Kevin Knudson: Welcome to My Favorite Theorem, a podcast about math and…I don't even know what it's going to be today. We'll find out. I'm one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida. Here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. So yeah, how are you today?

KK: I’ve had a busy week. Both of my PhD students defended on Monday.

EL: Wow, congratulations.

KK: Yeah, and so, through some weird quirk of my career, these are my first two PhD students. And it was a nice time, slightly nerve-wracking here and there. But everybody went through, everything's good.

EL: Great.

KK: So we have two new professors out there. Well, one guy's going to go into industry. But yeah, how about you?

EL: I’m actually—once we're done with this, I need to go pack for a trip I'm leaving on today. I'm teaching a math writing workshop at Ohio State.

KK: Right. I saw that, yeah.

EL: I mean, if it goes well, then we'll leave this part in the thing. And if it doesn't, no one else will know. But yeah, I'm looking forward to it.

KK: Good. Well, Ellen and I are going to Seattle this weekend.

EL: Fun.

KK: She got invited to be on a panel at the Bainbridge Island Art Museum. And I thought. “I'm going along” because I like Seattle.

EL: Yeah, it's beautiful there.

KK: Love it there. Anyway, enough about us. Today, we are pleased to welcome Moon Duchin to the show. Moon, why don’t you introduce yourself?

Moon Duchin: Hi, I'm Moon. I am a mathematician at Tufts University, where I'm also affiliated with the College of Civic Life. It’s a cool thing Tufts has that not everybody has. And in math, my specialties are geometric group theory, topology (especially low-dimensional topology), and dynamics.

KK Very cool.

EL: Yeah, and so how does this civil life thing work?

MD: Civic life, yeah.

EL: Sorry.

MD: So that's sort of because in the last couple years, I've become really interested in politics and in applications—I think of it as applications of math to civil rights. So that's that's sort of mathematics engaging with civics, it’s kind of how we do government. So that's become a pretty strong locus of my energy in the last couple years.

EL: Yeah.

KK: And I'll vouch for Moon's work here. I mean, I've gone to a couple of the workshops that she's put together. Big one at Tufts in 2017, I guess it was, and then last December, this meeting at Radcliffe. Really cool stuff. Really important work. And I've gotten interested in it too. And let's hope we can begin to turn some tides here. But anyway, enough about that. So, Moon, what's your favorite theorem?

MD: Alright, so I want to tell you about what I think is a really beautiful theorem that is known to some as Gromov’s gap.

KK: Okay.

EL: Which also sounds like it could be the name of a mountain pass in the Urals or something.

MD: I was thinking it sounds like it could be, you know, in there with the Mines of Moria in Middle Earth.

EL: Oh, yeah, that too.

MD: Just make sure you toss the dwarf across the gap. Right, it does sound like that. But of course, it's Mischa Gromov, who is the very prolific Russian-French mathematician who works in all kinds of geometry, differential geometry, groups, and so on.

So what the theorem is about is, what kinds of shapes can you see in groups? So let me set that up a little with—you know, let me set the stage, and then I'll tell you the result.

EL: Okay.

MD: So here's the setting. Suppose you want to understand—the central objects in geometric group theory are, wel,l groups. So what are groups? Of course, those are sets where you can do an operation. So you can think of that as addition, or multiplication, it's just some sort of composition that tells you how to put elements together to get another element. And geometric group theory is the idea that you can get a handle on the way groups work—they’re algebraic objects, but you can study them in terms of shape, geometrically. So there are two basic ways to do that. Either you can look at those spaces that they act on, in other words, spaces where that group tells you how to move around. Or you can look at the group itself as a network, and then try to understand the shape of that network. So let's stick with that second point of view for a moment. So that says, you know, the group has lots of elements and instructions for how to put things together to move around. So I like to think of the network—a really good way to wrap your mind around that is to think about chess pieces. So if I have a chessboard, and I pick a piece—maybe I pick the queen, maybe I pick the knight—there are instructions for how it can move. And then imagine the network where you connect two squares if your piece can get between them in one step. Right?

KK: Okay.

MD: So, of course that's going to make a different network for a knight than it would for a queen and so on, right?

EL: Yeah.

MD: Okay. So that's how to visualize a group, especially an infinite—that works particularly well for infinite groups. That's how to visualize a group as a bunch of points and a bunch of edges. So it's some big graph or network. And then GGT, geometric group theory, says, “What's the shape of that network?” Especially when you view it from a distance, does it look flat? Does it look negatively curved, like a saddle surface? Or does it kind of curve around on itself like a sphere? You know, what's the shape of the group?

And actually, just a cool observation from, you know, a hundred plus years ago, an observation of Felix Klein is that actually the two points of view—the spaces that the group acts on or the group itself—those really are telling the same story. So the shape of the space is about the same as the shape of the group. That's become codified as kind of a fundamental observation in GGT. Okay, great. So that's the space I want to think about. What is the shape—what are the possible shapes of groups? Okay, and that's where Gromov kicks in. So the theorem is about the relationship of area to perimeter. And here's what I mean by that.

Form a loop in your space, in your network. And here, a loop just means you start at a point, you wander around with a path, and you end up back where you started. Okay? And then look at the efficient ways to fill that in with area. So visualize that, like, first you have an outline, and then you try to fill it in with maybe some sort of potato chip-y surface that kind of interpolates around that boundary. Okay, so the question is, if you look at shapes that have the extremal relationship of area to perimeter, then what is the relationship of area to parameter?

So let's do that in Euclidean space first, because it's really familiar. So we know that the extremal shapes there are circles, and you fill those in with discs. And the relationship is that area looks about like perimeter squared, right?

KK: Right.

MD: Okay, great. So now here's the theorem, then. Get ready for it. I love this theorem. In groups, you can find groups where area looks like perimeter to the K power. It can look like perimeter to the 1, it can look like perimeter to the 1, or 3, or 4, and so on. You can build designer groups with any of those exponents. But furthermore, you can also get rational exponents. You can get pretty much any rational exponent you want. You can get 113/5, you can get, you know, 33/10. Pick your favorite exponent, and you can do it.

EL: Can you get less than one?

MD: Well, let's come back to that.

EL: Okay. Sorry.

MD: So let me state Gromov’s theorem in this level of generality. So here's the theorem. You can get pretty much any exponent that you want, as long as it's not between 1 and 2.

KK: Wow.

EL: Oh.

MD: Isn’t that cool? That's Gromov’s gap.

KK: Okay.

EL: Okay.

MD: So there's this wasteland between 1 and 2 that's unachievable.

KK: Wow.

MD: Yeah. And then you can, see—past 2, you can see anything. Um, it actually turns out, it's not just rationals. You can see lots of other kinds of algebraic numbers too.

KK: Sure.

MD: And the closure up there is everything from two to infinity! But nothing between one and two. It's a gap.

EL: Oh, wow. That's so cool!

MD: That’s neat, right? Evelyn, to answer your question under one turns out not to really be well defined for reasons we could talk about. But yeah.

KK: This is remarkable. This sounds like something Gromov would prove, right? I mean, just these weird theorems out of nowhere. I mean—how could that be true? And then there it is. Yeah.

MD: Or that Gromov would state and leave other people to prove.

KK: That—yeah, that's really more accurate. Yeah. So. Okay, so you can't get area to the—I mean, perimeter to the 3/2. I mean, that's, that's really…Okay. Is there any intuition for why you can't get things between one and two?

MD: Yeah, there kind of is, and it's beautiful. It is that the stuff that sits at the exponent 1, in other words, where area is proportional, the perimeter is just really qualitatively different from everything else. Hence the gulf. And what is that stuff? That is hyperbolic groups. So this comes back to Evelyn's wheelhouse, I believe.

EL: It’s been a while since I thought in a research way about this, but yes, vaguely at the distance of my memory.

MD: Let me refresh your memory. Yes, so negatively curved things, things that are saddled shaped, those are the ones where area is proportional to parameter. And everything else is just in a different regime. And that's really what this theorem is telling you.

So that's one beautiful point of view, and kind of intuition, that there's this qualitative difference happening there. But there's something—there’s so many things I love about this theorem. It's just the gateway to lots of beautiful math. But one of the things I love about this theorem is that it fails in higher dimensions, which is really neat. So if you, instead of filling a loop with area, if you were to fill a shell with volume, there would be no gap.

EL: Oh.

MD: Cool, right?

EL: So this is, like, the right way to measure it if you want to find this difference in how these groups behave.

MDL Absolutely. And, you know, another way to say it, is this is an alternative definition of hyperbolic group from the usual one. It's like, the right way to pick out these special groups from everything else is specifically to look at filling loops.

KK: Right. And I might be wrong here, but aren't most groups hyperbolic? Is that?

MD: Yeah, so that's definitely the kind of religious philosophy that’s espoused. But you know, to talk about most groups usually the way people do that is they talk about random constructions of groups. And a lot of that is pretty sensitive to the way you set up what random means. But yeah, that's definitely the, kind of, slogan that you hear a lot in geometric group theory, is that hyperbolic groups are special, but they're also generic.

EL: Yeah.

KK: So are there explicit constructions of groups with say, exponent 33/10, to pick an example?

MD: Yes, there are. Yeah. And actually, if you're going to end up writing this up, I can send you some links to beautiful papers.

EL: Yeah, yeah. But there’s, like, a recipe, kind of, where you're like, “Oh, I like this exponent. I can cook up this group.”

MD: Yeah. And that's why I kind of call them designer groups.

EL: Right, right. Yeah. Your bespoke groups here.

MD: Yeah, there are constructions that do these things.

KK: That’s remarkable. So I was going to guess that your favorite term was the isoperimetric inequality. But I guess this kind of is, right?

MD: I mean, exactly. Right? So the isometric inequality is all about asking, what is the extremal relationship of area to parimeter? And so this is exactly that, but it's in the setting of groups.

KK: Yeah, yeah.

EL: So how did you first come across this theorem?

MD: Well, I guess, in—when you're in the areas of geometric topology, geometric group theory, there's this one book that we sometimes call the Bible—here I'm leaning on this religious metaphor again—which is this this great book by Bridson and Haefliger called Metric Spaces of Non-Positive Curvature. And it really does feel like a Bible. It's this fat volume, you always want it around, you flip to the stuff you need, you don't really read it cover to cover.

KK: Just like the Bible. Yeah,

MD: Exactly. Great. And that's certainly where I first saw it proved. But, yeah, I mean, the ideas that circulate around this theorem are really the fundamental ideas in GGT.

KK: Okay, great. Does this come up in your own work a lot? Do you use this for things you do? Or is this just like, something that you love, you know, for its own sake?

MD: Yeah, no, it does come up in my own work in a couple of ways. But one is I got interested in the relationship between curvature—curvature in the various senses that come from classical geometry—I got interested in the relationship between that and other notions of shape in networks. So this theorem takes you right there. And so for instance, I have a paper of a theorem with Lelièvre, and Mooney, where we look at something really similar, which is, we call it sprawl. It's how spread out do you get when you start at a point and you look at all the different positions you can get to within a certain distance. So you look at a kind of ball around the point. And then you ask how far apart are the points in that ball from each other? So that's actually a pretty fun question. And it turns out, here's another one of these theorems where hyperbolic stuff, there's just a gap between that and everything else.

KK: Right.

MD: So let’s follow that through for a minute. So suppose you start at a base point, and you take the ball of radius R around that base point. And then you ask, “How far apart are the points in that ball from each other?” Well, of course, by the triangle inequality, the farthest apart that could possibly be is 2R because can connect them through the center to each other, right, 2R. Okay, so then you could ask, “Hm, I wonder if there's a space that’s so sprawling, so spread out, so much like, you know, Houston, right, so sprawling that the average distance is actually the maximum?”

EL: Yeah.

MD: Right. What if the average distance between two points is actually equal to 2R?

And that’s, so that's something that we proved. We proved that when you're negatively curved, and you have, you know, a few other mild conditions, basically—but certainly true for negatively curved groups, just like the setting of Gromov’s theorem—so for negatively curved groups, the average is the maximum. You’re as sprawling as you can be. Yeah, isn't that neat? So that's very much in the vein of this kind of result.

KK: Oh, that's very cool. All right.

EL: Yeah. Kind of like the SNCF metric, also, where you have to go to Paris to go anywhere else. Slightly different, but still, that you basically you have to go in to the center to get to the other side.

MD: It’s exactly the same collection of ideas. And I'm just back from Europe, where I can attest that it's really true. You want to get from point to point on the periphery of France, you’d better be going through Paris if you want to do it fast. But yeah, it’s precisely the same idea, right? So the average distance between points on the periphery of France will be: get to Paris and get to the other point. So there's a max there that's also realized.

KK: All right, so France is hyperbolic.

MD: France is hyperbolic. Yup, in terms of travel time.

EL: Very appropriate. It’s such a great country. Why wouldn't it be hyperbolic?

KK: All right, so the other fun thing on this podcast is we ask our guests to pair their theorem with something. So what pairs well with Gromov’s gap theorem?

MD: So I'm actually going to claim that it pairs beautifully with politics. Right? True to form, true to form.

EL: Okay.

KK: Right, yeah, sure.

MD: All right, so let me try and make that connection. So, well, I got really interested in the last few years in gerrymandering in voting districts. And classically, one of the ways that we know that a district is problematic is exactly this same way, that it's built very inefficiently. It has too much perimeter, it has too much boundary, a long, wiggly, winding boundary without enclosing very much area. That's been a longstanding measurement of kind of the fairness or the reasonableness of the district. So I got interested in that through likes of this kind of network curvature stuff, with the idea that maybe the problem is in the relationship between area and perimeter.

And so what does that make you want to do, if you're me? It makes you want to take a state and look at it as a network. And you can do that with census data. You sort of take the little chunks of census geography and connect the ones that are next to each other and presto! You have a network. And it's a pretty big network, but it's finite.

KK: Yeah.

MD: So Pennsylvania's got about 9000 precincts. So you can make a graph out of that. But it's got a whole lot more census blocks. Virginia—we were just looking at Virginia recently—300,000 census blocks. So that's a pretty big network, but you know, still super duper finite, right?

EL: Yeah.

MD: And so you can sort of ask the same question, what's the shape of that network? And does that—you know, maybe the idea is, if the network itself, which is neutral, no one's doing any gerrymandering, that's just where the people are.

EL: Yeah.

MD: If the network network itself is is negatively curved, in some sense, then maybe that explains large perimeters in a reason that isn't due to political malfeasance, you know?

EL: Right.

MD: So I think this is a way of thinking about shape and possibility that lends itself to lots of problems. But I like to pair everything with politics these days.

EL: Yeah, well, I really I think—so I went to your talk at the joint mass meetings a couple of years ago, which I know you've probably given similar talks, with talking about gerrymandering, I think it's really important for people not to take too simplistic a view and just say, “Oh, here's a weird shape.” And you were, you did a really great job of showing, like, there are sometimes good reasons for weird shapes. Obvious things like theres a river here and people end up grouping like this around the river for this reason. But there are a lot of different reasons for this. And if we want to talk about this in a way that can actually be productive, we have to be very nuanced about this and understand all of those subtleties, which are mixing the math—we can't just divorce it from the world. We we mix the math from, you know, the underlying civil rights and, you know, politics, historic inequalities in different groups and things like that.

MD: Yeah, absolutely. That's definitely the point of view that I've been preaching, to stick with the religious metaphor. It’s the one that says, if you want to understand what should be out of bounds, because it's unreasonable when it comes to redistricting, first you have to understand the lay of the land. You have to spec out the landscape of what's possible. And like you're saying, you know, that landscape can have lots of built and structure that districting has to respect. So, yeah, you should really—that could be physical geography, like you mentioned rivers—but it could also be human geography. People distribute themselves in very particular ways. And districting isn't done with respect to like, imaginary people, it’s done with respect to the real, actual people and where they live.

EL: Yeah.

MD: And that's why I really, you know, I think more and more that some of those same tools that we use to study the networks of infinite groups, we can bring those to bear to study the large finite networks of people and how they live and how we want to divide them up.

EL: Yeah, that's, that's a nice pairing, maybe one of the weightier pairings we’ve had.

KK: Yeah, right.

MD: It was either that or a poem, but. I was thinking Gromov’s gap, maybe I could pair that with The Wasteland.

EL: Oh.

MD: Because you can’t get in the wasteland between exponents one and two. Nah, let's go politics.

EL: Well, I've tried to read that poem a few times, and I always feel like I need someone to hold my hand and, like, point everything out to me. It's like, I know there's something there but I haven't quite grabbed on to it yet.

MD: Yeah. Poetry is like math, better with a tour guide.

EL: Yes.

KK: Well, we also like to give our guests a chance to plug things. You want to tell everyone where they can find you online and and maybe about the MGGG?

MD: Sure, yeah, absolutely. So I co-lead a working group called MGGG, the Metric Geometry and Gerrymandering Group, together with a computer scientist named Justin Solomon, who's over at MIT. You can visit us online at mggg.org, where we have lots of cool things to look at, such as the brief we filed with the Supreme Court a couple weeks ago, which just yesterday was actually quoted in oral argument, which was pretty exciting, if quoted in a surprising way.

You can also find cool software tools that we've been developing, like our tool called Districtr lets you draw your own districts and kind of see—try your own hand at either gerrymandering or fair district thing and gives you a sense of how hard that is. We think it's one of the more user-friendly districting tools out there. Lots of different research links and software tools and resources on our site. So that would be that'd be fun if people want to check that out and give us feedback.

Other things I want to mention: Oh, I guess I'm going to do the 538 Politics podcast tomorrow, talking about this new Supreme Court case.

EL: Nice.

MD: Yeah. So I think that'll be fun. Those are some smart folks over there who’ve thought a lot about some of the different ways of measuring gerrymandering, so I think that'll be a pretty high-level conversation.

KK: Yeah, I'm sure. They turn around real fast, like this will be months from now.

MD: Right. I see. Okay, cool. Yeah, by the time this comes out, maybe we'll maybe we'll have yet another Supreme Court decision on gerrymandering that will…

KK: Yeah, fingers crossed.

MD: We’ll all be handling the fallout from.

EL: Yeah.

KK: All right. Well, this has been great fun, Moon. Thanks for joining us.

MD: Oh, it's a pleasure.

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