. I had a fantastic weekend. Basically spent the whole thing reading and singing, so yeah, it was great. 

KK: Good for you. 

EL: Yeah. 

KK: I didn’t do much. I mopped the floors. 

EL: That’s good too. My floors are dirty. 

KK: That’s okay. Dirty floors, clean…something. So today we are pleased to have Chawne Kimber on the show. Chawne, do you want to introduce yourself? 

Chawne Kimber: Sure. Hi, I’m a professor at Lafayette College. I got my Ph.D. a long time ago at University of Florida. 

KK: Go Gators! 

CK: Yay, woo-hoo. I work in lattice-ordered groups. 

KK: Lattice-ordered groups, very cool. I should probably know what those are, but maybe we’ll find out what they are today. So yeah, let’s get into it. What’s your favorite theorem, Chawne? 

CK: Okay, so maybe you don’t like this, but it’s a suite of theorems. 

KK: Even better. 

EL: Go for it. 

CK: So, right, a lattice-ordered group is a group, to begin with, in which any two elements have a sup and an inf, so that gives you your lattice order. They’re torsion-free, so they’re, once you get past countable ones, they’re enormous groups to work with. So my favorite theorems are the representation theorems that allow you to prove stuff because they get unwieldy due to their size. 

EL: Oh cool. One of my favorite classes in grad school was a representation class. I mean, I had a lot of trouble with it. It was just representations of finite groups, and those were still really out there, but it was a lot of fun. Really algebraic thinking. 

CK: Well actually these representations allow you to translate problems from algebra to topology, so it’s pretty cool. The classical theorem is by Hahn in 1909. He proved the special cases that any totally ordered Archimedean group can be embedded as a subgroup of the reals, and it kind of makes sense that you should be able to do that. 

KK: Sure. 

CK: And then he said that any ordered abelian group, so not necessarily lattice-ordered, can be embedded in what’s called a lexicographical product of the reals. So we could get into what that is, but those are called Hahn groups. They’re just huge products of the reals that are ordered in dictionary order that only live on well-ordered sets. So this conjecture, it’s actually a theorem, but then there’s a conjecture that that theorem is actually equivalent to the axiom of choice. 

KK: Wow. 

EL: Oh wow. 

CK: Right? 

EL: Can we maybe back up a little bit, is it possible to, for me, I really like concrete examples, so maybe can you talk a little bit about a concrete example of one of these archimedean groups? I don’t know how concrete the concrete examples are. 

CK: No, they’re just really weird ways of hacking at the reals, basically, so they’re just subgroups of the reals. Think of your favorite ones, and there you go, the ones that are archimedean. And as soon as you add two dimensions of ordering, it’s even more complex, right? So the classical example that I work with would be rings of continuous functions on a topological space, and then you can build really cool examples because we all understand continuous functions, so C(X), real-valued continuous functions on a Tychonoff space, so T-3 1/2, whatever. 

KK: Metric space. 

CK: The axioms so you have enough continuous functions. So Gillman and Jerison in the 1950s capitalized on a theorem from the 1930s by Gelfand and Kolmogorov that said that the maximal ideals of C(X), if you take them in the hull-kernel topology, are isomorphic to the Stone-Čech compactification of the space that you’re working on. And so if you have a compact space to begin with, then your space is isomorphic to your maximal ideals. So then, just build your favorite—so C(X) is lattice-ordered, if you take the pointwise ordering, and then since the reals have a natural order on then, you pick up your sups and infs pretty easily. So there you’re starting to touch some interesting examples of these groups. Have I convinced you, Evelyn? 

EL: Yeah, yeah. 

CK: Okay, good. So they’re huge. You have to have some complexity in order to be able to prove anything interesting about them. So then there the Hahn embedding is pretty obvious. You just take the images of the functions. There’s too much structure in a ring like that, so maybe you want to look at just an ordered group to get back to the Hahn environment. So how can you mimic Hahn in view of Gelfand-Kolmogorov? So can we get continuous functions as the representation of an ordered group? Because the lex products that Hahn was working with are intractable in a strong way. And so then you have to start finding units because you have to be able to define something called a maximal sub-object, so you want it to be maximal with respect to missing out on some kind of unit. And so then we get into a whole series of different embedding theorems that are trying to get you closer to being able to deal with the conjecture I mentioned before, that Hahn’s embedding theorem is equivalent to the axiom of choice. 

EL: Yeah, I’m really fascinated by this conjecture. It kind of seems like it comes out of nowhere. Maybe we can say what the axiom of choice is and then, is there a way you can kind of explain how these might be related? 

CK: Yes and no. 

KK: Let’s start with the axiom of choice. 

CK: Yeah, so the axiom of choice is equivalent to Zorn’s lemma, which says that maximal objects exist. So that’s the way that I deal with it. It allows me to say that maximal ideals exist, and if they didn’t exist, these theorems wouldn’t exist. You use this everywhere in order to prove Hahn’s theorem, so that’s why it’s assumed to be possibly equivalent. This isn’t the part that I work on. I’m not a logician. 

KK: So many things are equivalent to the axiom of choice. For example, the Tychonoff product theorem, which is that the product of compact spaces is compact. That’s actually equivalent to the axiom of choice, which seems a bit odd. I was actually reading last night, so Eugenia Cheng has this book Beyond Infinity, her most recent book, good bedtime reading. I learned something last night about the axiom of choice, which is that you need the axiom of choice to prove that if you have two infinities, two countable infinities, you want to think [they’re the same], it’s countable somehow. If they come with an order, then fine, but if you have two, like imagine pairs of socks, like an infinite collection of pairs of socks, is that countable? Are the socks countable? It’s an interesting question, these weird slippery things with the axiom of choice and logic. They make my head hurt a little bit. 

CK: Mine too. 

EL: So yeah, you’re saying that looking at the axiom of choice from the Zorn’s lemma point of view, that’s where these maximal objects are coming in in the Hahn conjecture, right? 

CK: Absolutely. 

KK: That makes sense. 

CK: That’s kind of why I drew the parallel with this theorem about C(X), these maximal ideals being equivalent to the space you’re on. Pretty cool. 

KK: Right. Because even to get maximal ideals in an arbitrary ring, you really need Zorn’s lemma. 

CK: Right. And there’s a whole enterprise of people working to see how far you can peel that back. I did take a small foray into trying to understand gradations of the axiom of choice, and that hurts your head, definitely. 

KK: Right, countable axiom of choice, all these different flavors. 

CK: Williams prime ideal theorem, right. 

KK: Yeah, okay. 

EL: So what drew you to these theorems, or what makes you really excited about them? 

CK: Well, you know, as a super newbie mathematician back in the day, I was super excited to see that these disparate fields of algebra and topology that everyone had told me were totally different could be connected in a dictionary way. So a characteristic of a ring can be connected is equivalent to a characteristic on a topological space. So all kinds of problems can be stated in these two different realms. They seem like different questions, but they turn out to be equivalent. So if you just know the way to cross the bridge, then you can answer either question depending on which realm gives you the easier approach to the theorem. 

KK: I like that interplay too. I’m a topologist, but I’m a very algebraic one for exactly that reason. I think there are so many interesting ideas out there where you really need the other discipline to solve it, or looking through that lens makes it a lot clearer somehow. 

EL: And was this in graduate school that you saw these, or as a new professor? 

CK: Definitely grad school. I was working on my master’s. 

KK: So I wonder, what does one pair with this suite of theorems? 

CK: It’s a very hard question, actually. 

KK: That’s typical. Most people find this the more difficult part of the show. 

CK: Yeah. I think that if you were to ask my Ph.D. advisor Jorge Martinez what he would pair, he is very much a wine lover and an opera lover. So it would be both. You’d probably see him taking a flask into Lincoln center while thinking about theorems. So he loved to go to Tuscany, so I assume that’s where you get chianti. I don’t know, I could be lying. 

KK: You do, yeah. 

CK: Yeah, so let’s go with a good chianti, although that might make me sound like Hannibal Lecter. 

KK: No fava beans. 

CK: So we’ve got a chianti, and maybe a good opera because it’s got to be both with him. It’s hard for me to say. So he comes up to New York to do an opera orgy, just watching two operas per day until he falls down. I sometimes join him for that, and the last one I went to was Così fan tutte, and so let’s go with that because that’s the one I remember. 

EL: If I remember correctly—it’s been a while since I saw or listened to that opera—there are pairs of couples who end up in different configurations, and it’s one of these “I’ll trick you into falling in love with the other couple’s person” that almost seems like the pairs being topology and algebra, and switching back and forth. I don’t know, maybe I’m putting ideas in your mind here. 

CK: Or sort of the graph of the different couplings, the ordered graph could be the underlying object here. You never know. 

EL: An homage to your advisor here with this pairing. 

CK: Yeah, let’s do that. 

EL: Well I must admit I was kind of hoping that you might pair one of your own quilt creations here. So I actually ran into you through a quilting blog you have called completely cauchy. Do you mind talking to us a little bit about how you started quilting and what you do there because it’s so cool. 

CK: Yeah. Of course I chose that name because Cauchy is my favorite mathematician, and as a nerd there would be no other quilt blog named after a dead mathematician. So I am a little mortified that when you google “Cauchy complete,” as many students do, mine is actually the first entry that comes up on google. 

KK: Excellent. 

CK: I don’t know what that means, but okay. So yeah, when I applied for tenure, which is kind of a hazing process no matter where you are, no matter how good of a faculty member you are, I really wanted to have control, and you don’t have control at that point. And so I started sewing for fun, late at night, at 1 am, after everything kind of felt done for the day. I never imagined that I’d be doing what I’m doing today, which is using quilting to confront issues of social justice in the United States, and they’ve been picked up by museums and other venues. It’s this whole side hustle out there that I kept quiet for a long, long time. And then once I got promoted to full professor I came out of the closet. 

KK: Were you concerned that having a side hustle, so to speak, would compromise your career? Because it shouldn’t. 

CK: Yeah, I think something so gender-specific as quilting, something you associate with grandmas. At the end of the day, the guys I work with, I must say half of my quilts have four-letter words on them, you know, the more interesting four-letter words, so as soon as my guys saw them, they were totally on board with this enterprise, so I didn’t really need to be into the closet, but I didn’t want anybody to ever say, “Oh, she should have proved one more theorem instead of making that quilt.” 

EL: Yeah. 

KK: It’s unfortunate that we feel that way, right? I think that’s true of all mathematicians, but I imagine it’s worse for women, this idea that you have to work twice as hard to prove you’re half as good or something like that? 

CK: Do we need to mention I’m also a black woman? So that’s actually how I was raised, you need to do three times as much to be seen as half as good, and that’s the way that I’ve lived my life, and it’s not sustainable in any way. 

KK: No, absolutely not. 

EL: But yeah, they are really cool quilts, so everyone should look at completely cauchy, and that’s spelled cauchy. There’s a mathematician named Cauchy. I actually have another mathematician friend with a cat named Cauchy, or who had a cat named Cauchy. I think the cat has passed away. Yeah, and I actually sew as well. I’ve somehow never had the patience for quilting. It just feels somehow like too little. I like the more immediate gratification of making a whole panel of a skirt or something. You do really intricate little piecing there, which I admire very much, and I’m glad people like you do it so I don’t have to. 

KK: Sure, but Evelyn, you don’t have to make it little. 

CK: You don’t. 

KK: I’m sure you’ve seen these Gee’s Bend quilts, right, they’re really nice big pieces, and that can have a very dramatic effect too. But yeah, the intricate work is really remarkable. My wife has done a little quilting, and she always gets tired of it because of the fine stuff, but then she’s a book artist. She sets lead typing in her printing press by hand, and that’s fine, but piecing together little pieces of cloth somehow doesn’t work. 

CK: It seems more futile, you take a big piece of fabric and cut it into small pieces so that you can sew it back together. That is kind of dumb when you think about it. 

KK: Well, but I don’t know, you’ve got this whole Banach-Tarski thing, maybe. 

EL: Bring it back around to the axiom of choice again. 

CK: You guys are good at this. 

KK: It’s not our first podcast. Well this has been great fun. Anything else you want to promote? 

CK: No, I’m good. 

KK: Thanks for joining us, Chawne. This has really been interesting, and we appreciate you being on. 

CK: Great. Thank you. 

EL: Thanks. 
 
 

 




 <a href="https://blogs.scientificamerican.com/roots-of-unity/math-quilting-and-activism/" target="_blank">Show notes</a>

<description>Join mathematician Chawne Kimber for a journey into Archimedean groups, lattice-ordered groups, and quilting.</description>

In which we embed lattice-ordered groups in things and also make some quilts.

Join mathematician Chawne Kimber for a journey into Archimedean groups, lattice-ordered groups, and quilting.

Episode 28 - Chawne Kimber

<input onclick="this.value=this.value=='Episode Transcript (click to expand)'?'Hide Transcript':'Episode Transcript (click to expand)';" type="button" class="spoilerbutton" value="Episode Transcript (click to expand)">
 
 
 
 
 
 
 Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m one of your hosts, 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. How are you doing, Evelyn? 

EL: I’m all right. I’m excited because we’re trying a different recording setup today, and a few of our recent episodes, I’ve had a few connection problems, so I’m hoping that everything goes well, and I’ve probably jinxed myself by saying that. 

KK: No, no, it’s going to be fine. Positive thinking. 

EL: Yeah, I’m hoping that the blips that our listeners may have heard in recent episodes won’t be happening. How about you? Are you doing well? 

KK: I’m fine. Spring break is next week, and we’ve had the air conditioning on this week. This is the absurdity of my life. It’s February, and the air conditioning is on. But it’s okay. It’s nice. My son is coming home for spring break, so we’re excited. 

EL: Great. We’re very happy today to have Jana Rodriguez-Hertz on the show. So, Jana, would you like to tell us a little bit about yourself? 

Jana Rodriguez-Hertz: Hi, thank you so much. I’m originally from Argentina, I have lived in Uruguay for 20 years, and now I live in China, in Shenzhen. 

EL: Yeah, that’s quite a big change. When we were first talking, first emailing, I mean, you were in Uruguay then, you’re back in China now. What took you out there? 

JRH: Well, we got a nice job offer, and we thought we’d like to try. We said, why not, and we went here. It’s nice. It’s a totally different culture, but I’m liking it so far. 

KK: What part of China are you in, which university? 

JRH: In Southern University of Science and Technology in Shenzhen. It’s in Shenzhen. Shenzhen is in mainland China in front of Hong Kong, right in front of Hong Kong. 

KK: Okay. That’s very far south. 

EL: I guess February weather isn’t too bad over there. 

JRH: It’s still winter, but it’s not too bad. 

EL: Of course, that will be very relevant to our listeners when they hear this in a few months. We’re glad to have you here. Can you tell us about your favorite theorem? 

JRH: Well, you know, I live in China now, and every noon I see a dynamical process that looks like the theorem I want to talk to you about, which is the dynamical properties of Smale’s horseshoe. Here it goes. You know, at the canteen of my university, there is a cook that makes noodles. 

EL: Oh, nice. 

JRH: He takes the dough and stretches it and folds it without mixing, and stretches it and folds it again, until the strips are so thin that they’re ready to be noodles, and then he cuts the dough. Well, this procedure can be described as a chaotic dynamical system, which is the Smale’s horseshoe. 

KK: Okay. 

JRH: So I want to talk to you about this. But we will do it in a mathematical model so it is more precise. So suppose that the cook has a piece of dough in a square mold, say of side 1. Then the cook stretches the dough so it becomes three times longer in the vertical sense but 1/3 of its original width in the horizontal sense. Then he folds it and puts the dough again in the square mold, making a horseshoe form. So the lower third of the square is converted into a rectangle of height 1 and width 1/3 and will be placed on the left side of the mold. The middle third of the square is going to be bent and will go outside the mold and will be cut. The upper third will be converted to another rectangle of height 1 and width 1/3 and will be put upside down in the right side of the mold. Do you get it? 

KK: Yeah. 

JRH: Now in the mold there will be two connected components of dough, one in the left third of the square and one in the right third of the square, and the middle third will be empty. In this way, we have obtained a map from a subset of the square into another subset of the square. And each time this map is applied, that is, each time we stretch and fold the dough, and cut the bent part, it’s called a forward iteration. So in the first forward iteration of the square, we obtain two rectangles of width 1/3 and height 1. Now in the second forward iteration of the square, we obtain four rectangles of width 1/9 and height 1. Two rectangles are contained in the left third, two rectangles in the right third. These are four noodles in total. 

Counting from left to right, we will see one noodle of width 1/9, one gap of width 1/9, a second noodle of width 1/9, a gap of 1/3, and two more noodles of width 1/9 separated by a gap of width 1/9. Is that okay? 

KK: Yes. 

JRH: So if we iterate n times, we will obtain 2n noodles of width (1/3)n. And if we let the number of iterations go to infinity, that is, if we stretch and fold infinitely many times, cutting each time the bent part, we will obtain a Cantor set of vertical noodles. 

KK: Yes. 

EL: Right. So as you were saying the ninths with these gaps, and this 1/3, I was thinking, huh, this sounds awfully familiar. 

KK: Yeah, yeah. 

EL: We’ll include a picture of the Cantor set in the show notes for people to look at. 

JRH: When we iterate forward, in the limit we will obtain a Cantor set of noodles. We can also iterate backwards. And what is that? We want to know for each point in the square, that is, for each flour particle of the dough in the mold, where it was before the cook stretched vertically and folded the dough the first time, where it came from. Now we recall that the forward iteration was to stretch in the vertical sense and fold it, so if we zoom back and put it backwards, we will obtain that the backward sense the cook has squeezed in the vertical sense and stretched in the horizontal sense and folded, okay? 

EL: Yes. 

JRH: Each time we iterate backwards, we stretch in the horizontal sense and fold it and put it in that sense. In this way, the left vertical rectangle is converted into the lower rectangle, the lower third rectangle. And the right side rectangle, the vertical rectangle, is converted into the upper third rectangle, and the bent part is cut. If we iterate backwards, now we will get in the first backward iteration four horizontal rectangles of width 1/9 and the gaps, and if we let the iterations go to infinity, we will obtain a Cantor set of horizontal noodles. 

When we iterate forward and consider only what’s left in the mold, we start with two horizontal rectangles and finish with two vertical rectangles. When we iterate backwards we start with two vertical rectangles and finish with two horizontal rectangles. Now we want to consider the particles that stay forever in the mold, that is, the points so that all of the forward iterates and all the backwards iterates stay in the square. This will be the product of two middle-thirds Cantor set. It will look more like grated cheese than noodles. 

KK: Right. 

JRH: This set will be called the invariant set. 

KK: Although they’re not pointwise fixed, they just stay inside the set. 

JRH: That’s right. They stay inside the square. In fact, not only will they be not fixed, they will have a chaotic behavior. That is what I want to tell you about. 

KK: Okay. 

JRH: This is one of the simplest models of an invertible map that is chaotic. So what is chaotic dynamics anyways? There is no universally accepted definition about that. But one that is more or less accepted is one that has three properties. These properties are that periodic points are dense, it is topologically mixing, and it has sensitivity to initial conditions. And let me explain a little bit about this. 

A periodic point is a particle of flour that has a trajectory that comes back exactly to the position where it started. This is a periodic point. What does it mean that they are dense? As close as you wish from any point you get one of these. 

Topologically mixing you can imagine that means that the dough gets completely mixed, so if you take any two small squares and iterate one of them, it will get completely mixed with the other one forever. From one iteration on, you will get dough from the first rectangle in the second rectangle, always. That means topologically mixing. 

I would like to focus on the sensitivity to initial conditions because this is the essence of chaos. 

EL: Yeah, that’s kind of what you think of for the idea of chaos. So yeah, can you talk a little about that? 

JRH: Yeah. This means that any two particles of flour, no matter how close they are, they will get uniformly separated by the dynamics. in fact, they will be 1/3 apart for some forward or backward iterate. Let me explain this because it is not difficult. Remember that we had the lower third rectangle? Call this lower third rectangle 0, and the upper third rectangle 1. Then we will see that for some forward or backward iterate, any two different particles will be in different horizontal rectangles. One will be in 1, and the other one will be in the 0 rectangle. How is that? If two particles are at different heights, than either they are already in different rectangles, so we are done, or else they are in the same rectangle. But if they are in the same rectangle, the cook stretches the vertical distance by 3. Every time they are in the same horizontal rectangle, their vertical distance is stretched by 3, so they cannot stay forever in the same rectangle unless they are at the same height. 

KK: Sure. 

JRH: If they are at different heights, they will get eventually separated. On the other hand, if they are in the same vertical rectangle but at different x-coordinates, if we iterate backwards, the cook will stretch the dough in the horizontal sense, so the horizontal distance will be tripled. Each time they are in the same vertical rectangle, they cannot be forever in the same vertical rectangle unless they are in the same, unless their horizontal distance is 0. But if they are in different positions, then either their horizontal distance is positive or the vertical distance is positive. So in some iterate, they will be 1/3 apart. Not only that, if they are in two different vertical rectangles, then in the next backwards iterate, they are in different horizontal rectangles. So we can state that any two different particles for some iterate will be in different horizontal rectangles, no matter how close they are. So that’s something I like very much because each particle is defined by its trajectory. 

EL: Right, so you can tell exactly what you are by where you’ve been. 

JRH: Yeah, two particles are defined by what they have done and what they will do. That allows something that is very interesting in this type of chaotic dynamics, which is symbolic dynamics. Now you know that any two points in some iterate will have distinct horizontal rectangles, so you can code any particle by its position in the horizontal rectangles. If one particle is in the beginning in the 0 rectangle, you will assign to them a sequence so that its zero position is 0, a double infinite sequence. If the first iterate is in the rectangle 1, then in the first position you will put a 1. In this way you can code any particle by a bi-infinite sequence of zeroes and ones. So in dynamics this is called conjugation. You can conjugate the horseshoe map with a sequence of bi-infinite sequences. This means that you can code the dynamics. Anything that happens in the set of bi-infinite sequences, happens in the horseshoe and vice versa. This is very interesting because you will find particles that describe any trajectory that you wish because you can write any sequence of zeroes and ones as you wish. You will have all Shakespeare coded in the horseshoe map, all of Donald Trump’s tweets will be there too. 

KK: Let’s hope not. Sad! 

JRH: Everything will be there. 

EL: History of the world, for better and worse. 

KK: What about Borges’s Library of Babel? It’s in there too, right? 

JRH: If you can code it with zeroes and ones, it’s there. 

EL: Yeah, that’s really cool. So where did you first run into this theorem? 

JRH: When I was a graduate student, I ran into chaos, and I first ran into a baby model of this, which is the tent map. A tent map is in the interval, and that was very cool. Unlike this model, it’s coded by one-sided sequences. And later on, I went to IMPA [Instituto de Matemática Pura e Aplicada] in Rio de Janeiro, and I learned that Smale, the author of this example, had produced this example while being at IMPA in Rio. 

KK: Right. 

JRH: It was cool. I learned a little more about dynamics, about hyperbolic dynamics, and in fact, now I’m working in partially hyperbolic dynamics, which is very much related to this, so that is why I like it so much. 

KK: Yeah, one of my colleagues spends a lot of time in Brazil, and he’s still studying the tent map. It’s remarkable, I mean, it’s such a simple model, and it’s remarkable what we still don’t know about it. And this is even more complicated, it’s a 2-d version. 

EL: So part of this show is asking our guests to pair their theorem with something. I have an idea of what you might have chosen to pair with your theorem, but can you tell us what you’ve chosen? 

JRH: Yeah, I like this sensitivity to initial conditions because you are defined by your trajectory. That’s pretty cool. For instance, if you consider humans as particles in a system, actually nowadays in Shenzhen, it is only me who was born in Argentina, lived in Uruguay, and lives in Shenzhen. 

EL: Oh wow. 

JRH: This is a city of 20 million people. But I am defined by my trajectory. And I’m sure any one of you are defined by your trajectory. If you look at a couple of things in your life, you will discover that you are the only person in the world who has done that. That is something I like. You’re defined, either by what you’ve done, or what you will do. 

EL: Your path in life. It’s interesting that you go there because when I was talking to Ami Radunskaya, who also chose a theorem in dynamics, she also talked about how her theorem related to this idea of your path in life, so that’s a fun idea. 

JRH: I like it. 

KK: Of course, I was thinking about taffy-pulling the whole time you were describing the horseshoe map. You’ve seen these machines that pull taffy, I think they’re patented, and everything’s getting mixed up. 

EL: Yeah. 

JRH: All of this mixing is what makes us unique. 

EL: So you can enjoy this theorem while pondering your life’s path and maybe over a bowl of noodles with some taffy for dessert. 

KK: This has been fun. I’d never really thought too much about the horseshoe map. I knew it as this classical example, and I always heard it was so complicated that Smale decided to give up on dynamics, and I’m sure that’s false. I know that’s false. He’s a brilliant man. 

JRH: Actually, he’s coming to a conference we’re organizing this year. 

EL: Oh, neat. 

KK: He’s still doing amazingly interesting stuff. I work in topological data analysis, and he’s been working in that area lately. He’s just a brilliant guy. The Fields Medal was not wasted on him, for sure. 

EL: Well thanks a lot for taking the time to talk to us. I really enjoyed talking with you. 

JRH: Thank you for inviting me. 
[outro]

 






 




 <a target="_blank" href="https://blogs.scientificamerican.com/roots-of-unity/jana-rodriguez-hertz-favorite-theorem/">Show notes (click here)</a>

<description>Mathematician Jana Rodriguez Hertz tells us about the Smale horseshoe map, symbolic dynamics, noodles, and all kinds of other fun stuff.</description>

In which we toss around some horseshoes.

Mathematician Jana Rodriguez Hertz tells us about the Smale horseshoe map, symbolic dynamics, noodles, and all kinds of other fun stuff.

Episode 21 - Jana Rodriguez Hertz

Episode 13 - Patrick Honner
 
 Kevin Knudson & Evelyn Lamb
 
 
 
 
 
 
 
 <a href="https://static1.squarespace.com/static/50e8abdce4b0404f376dbe62/t/5a88df5e71c10b5845ccefde/1518919596014/Honner+-+8%3A11%3A17%2C+9.26+PM.mp3" download="Episode 13 - Patrick Honner" class="download" target="_blank">Download</a>
 
 
 
 
 
 

 
 <input onclick="this.value=this.value=='Episode Transcript (click to expand)'?'Hide Transcript':'Episode Transcript (click to expand)';" type="button" class="spoilerbutton" value="Episode Transcript (click to expand)">
 
 
 
 
 
 
Evelyn Lamb: Welcome to My Favorite Theorem. I’m Evelyn Lamb, a freelance math and sci-ence writer in Salt Lake City. And this is my cohost. 

Kevin Knudson: Hi. I’m Kevin Knudson, professor of mathematics at the University of Florida. How are you doing, Evelyn? 

EL: Pretty good. It’s hot here, but it gets cool enough at night that it’s survivable. It’s not too bad. 

KK: It’s just hot here. It’s awful. 

EL: Yeah, there’s really something about that dry heat. I lived in Houston for a while. It’s differ-ent here. So on each episode we invite someone on to tell us about their favorite theorem, and today we’re delighted to have Patrick Honner. Hey! Can you tell us a little bit about yourself? 

Patrick Honner: Hi I’m happy to be here. Great to see you, Evelyn and Kevin. I’m in Brooklyn. It’s hot and muggy here. It’s never survivable in New York. I’ve got that going for me. I’m really excited to be here. I’m a high school math teacher. I teach at Brooklyn Technical High School. I studied math long ago, and I’m excited to talk about my favorite theorem today. 

KK: Cool. 

EL: Great. 

KK: So what do you have for us? 

PH: In thinking about the prompt of what my favorite theorem was, I guess I came to thinking about it from the perspective of a teacher, of course, because that’s what I’ve been doing for the last almost 20 years. So I was thinking about the kinds of theorems I like to teach, that are fun, that I think are really engaging, that are essential to the courses that i teach. A couple came to mind. I teach calculus occasionally, and I think the intermediate value theorem is probably my favorite theorem in calculus. I feel like the mean value theorem gets all the love in calculus. Eve-ryone thinks that’s the most important, but I really like the intermediate value theorem. I really love De Moivre’s theorem as a connection between complex numbers and geometry and alge-bra, and a little bit of group theory in there. But what really stuck out when thinking about what my favorite theorem is was Varignon’s theorem. 

KK: I had to look this up. 

PH: Well I think a lot of people, they know it when you show it to them, but they don’t know the name of it. That’s also part of why I like it. The name is sort of exotic sounding. It transports them to France somehow. 

EL: Nice. 

KK: Varignon’s theorem is a theorem of Euclidean geometry. It’s not that deep or powerful or exciting, but there’s just something about the way you can interact with it and play with it in class, and the way you prove it and the different directions it goes that really makes it one of my favorite theorems. 

KK: Now we’re intrigued. 

EL: Yeah. What is this theorem? 

PH: Imagine, so Varignon’s theorem is a theorem about quadrilaterals. If you imagine a quadrilateral in the plane, you’ve got the four sides. If you construct the midpoints of each of the four sides, and then connect them in a consistent orientation, so clockwise or counterclockwise, then you will get another quadrilateral. You start with the four sides, take the midpoints and connect them. Now you’ve got another quadrilateral. So if you start with a square, you can imagine those mid-points appearing, and you connect them, then that new quadrilateral would be a square. So you have a square inside of a square. This is a picture I think a lot of people can see. 

If you started with a rectangle and you constructed those midpoints, if the rectangle were a non-square rectangle, so longer than it was wide, you can think about it for a moment and maybe draw it, and you’d see a rhombus. Sort of a long, skinny rhombus, depending on the nature of the rectangle. Varignon’s theorem says that regardless of whatever quadrilateral you start with, the quadrilateral you form from those midpoints will be a parallelogram. And I just think that this is so cool. 

KK: It’s always a parallelogram. 

EL: Yeah, that’s really surprising. By every quadrilateral, do you mean only convex ones, is this for all quadrilaterals? 

PH: That’s part of the reason why it’s so much fun to play around with this theorem. It’s true for every quadrilateral, and in fact in some ways, it’s true even for things that aren’t quadrilaterals. In some ways it’s this continual intuition-breaking process with kids when you’re playing around with them. The way you can engage a class with this is you can just tell every student to draw their own quadrilateral and then perform this procedure where they construct the midpoints and connect them. Then you can tell them, ‘Look around. What do you see?’ The first thing the kids see is that everybody drew a square and everybody has a square inscribed, right? 

So this is a nice opportunity to confront kids about their mathematical prejudices. Like if you ask them to draw a quadrilateral, they draw a square. If you ask them to draw a triangle, they draw an equilateral triangle. But then there will always be a couple of kids who drew something a little bit more interesting. You can get kids thinking about what all of those things have in common and start looking for a conjecture. You can kind of push the and prod them to maybe do some different things. So maybe on the next interaction of this activity, we’ll get some rectangles or some arbitrary, some non-special quadrilaterals. 
Even after a couple rounds of this, you’ll still see that almost all the quadrilaterals drawn are convex. Then you can start pushing the kids to see if they understand that there’s another way to draw a quadrilateral that might pose a problem for Varignon’s theorem. It’s so cool that when you get to that convex one, kids never believe that it’ll still form a parallelogram. 

EL: In the non-convex one. 

PH: That’s right, the concave one, the non-convex. I always get the two words mixed up. Maybe that’s why the kids are so confused. Yeah, the kids will never believe in the non-convex case that it’ll still form a parallelogram. Wow, I can’t believe that. 

KK: It seems like, I looked this up, even if the thing isn’t really a quadrilateral, if you take four points in the plane and draw two triangles that meet where the lines cross, it still works, right? 

PH: Yeah. There’s yet another level to go with this. Now you’ve got the kids like, wait, so for concave, this works? It’s kind of mind-blowing. Then you can start messing around with their idea of what a quadrilateral actually is. If you show them, well, what if I drew a complex quadri-lateral. I don’t use that terminology right away, but just this idea of connecting the vertices in such a way that two sides appear to cross. It can’t possibly work there, can it? The kids don’t know what to think at this point. They think something weird is going around. Amazingly, even if the quadrilateral crosses itself like that, as long as it’s the non-degenerate case, the four points will still make a parallelogram. It’s really remarkable. 

KK: Is there a slick proof of this, or is it one of these crazy things, and you have to construct and construct and construct, and before you know it you’ve lost track of what you’re doing? 

PH: No, that’s another reason why this is such a great high school activity. The proof is really accessible. In fact there are several proofs.
But before we talk about my favorite proof of my favorite there, there’s another case, another level you can go with Varignon’s theorem. Often I’ll leave this with students as something to think about, a homework problem or something like that. Varignon’s theorem actually works even if the four points don’t form a quadrilateral, so if the four points aren’t coplanar, say. This process of connecting the midpoints will still form a parallelogram. It’s amazing just that the four points are coplanar. You wouldn’t necessarily expect that the four midpoints would be in the same plane if the four starting points aren’t in the same plane. Moreover, those four points form a parallelogram. It’s such an amazing thing. 

EL: What is your favorite proof, then? 

PH: My favorite proof of Varignon’s theorem is something that connects to a couple of key ideas that we routinely explore in high school geometry. The first is one of the first important theorems about triangles that we prove, that’s simple but has some power. It’s that if you connect the mid-points of two sides of a triangle, that line segment is parallel to the third side. And it’s also half the length. But the parallelism is important. 
 
The other idea, and I think this is one of the most important ideas that i try to emphasize with students across courses, is the idea of transitivity, of equality or congruence, or in this case par-allelism.
The nice proof of Varignon’s theorem is that you imagine all the quadrilaterals and midpoints. And you draw one diagonal. You just think about one diagonal. Now if you cover up half of the quadrilateral, you’ve got a triangle. The line segment connecting those two midpoints is parallel to that diagonal because that’s just that triangle theorem. Now if you cover up the other half of the quadrilateral, you have a second triangle. And that segment is parallel to the diagonal. So both of those line segments are parallel to that diagonal, and therefore by transitivity, they’re parallel to each other, and now you have that the two opposite sides are parallel. And the exact same argument works for the other sides using the other diagonal. 

KK: I like that. My first instinct would be to do some sort of vector analysis. You realize all the sides as vectors and then try to add them up and show that they’re parallel or something. 

PH: Yeah, and in some of the courses i teach, I do some work with vectors, and this is definitely something we do. We explore that proof using vectors, or coordinate geometry. Maybe later in the year we’ll do some work with coordinate geometry. We can prove it that way too. 

EL: Yeah, I think I would immediately go to coordinates. Of course, I would have assumed they were coplanar in the first place. If you tell me it’s a quadrilateral, yeah, it’s going to be there in the plane and not in 3-space. 

PH: I love coordinate geometry, and I definitely have that instinct to run to coordinates when I want to prove something. One of those things you have to be careful of in the high school class is making sure they understand all the assumptions that underly the use of coordinates, and un-derstanding the nature of an arbitrary figure. Going back to one of the first things I said, if you ask kids to draw a quadrilateral, they’re going to draw a square, or if you ask them to draw an arbitrary quadrilateral, they’re often going to draw a square or rectangle. If you ask them to draw an arbitrary quadrilateral in the plane, they might make assumptions about where those coordi-nates are likely to be. 

EL: Yeah. 

KK: Your students are lucky to have you. 

PH: That’s what I tell them! 

KK: Really, to give this much thought to something like this and show all these different per-spectives and how you might come at it in all these different ways, my high school geometry class, I mean I had a fine teacher, but we never saw anything with this kind of sophistication at all. 

PH: It’s fun. I would like to present it as if I sat around and thought deeply about it and had this really thoughtful approach to it, but it just kind of happened. I think, again, that’s why this is one of my favorite theorems. You can just put this in front of students and have them play and just run with this. It’ll just go in so many different directions. 

EL: So what have you chosen to pair with this theorem? What do you enjoy experiencing along with the glory of this theorem? 

PH: This was a tricky one. I feel like when I think of Varignon’s theorem, really focusing on the name, it really transports me to France. I feel like it’s a hearty stew, like boeuf Varignon or something like that. I think you need some crusty bread and a glass of red wine with Varignon’s theorem]. Not my students. 

EL: Crusty bread and grape juice for them. Yeah, I just got back from living in France for six months, and actually I didn’t have any boeuf bourgignon, or Varignon, while I was there, but I did enjoy quite a few things with crusty bread and a glass of red wine. I highly recommend it. 

KK: This has been great fun. 

PH: Yeah, I’ve enjoyed this. You seem to enjoy talking about this more than my students, so this was great for me. 

KK: It helps to be talking to a couple of mathematicians, yeah. 

EL: So, we like to let guests plug websites or anything. So would you like to tell people about your blog or any things you’re involved in that you’d like to share? 

PH: Yeah, sure. I blog, less frequently now than I used to, but still pretty regularly. I blog at mrhonner.com. You can generally find out about what I’m doing at my personal website, pat-rickhonner.com. I’m pretty active on Twitter, @mrhonner. 

KK: Lots of good stuff on Patrick’s blog, especially after the Regents exams. You have a lot to say. 

PH: Not everybody thinks it’s good stuff. I’m glad some people do. 

KK: I don’t live in New York. It’s fine with me. 

EL: Yeah, he has a series kind of taking apart some of the worst questions on the New York Regents exams for math. It can be a little frustrating. 

PH: We just wrapped up Regents season here. Let’s just say there are some posts in the works about what we’re facing. You know, I enjoy it. It always sparks interesting mathematical conver-sations. My goal is just to raise awareness about the toll of these tests and how sometimes it seems like not enough attention is given to making sure these tests are of high quality and are valid. 

KK: I don’t think it’s just a problem in New York, either. 

PH: It is not just a problem in New York. 

KK: Well thanks for joining us, Patrick. This was really great. I learned something today. 

EL: Yeah, me too. 

PH: It was my pleasure. Thanks for having me. Thanks for giving me an opportunity to think about my favorite theorem and come on and talk about it. And maybe Varignon’s theorem will appear in a couple more geometry classes next year because of it. 

KK: Let’s hope. 

EL: Yeah, I hope so. 

KK: Take care. 

PH: Thanks. Bye. 
 
 

 
 




 <a target="_blank" href="https://blogs.scientificamerican.com/roots-of-unity/patrick-honners-favorite-theorem/">Show notes (click here)</a>

<description>Our guest Patrick Honner tells us about Varignon's Theorem about the midpoints of quadrilaterals. Spoiler alert: if you connect them you always (!) get a parallelogram.</description>

In which midpoints lead to parallelograms...

Our guest Patrick Honner tells us about Varignon's Theorem about the midpoints of quadrilaterals. Spoiler alert: if you connect them you always (!) get a parallelogram.

Episode 13 - Patrick Honner

Episode 12 - Candice Price
 
 Kevin Knudson & Evelyn Lamb
 
 
 
 
 
 
 
 <a href="https://static1.squarespace.com/static/50e8abdce4b0404f376dbe62/t/5a760ec0ec212dda580cf481/1517686519089/Price+-+1%3A1%3A18%2C+2.18+PM.mp3" download="Episode 12 - Candice Price" class="download" target="_blank">Download</a>
 
 
 
 
 
 

 
 <input onclick="this.value=this.value=='Episode Transcript (click to expand)'?'Hide Transcript':'Episode Transcript (click to expand)';" type="button" class="spoilerbutton" value="Episode Transcript (click to expand)">
 
 
 
 
 
 Kevin Knudson: Welcome to My Favorite Theorem. I am Kevin Knudson, professor of mathematics at the University of Florida, and I am joined by my cohost. 

Evelyn Lamb: Hi. I’m Evelyn Lamb. I’m a math and science writer in Salt Lake City, Utah. 

KK: How’s it going? 

EL: Yeah, it’s going okay. It’s a bit smoky here from the fires in the rest of the west. A few in Utah, but I think we’re getting a lot from Montana and Oregon and Washington, too. You can’t see the mountains, which is a little sad. One of the nice things about living here. 

KK: Yeah. Well, Hurricane Irma is bearing down on Florida. I haven’t been to the grocery store yet, but apparently we’re out of water in town. So I might have waited a couple days too late. 

EL: Fill up those bathtubs, I guess. 

KK: I guess. I don’t know. I’m dubious. You know, I lived in Mississippi when Katrina happened, and the eye came right over where we lived, and we never even lost Direct TV. I’m trying not to be cavalier, but we’ll see. Fingers crossed. It’s going to be bad news in south Florida, for sure. I really hope everybody’s OK. 

EL: Yeah, definitely. 

KK: Anyway. 

EL: Fire, brimstone, and water recently. 

KK: Anyway, we’re not here to talk about that. We’re here to talk about math. Today we’re thrilled to have Candice Price with us. Candice, want to say hi? 

Candice Price: Hi everyone! 

KK: Tell us a little bit about yourself. 

CP: Sure. I’m currently an assistant professor of mathematics at the University of San Diego. I got my Ph.D. at the University of Iowa, and I study DNA topology, so knot theory applied to DNA, applied to biology. 

EL: So that’s knot with a ‘k.’ 

CP: Yeah, knot. 

KK: San Diego is a big switch from Iowa. 

CP: Yeah, it is. In fact, I had a stopover in New York and a stopover in Texas before getting here. All over. 

EL: You’ve really experienced a lot of different climates and types of people and places. 

CP: Yeah. American culture, really. 

KK: All right. You’ve told us. Evelyn and I know what your favorite theorem is, and I actually had to look this up, and I’m intrigued. So, Candice, what’s your favorite theorem? 

CP: Sure. My favorite theorem is actually John H. Conway’s basic theorem on rational tangles. It’s a really cool theorem. What Conway states, or shows, is that there’s a one-to-one correspondence between the extended rational numbers, so rational numbers and infinity, and what are known as rational tangles. What a rational tangle basically is, is you can take a 3-ball, or a ball, an open ball, and if you put strings inside the ball and attach the strings to the boundary of the ball, so they’re loose in there but fixed, and you add these twists to the strings inside, if you take a count to how many twists you’ve added in these different directions, maybe the direction of west and the direction of south, and if you just write down how many twists you’ve done, first going west and then going south, and then going west, going south, all of those, all the different combinations you can do, you can actually calculate a rational number, and that rational number is attributed to that tangle, to that picture, that three-dimensional object. 

It’s pretty cool because as you can guess, these tangles can get very complicated, but if I gave you a rational number, you could draw that tangle. And you can say that any tangle that has that same rational number, I should be able to just maneuver the strings inside the ball to look like the other tangles. So it’s actually pretty cool to say that something so complicated can just be denoted by fractions. 

EL: Yeah. So how did you encounter this theorem? I encountered it from John Conway at this IAS program for women in math one year, and I don’t think that’s where we met. I don’t remember if you were there. 

CP: I don’t think so. 

EL: Yeah, I remember he did this demonstration. And of course he’s a very engaging, funny speaker. So yeah, how did you encounter it? 

CP: It’s pretty cool, so he has this great video, the rational tangle dance. So it’s fun to show that. I started my graduate work as a master’s student at San Francisco State University, and I had learned a little bit about knot theory (with a ‘k’) as an undergrad. And so when I started my master’s I was introduced to Mariel Vazquez, who studies DNA topology. So she actually uses rational tangles in her research. That was the first time I had even heard that you could do math and biology together, which is a fascinating idea. She had introduced to me the idea of a rational tangle and showed me the theorem, and I read up on the proof, and it’s fascinating and amazing that those two are connected in that way, so that was the first time I saw it. 

KK: Since I hadn’t heard of this theorem before, I looked it up, and I found this really cool classroom activity to do with elementary school kids. You take four kids, and you hand them two ropes. You allow them to do twists, the students on one end of the ropes interchange, and there’s a rotation function. 

CP: Yeah. 

KK: And then when you’re done you get a rational number, and it leads students through these explorations of, well, what does a twist do to an integer? It adds one. The rotate is a -1/x kind of thing. 

CP: Right. 

KK: So I was immediately intrigued. This really would be fun. Not just for middle school kids, maybe my calculus students would like it. Maybe I could find a way to make it relevant to my undergrads. I thought, what great fun. 

CP: Yeah. I think it’s even a cool way to show students that with just a basic mathematical entity, fractions or rational numbers, you can perform higher mathematics. It’s pretty cool. 

KK: This sort of begs the question: are there non-rational tangles? There must be. 

CP: Yes there are! It categorizes these rational tangles, but there is not yet a categorization for non-rational tangles. There are two types. One is called prime, and one is called locally knotted. So the idea of locally knotted is that one of the strands just has a knot on it. A knot is exactly what you think about where you have a knot in your shoestring. Then prime, which is great, is all of the tangles that are not rational and not locally knotted. So it’s this space where we’ve dumped the rest of the tangles. 

KK: That’s sort of unfortunate. 

CP: Yeah, especially the choice of words. 

KK: You would think that the primes would be a subset, somehow, of the rational tangles. 

CP: You would hope. 

EL: So how do these rational tangles show up in DNA topology? 

CP: That’s a great question. So your DNA, you can think of as long, thin strings. That’s how I think about it. And it can wrap around itself, and in fact your DNA is naturally coiled around itself. That’s where that twisting action comes, so you have these two strings, and each string, we know, is a double helix. But I don’t care about the helical twist. I just care about how the DNA wraps around itself. These two strings can wind around, just based on packing issues, or a protein can come about and add these twists to it, and naturally how it just twists around. Visually, it looks like what is happening with rational tangles. Visually, the example that Kevin was mentioning, that we have the students with the two ropes, and they’re sort of twisting the ropes around, that’s what your DNA is doing. It turns into a great model, visually and topologically, of your DNA. 

KK: Very cool. 

CP: I like it. 

KK: Wait, where does infinity come from, which one is that? It’s the inverse of 0 somehow, so you rotate the 0 strand? 

CP: Yes, perfect. Very good. 

KK: So you change your point of view, like when I’m proving the mean value theorem in calculus, I just say, well, it’s Rolle’s theorem as Forrest Gump would look at it, how he tilts his head. 

CP: Right. I’m teaching calculus. I might have to use that. That’s good. I mean, hopefully they’ll know who Forrest Gump is. 

KK: Well, right. You’re sort of dating yourself. 

CP: That’s also a fun conversation to have with them. 

KK: Sure. So another fun conversation on this podcast is the pairing. We ask our guests to pair their theorem with something. What have you chosen to pair Conway’s theorem with? 

CP: So I thought a lot about this. So being in California, right, what I paired this with is a Neapolitan shake from In n Out burger. And the reason for that is, you’ve sort of taken these three different flavors, equally delicious on their own, right, rational numbers, topology, and DNA, and you put them together in this really beautiful, delicious shake. So the Neapolitan shake from In n Out burger is probably my favorite dessert, so for me, it’s a good pairing with Conway’s rational tangle theorem. 

KK: I’ve only eaten at In n Out once in my life, sadly, and I didn’t have that shake, but I’m trying to picture this. So they must not mix it up too hard. 

CP: They don’t, not too hard. So there’s a possibility of just getting strawberry, just getting vanilla, just getting chocolate, but then you can at some point get all three flavors together, and it’s pretty amazing. 

KK: So I can imagine if you mix it too much, it would just be, like, tan. It would just be this weird color. 

CP: Maybe not as delicious looking as it is tasting. 

KK: That’s an interesting idea. 

CP: It’s pretty cool. 

KK: So we also like to give our guests a chance to plug anything they’re working on. Talk about your blog, or anything going on. 

CP: Sure. I am always doing a lot of things. I am hoping I can take this time to plug, in February we have a website—we is myself, Shelby Wilson, Raegan Higgins, and Erica Graham—a website called Mathematically Gifted and Black where we showcase or spotlight every day a contemporary black mathematician and their contributions to mathematics, and we’re working on that now. We’ll have an article in the AMS Notices in February coming up. It’s up now so you can see it. We launched in February 2017. It’s a great website. We’re really proud of it. 

EL: Yeah. Last year it was a lot of fun to see who was going to be coming on the little calendar each time and read a little bit about their work. You guys did a really nice job with that. 

CP: Thanks. We’re very proud, and I think the AMS will put a couple of posters around the website as well. 

KK: Great. Well, Candice, thanks for joining us. 

CP: Thank you. 

KK: This has been good fun. I like learning new theorems. Thanks again. 

CP: Yeah, of course. Thank you. I enjoyed it. 

[outro]
 

 




 <a target="_blank" href="https://blogs.scientificamerican.com/roots-of-unity/candice-prices-favorite-theorem/">Show notes (click here)</a>

<description>Our guest Candice Price tells us about Conway's rational tangles and how they relate to the topology of DNA. Also, shakes from In 'N' Out.</description>

In which we get tangled up, rationally.

Our guest Candice Price tells us about Conway's rational tangles and how they relate to the topology of DNA. Also, shakes from In 'N' Out.

Episode 12 - Candice Price

Join us as we spend each episode talking with a mathematical professional about their favorite result. And since the best things in life come in pairs, find out what our guest thinks pairs best with their theorem.

My Favorite Theorem

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