Episode 91 - Karen Saxe

My Favorite Theorem

Kevin Knudson: Welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined, as always, by my fabulous co-host.
Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, trying to remember how to do this. It's been a minute since we've recorded one of these. We kind of went dormant for the winter.
KK: Yeah, a little bit, a little bit. Yeah. But Punxsutawney Phil told us — I don’t, what did he say? Let's pretend he said six more weeks of winter.
EL: I think he usually does. I don’t know.
KK: I mean, objectively, there are always six more weeks of winter. Like, the calendar says so, right?
EL: Yeah.
KK: Anyway, yeah.
EL: And, you know, he probably is pretty good at seeing shadows if he's a prey animal because he'd be used to seeing, like, a bird coming overhead.
KK: That’s an interesting question.
EL: Do birds eat groundhogs?
KK: That’s what I was going to wonder. I mean, like, eagles, maybe, but groundhogs are pretty large, right? I mean,
EL: Yeah. What eats groundhogs?
KK: Well, that's something to investigate later.
EL: Yeah.
KK: So it is Pi Day, right?
EL: It is! Well…
KK: We’re actually, we're recording this on Pi Day. When our listeners hear this, it won't be, but we're recording.
EL: And, I always have to put in a plug for my calendar.
KK: That’s right.
EL: The AMS math page-a-day calendar on which Pi Day does not occur on this day.
KK: That’s right.
EL: There are other Pi days on this calendar, none of which is this day, my little joke here. So you can find that in the AMS bookstore.
KK: Right. Are you Team Pi or Team Tau?
EL: I’m Team whichever one works for the calculation that you’re doing. It’s not that big a deal.
KK: That’s right. That's right. Okay. All right. Enough of us, enough of our useless banter, although we did discuss what's the ratio of banter to actual talk, right, that there's, there's like a perfect ratio. But we are pleased today to welcome Karen Saxe. Karen, why don't you introduce yourself and let us know all about you?
Karen Saxe: Hi, there, everybody. So first of all, happy Pi Day. If listeners know who I am, I was a professor at Macalester College for about for over 25 years. And then about seven years ago came to work at the American Mathematical Society, where I am very happy to be the director of the Government Relations Office. So I work in DC with Congress and federal agencies. And could quite a bit about this. I'm also happy to be here because it's Women's History Month. And it will be appropriate that it is Pi Day when you hear what my favorite theorem is.
KK: Okay, good to know. So, I'm curious to know more about this government relations business. So I mean, I know that the AMS does a lot of work on Capitol Hill, but maybe some of our listeners don’t. Can you explain a little more about what your office does?
KS: Yeah, so we do a lot of things. So first of all, we communicate — I sort of view the work of our office as going two ways. One is to communicate to Congress why mathematics is important to almost everything they make decisions about, you know, our national security, health care, you know, modeling epidemics, thinking, like you’re in Florida, thinking about how to model severe weather and things they care about, and then why they should fund fundamental research in mathematics and all sciences. And then also you know, how they make decisions about education. So we tell Congress, we give them advice and feedback on our view about what they should do in those realms. And then on the sort of flip side, I tell the AMS community, the whole math community about what Congress is doing and what's happening at the agencies like the NSF, and Department of Defense and Department of Energy, that that they might care about things, things that would affect their lives. So that’s sort of it in a nutshell. I spend a lot of time on the hill. I just came this morning, I went to a briefing put on by the National Science Board, which is the presidentially-appointed board that oversees the NSF. And they put out a congressionally mandated report every few years on the state of, it's called the indicators report. I'm sure I found it more interesting than everybody else, but it's pretty fascinating. You know, it covers everything from publications around the world, like which countries are are putting out the most science publications, what the collaborator network looks like around the world, and that to sort of US demographic information about education, you know, who's getting undergraduate degrees? Who's getting two year degrees? Who's getting PhDs, that that sort of thing. It covers a lot, actually. Pretty interesting.
KK: Yeah, yeah. All that in like two hours, right, and then it's over.
KS: Yeah, all that in two hours. And then they give you the big report that you can. And I've got them sitting in front of me. But given that this is a podcast, showing things doesn't work.
KK: Well, we do it all the time.
KS: Here’s one of the reports I picked up this morning. Actually, one really, so they're, you know, they're one thing. And you might end up cutting this, but one thing that's sort of fascinating to me is they always list barriers for getting into STEM degrees. And you know, there are things listed, like college accessibility, things that — and even going back. So like, you know, school kids who say they don't have science teachers in their schools, they don't have math teachers, but they've added to this list. “I can't support my family on a graduate student stipend.” So this is something.
EL: Yeah.
KK: That’s real.
KS: And we are, we've endorsed a bill in Congress that would look that would help to improve the financial stability, I guess, you would say, or the ability to be a grad student or a postdoc. So it's looking at stipends, it's looking at benefits, you know, leave time, all that sort of stuff, making it a job that you can choose to take when you're 23, and have a family to support and could make a hell of a lot more money doing something else with a math undergraduate degree.
EL: Yeah, and not see it as something where it's like, you're kind of putting off real life for a little longer, which I think maybe in the past was more of the model, like, oh, yeah, you'll have a real career later. But you know, in your mid-20s, you'll just keep being a student and not have kids or, you know, things, you know, not have parents to support or things like that.
KS: Exactly.
KK: Yeah. Okay. That's, that's good to know. Thank you for all that hard work you do, Karen. So but this is a math podcast.
KS: Right.
KK: So what’s your favorite theorem?
KS: Okay, so first, I'm going to tell you about the three theorems that I didn't choose.
KK: Cool.
EL: Great.
KS: So — I'm sure everybody goes through this — and thinking about my research, it would probably have to be the Riesz-Thorin interpolation theorem, which basically tells you that if you've got a bounded linear operator on two Lp spaces, then it's bounded on every Lp space in between those two values of p, so I used that all the time when I did research on that sort of thing. Then, but I was primarily a teacher of undergraduates, and kind of my two favorite theorems to teach are always Liouville’s theorem and, and then the uncountability of the real numbers.
EL: Yeah.
KS: And Liouville, they’re the one that says, you know, that there's a bounded — if you have a bounded entire function function, it's got to be constant. And the result is so stunning, and it gives a great proof of the fundamental theorem of algebra, that every non-constant polynomial has a root. So I always love teaching that. And then of course, like, Cantor’s diagonal argument about the real numbers, nothing beats that proof in terms of like, cool proof, in my opinion.
EL: Yeah. All-time great.
KS: Yeah, all-time great, right. And I think it's been mentioned on your podcast before. But what I picked was this theorem that says that if you have a given fixed perimeter, then the circle maximizes the two-dimensional shape you can make, so the isoperimetric theorem.
EL: Nice! And as you said, very appropriate for Pi Day.
KS: Yeah, which, I hadn’t even thought about that, which is sort of also embarrassing. But until we started acknowledging Pi Day, I hadn't thought about that. So another way to say it, or the way you might see it in a textbook, is if you have a perimeter P and an area A, then P2−4πA is greater than or equal to 0, with equality if and only if you have a circle. So this theorem has a very long, fascinating history. Lots of great applications. And for all those reasons, I love it. I love history.
KK: Yeah.
KS: I love math.
KK: Yeah. Do you have a favorite proof of this theorem?
KS: I do, actually. Yeah. Well, I didn't know you'd ask that. So there are a lot of proofs. And the one that I like, and this comes from being an analyst probably, is in the early 1900s. Hurwitz gave a proof using Fourier series. I love that proof. And proofs are quite old, going back thousands of years to the Greeks. And then in 1995, Peter Lax actually gave a new short calculus-based proof. But I like the Fourier series proof, just because I like Fourier series.
EL: Yeah, that's a topic that I wish I understood better. Somehow I kind of missed really, ever feeling like I've really got my teeth into Fourier series. Maybe that's a little embarrassing to admit on a math podcast.
KK: I don’t know. I took that one PDEs class as an undergrad and, like, that's where you see it, you know, doing the — whichever, the wave or the heat equation, whichever one it is — maybe both? I don't know. And then that’s it, that shows you how much I remember, too.
KS: Yeah. Good. So you're not gonna dare ask me to give you that proof or anything?
EL: Yeah, generally, a proof like that on audio is not the ideal medium.
KS: It doesn’t work.
EL: Actually, you brought up these ancient proofs. So yeah. Yeah, I guess how long has humanity known this fact, do you think, or do you know?
KS: So it's considered that the Greeks knew the proof. And then it was proved around 200 BCE. It even features in Virgil's version of the tale of Dido, Queen Dido.
EL: Oh, that’s right.
KS: So yeah, I think that was around 50 or 100 BCE, after the Greeks knew the theorem. So can I say what that story is?
EL: Yeah.
KK: Yeah, please.
KS: So she apparently fled her home after her brother had killed her husband. Okay, so we're already in an interesting phase. She somehow ended up on the north coast of Africa after that, and she was bargaining to get some land. And they told her, oddly, that that somehow she could get as much land as she could enclose with an oxhide.
KK: Okay.
KS: And so she took this oxide and cut it into very thin strips, and then enclosed an area, that was the largest she could conceive of, with the given per perimeter.
KK: Okay.
KS: So there's that. So it appeared, like, 2000 years ago, or more, and then you sort of we sort of jumped into the early 1800s when Steiner gave geometric proofs. But what's kind of fascinating is his proofs all assumed that a solution existed. And I haven't looked at these proofs, at least not in a long time. But then later in that century, Weierstrass is credited with giving a proof that, well, first, he proves that a solution does in fact exist. And he did use the calculus of variations to get this proof. So that's, that's sort of the story of the, of the theorem.
EL: Yeah, this actually — you know, we say the Greeks knew this, but I kind of wonder if this is one of those things that humans would kind of intuitively know, even if they're not in a framework where they have language about proving mathematical theorems, even if that's not an aspect of, of their culture, but it seems like you're trying to get into the mentality of like, what is really intuitive or innate about mathematics for humans? And I wonder if that, you know, we kind of would understand, well, if I took a square or something, I could sort of bow it out a little bit, and get a little more area with the same string.
KS: Actually, I mean, one reason I love this theorem is you can give string to kids, and I used to do this, like in elementary schools, and tell them make the biggest shape. And you have to tell them what closed is, no, you have to describe that the string has to come back to where it started. And they all come up with a circle. And this is, you know, second, third grade kids. So it is really intuitive. Yeah. So what it's meant by the Greeks knew this theorem is not 100 percent clear.
KK: Because they didn’t even use pi, right?
KS: And then actually, Evelyn to what you just said, you know, there's something that's quite interesting to me, which is that, you know, if you think about, you know, shapes of constant width, you know what I'm talking about?
EL: Yeah.
KS: So, if you take the fixed perimeter, there's an infinite number of these, the circle’s the largest one and those Reuleaux, I think that's how you say his name, those triangles are the ones of smallest area.
EL: Okay.
KS: And you were just kind of alluding to that, like take a triangle and go puff out the sides, or something.
KK: And you can push in.
KS: Yeah. Right. And you can do it for any regular polygon.
EL: Yeah. Well, British money has a couple of these that are I think heptagons, Reuleaux heptagons? Are they all called Reuleaux? Or just the triangles? I don't know.
KS: No, but you’re right about that, they do. And so it's kind of funny, I saw something that was talking about these points, like, what possessed them to make those points? And if you have a machine that has a hole size, and you know, it could fit a circle, it has a diameter, right, but it can also obviously fit one of these other shapes. Yeah. So that works. And I think you're right. It's a heptagon, heptagonal version of those.
EL: Yeah. The first time I went to the UK, this was, I think, the most exciting things on my trip to me, was these coins. Like, who thought to make these? And I actually, I remember, I wrote a blog post about it and discovered that it was a little hard to figure out if I had the rights to use a picture because all the images of these coins are like, technically property of the Crown.
KS: That’s funny.
EL: Abolish the monarchy, man.
KK: Her Majesty relented in the end?
EL: Yeah, so strange. I was like, well, I'm not gonna beg the queen for the right to post this on my math blog. So I don't remember what happened with that. Hopefully, I'm not opening myself to takedown.
KS: I think you’re probably okay.
EL: Hopefully the statute of limitations has run out on that. Anyway.
KK: I recently came across, I was going through an old notebook, and I found — I don't know why I tucked it in there — from the late 90s. I had one of these 10 Deutsche Mark notes that had Carl Gauss on it.
EL: Oh, nice.
KK: And so I put it on Instagram. And I'm now starting to worry. Wait a minute. Will the German government come after me? Although it's not really legal tender anymore.
EL: Yeah, the pre-2000, whenever they went to the Euro, government.
KK: It was pre-Euro. Yeah, I think I’m safe too.
EL: But anyway, getting back to the math, Karen. So, has this been a favorite of yours for a long time? I guess to me, this is one that I don't think the first time I saw it, I would have been super impressed by it. So what was your experience? What's your history with this theorem?
KS: Right. So like, why did I decide I liked it? Because yeah, it's sort of like, okay, I mean, it's appealing, because everybody can understand it, it’s very intuitive. It's got this, the proof has this interesting history. But why I like it is because you probably know that I'm pretty engaged with congressional redistricting. And when they do measures of compactness of districts, this is the theorem that kind of motivates all their measures.
KK: The Polsby-Popper metric, right?
KS: Yes, exactly. And so you take the Polsby-Popper measure, which was come up in 1991. So like, different states, should I say something about redistricting?
KK: Sure, yeah.
KS: So yeah, I mean, just like the very brief thing is every 10 years, we have to do the census. This is mandated in our Constitution, for the purposes of reapportionment of the House of Representative seats to the state so then after the census is done the seats, which we now have 435 of them, they're doled out to the states. And how that's done is a whole nother you know, interesting math problem, more interesting, probably. But then once the states get their number of seats, like how many in Florida?
KK: We’re up to 27? [Editor’s note: It’s actually 28.]
KS: So let's pretend there's 27 for a minute.
KK: I think that’s right. [Ron Howard voice: It wasn’t.]
KS: Okay. Then, you know, the Florida Legislature, probably, I don't know who does it in Florida, but somebody.
KK: Let’s not talk about that.
KS: Yeah, let’s not talk about that. Whoever’s in charge has to carve up the state geographically into 27 districts, one for each representative, and how they do that geographic carving up is extremely complicated. And to answer the question, “Has this been gerrymandered?” there are certain measures of what's called compactness, and this is like a whole nother thing I could talk for hours on. And compactness sort of measures the lack of convexity, sort of, so like, are there long skinny arms going out? And this is where obviously, like a podcast is, is not the best. But in any case, you know, are there long skinny arms going out, or does the thing look like a circle? So the Polsby-Popper measure tells you how close to a circle, or a disk because it's filled in, but in any case, your district is. Well, that's kind of weird, because if you think about tiling any state with circles, it’s just not going to happen.
EL: Right.
KS: Yeah. So just to sort of fetishize circles is bizarre. But I guess, like, what are your other options? Well, there are lots of other options. But the Polsby-Popper is the most common. There's a handful of states that require specific compactness measures in their process, and many other states that require compactness, but they don't specify the actual measure. In any case, the Polsby-Popper is the most common. And the other common measure is called the Reock measure, and that also fetishizes circles. It's a similar type thing. So with the Polsby-Popper, it's kind of interesting, because they they first published it in a law journal in 1991, in this context for redistricting, but it has actually been mentioned, as far back as the late ‘20s. And can I read you a funny a funny opening line?
EL: Yeah, sure.
KS: So the it first appeared, as far as I know, in a 1927 paper in the Journal of Paleontology. Okay. And how's this for the start of a paper? In quotes: “How round is a rock? This is a question that the geologist is often forced to ask himself.” Okay.
EL: Nice.
KS: So that's a great opening sentence. And then it kind of carries on: “when he wishes to consider the amount of erosion that a stone has received.” And then the paper is actually about measuring the roundness of grains of sand.
EL: Oh, cool.
KS: So there's a lot to say here that the paper is filled with hilarious hand drawings, you know, but also, of course, that geologists seem to be male is another observation.
EL: Yeah, well, and the grammar rules of the time.
KS: Yeah, exactly. But even just this past January, I ran into a paper that was published, and uses this to measure the aggressiveness. It's in, like, a cancer journal. I can't remember which one. And I wrote it down, but of course, what do you know, I can't see it. Anyways — oh, Cancer Medicine is the name of the journal — and it used the Polsby-Popper measure to measure aggressiveness of tumor growth. So you know, it has a life.
EL: That's so so interesting. When you said Journal of Paleontology, I was just like, how is that going to come up in paleontology? But what do you say? Yeah, how round is a rock? It's like, yeah, you do need to measure that. I actually, just the other day watched this interesting video about sand grains and like, certain beaches, or, and certain dunes have different acoustical properties. Due to, like, if they've got a lot of the same sized sand grains and if they pack really well, or if they don't, sometimes there can be the squeaking effect, like when you walk on it, or in a dune, like when there's wind, there can be these like deep, deep resonances, like almost a thunder sound that happens.
KS: Oh, that is interesting.
EL: And this this video went and looked under the microscope at the sand on these different beaches, and kind of showed how some of them packed together better or worse, and some of them are more uniform. So they might secretly be using that metric.
KK: They might.
KS: That’s fascinating. I mean, I heard I've heard that squeaky sound on beaches.
EL: I never have I'm not a huge beach person. So I guess, yeah, but I'm curious about going to one of these beaches someday now.
KS: Yeah. And when you said that I was thinking of the packing, like how they pack, but that would have to do with their shape, and their size. Well, I don't know.
KK: So this is a sphere packing question now. And it's yes.
EL: Or a “how sphere-y is your sphere”-packing question.
KS: How spherey is your sphere?
EL: Not quite as catchy.
KK: Right. So the other part of this podcast is we like to ask our guests to pair their theorem with something, so what pairs well with the isoperimetric inequality?
KS: So naturally, you know, a mathematician would ask, are there analogs in higher dimensions? Right? And then back to how spherey is your sphere, so I play tennis quite a bit. So I'm going to pair it with tennis.
EL: Excellent.
KS: The shape of the ball abides by the theorem.
KK: Yes. Right.
KS: And works for so many reasons.
EL: Yeah. Well, and you are not the the first My Favorite Theorem guest to pick tennis, actually.
KK: That’s right. Yeah.
EL: Yeah, we've had Dr. Curto.
KK: Carina.
EL: Yeah. Carina Curto, paired paired hers with tennis. It was it was about linear algebra. That's right. Yeah. Yeah. Hers was about how this thing kind of goes back and forth. When you're doing this thing in linear algebra. So you picked different aspects of tennis to pair with your theorem.
KK: Yep. Do you play much do you, you play, you play a lot?
KS: I play — it’s embarrassing to put on a very well listened-to podcast — that I do play a lot, because I don't know how good I am.
KK: That doesn’t matter.
KS: But I play a couple times a week.
KK: I used to play quite a bit. So as a teenager, certainly. And then in my 30s I played a lot. I played a little league tennis. This is when I lived in Mississippi. And actually, my team won the state championship two years running at our level.
KS: Oh, wow.
KK: But I'm not any good. This was like, you know, I'm like a 3.5. Like, you know, just a very intermediate sort of player.
KS: Yeah, that's what I am.
KK: Yeah, my shoulder won't take it anymore.
KS: I still, I feel lucky. Because physically, I can do it. Right now. I'm in a 40+ league, and that's good. But next season, whatever you call it, or next season, I guess, I'm in an 18+ League, and I've done this before. It means the other players are allowed to be as young as 18. It’s a little humbling, even if we can serve, you know, we have the technical skills, like they’re, you know, like the shots you use in the 40s, like, lobbing is not a good strategy in 18+ because they can run.
KK: Back when I was in my 30s and played, I played a lot of singles still, and I could still do it. But when I would come up against the 20-year-olds, it'd be a lot harder. But then I also learned, I used to play a lot of doubles with with these guys in their 70s. And they destroyed me every time. They were just —because they knew where to be. They had such skill and good instincts for where the ball was going to be. It was humbling in that way.
KS: Yeah, it's it's fun. And I prefer playing doubles these days. It's just more fun and different strategy.
KK: Yeah, and less court to cover. That helps.
KS: Less court to cover. And it’s more social. It's a lot of fun.
KK: Yeah, so you haven't succumbed to pickleball, have you?
KS: I played once, on my 60th birthday. Because no one would play tennis with me. And I got invited to a pickleball thing. And I was like, Okay, we're gonna do it. And, you know, it was fun, but I haven't really. It’s a challenge in Minnesota playing pickleball because it's so windy and the balls are so light, and it’s like whiffle ball.
KK: That’s what they are, basically.
KS: The ball kind of blows around all over the place. So yeah, I haven't I succumbed to doing that. In DC I'm lucky to have enough people to play tennis with. There's a lot of them.
KK: Cool. All right.
EL: Yeah. Great pairing.
KK: Yeah, yeah. So we also give our guests a chance to plug anything they're working on. You sort of already did that. I mean, you're doing all the work. Anything else you want to pitch?
KS: I mean, back to what I do, one reason I love this new job is I get to go in and and make connections to any Congressperson. You know, they have their own interests motivated by their own history, their own life, their own constituents. And this can be — there are obvious things we think about, like people, congressional members who are interested in their electric grid, or ocean modeling for the Hawaii delegation. But it's fun. And it's a fun challenge to think of things. So there's one newish member who was a truck driver before he was elected to Congress. And, we went in and their office was like, we can't make a connection to math. And we started talking about logistics, you know, truck routing. And it was great. It turned into a great conversation where they hadn't really thought about that. So this is what I really love about my job, trying to connect math to anything they’ve got. What they’re interested in, I'm gonna I'm gonna try to connect math, and there are very few issues that that can't be connected.
EL: Yeah, well I actually have a question, something that our listeners might be interested in is like if a mathematician is listening to this, and wonders, how can I get more connected to what's happening? How can I understand what math and science, you know, representatives do on the hill? Is there a newsletter or a website or something that you have that they could look at? And, you know, maybe find ways to get more involved? Or at least more informed?
KS: Yeah, definitely. So first of all, I used to write a blog, but I don't do that anymore for the AMS. The AMS Government Relations page — so my office is the Office of Government Relations. And I believe if you search, AMS government relations, you'll get to my webpage, you know, the one that I call mine, and you'll see a lot of different things there. There are ways to get engaged. We offer felt three fellowships. Two are for graduate students, one is for a person with a PhD in mathematics to come and to come here physically and do things. One is a boot camp for graduate students, a three-day graduate boot camp to come learn about legislative policy. And then the the biggest one is a year long fellowship and working in Congress. I do hill visits with people. And you know, I'm pretty willing to bring almost any mathematician to the hill, and that can be virtual these days. So we have volunteer members through our committee work who fly in and do these hill visits. We did this last Wednesday, we had about 25 AFS, volunteers fly in, and that was a fantastic day. But I can do them virtually. I've done them with big groups of grad students from departments, and people can email me if they want. And I think you guys have my email.
EL: Yeah. Thanks.
KS: So those are the big ways. And then for AMS members who are a little more advanced in their careers, you can volunteer for AMS committees. And there's the Committee on Science Policy, which really focuses on this one. And then I'm also in charge of the Human Rights Committee for the AMS, which can be of interest to a lot of people.
KK: Sure.
EL: For sure.
KK: Lots going on there.
KS: Yeah, lots going on.
KK: Well, Karen, this is terrific. Thanks so much for taking time out of your day, and thanks for joining us.
KS: Thank you.
[outro]

In this episode, we enjoyed talking with Karen Saxe about her work as the director of the American Mathematical Society's Office of Government Relations and her favorite theorem, the isoperimetric theorem. Below are a few links you might find relevant as you listen:
Saxe's website and the homepage of the AMS Office of Government Relations

A survey of the history of the isoperimetric problem by Richard Tapia
The 1995 proof by Peter Lax
Evelyn's blog post about 50 pence coins and other British objects of constant width
The Polsby-Popper test to measure gerrymandering
A public lecture by mathematician Moon Duchin about mathematics and redistricting
The 1927 Journal of Paleontology article that first uses the Polsby-Popper metric (though not with that name)
An Atomic Frontier video about squeaky sand
Our episode with fellow tennis-enjoyer Carina Curto

The 10 Deutsche Mark note