Evelyn Lamb: Hello and welcome to My Favorite Theorem. I’m Evelyn Lamb, one of your hosts. 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 good. I actually forgot to say what I do. In case anyone doesn’t know, I’m a freelance math and science writer, and I live in Salt Lake City, Utah, where it has been very cold recently, and I’m from Texas originally, so I am not okay with this.
KK: Everyone knows who you are, Evelyn. In fact, Princeton University Press just sent me a complimentary copy of the Best Math Writing of 2017, and you’re in it, so congratulations, it’s really very cool. [clapping]
EL: Well thanks. And that clapping you heard from the peanut gallery is our guest today, Jayadev Athreya. Do you want to tell us a little bit about yourself?
Jayadev Athreya: Yeah, so I’m based in Seattle, Washington, where it is, at least for the last 15 minutes it has not been raining. I’m an associate professor of mathematics at the University of Washington, and I’m the director of the Washington Experimental Mathematics Lab. My work is in geometry, dynamical systems, connections to number theory, and I’m passionate about getting as many people involved in mathematics as a creative enterprise as is possible.
KK: Very cool.
EL: And we actually met a while ago because my spouse also works in your field. I have the nice privilege of getting to know you and not having to learn too much about dynamical systems.
JA: Evelyn and I have actually known each other since, I think Evelyn was in grad school at Rice. I think we met at some conferences, and Evelyn’s partner and I have worked on several papers together, and I’ve been a guest in their wonderful home and eaten tons of great granola among other things. On one incredibly memorable occasion, a buttermilk pie, which I won’t forget for a long time.
KK: Nice. I’ve visited your department several times. I love Seattle. You have a great department there.
JA: It’s a wonderful group of people, and one of the great things about it is of course all departments recognize research, and many departments also recognize teaching, but this department has a great tradition of public engagement with people like Jim Morrow, who was part of the annual [ed. note: JA meant inaugural; see https://sites.google.com/site/awmmath/awm-fellows] class of AWM fellows and runs this REU and this amazing event called Math Day where he gets two thousand high school kids from the Seattle area on campus. It’s just a very cool thing for a research math department to seriously recognize and appreciate these efforts. I’m very lucky to be here.
KK: Also because I’m a topologist, I have to take a moment to give, well, I don’t know what the word is, but you guys lost a colleague recently.
JA: We did.
KK: Steve Mitchell. He was a great topologist, but even more, he was just a really great guy. Sort of unfailingly kind and always really friendly and helpful to me when I was just starting out in the game. My condolences to you and your colleagues because Steve really was great, and he’s going to be missed.
JA: Thank you, Kevin. There was a really moving memorial service for Steve. For any of the readers who are interested in learning more about Steve, for the last few years of his life he wrote a really wonderful blog reflecting on mathematics and life and how the two go together, and I really recommend it. It’s very thoughtful. It’s very funny, even as he was facing a series of challenges, and I think it really reflects Steve really well.
KK: His biography that he wrote was really interesting too.
JA: Amazing. He came with a background that was very different to a lot of mathematicians.
EL: I’ll have to check it out.
KK: Enough of that. Let’s talk about theorems.
EL: Would you like to share your favorite theorem?
JA: Sure. So now that I’m in the northwest, and in fact I’m even wearing a flannel shirt today, I’m going to state the theorem from the perspective of a lumberjack.
JA: So when trees are planted by a paper company, they’re planted in a fairly regular grid. So imagine you have the plane, two number lines meeting at a 90 degree angle, and you have a grid, and you plant a tree at each grid point. So from a mathematician’s perspective, we’re just talking about the integer lattice, points with integer coordinates. So let’s say where I’m standing there’s a center point where maybe there’s no tree, and we call that the origin. That’s maybe the only place where we don’t plant a tree. And I stand there and I look out. Now there are a lot of trees around me. Let’s say I look around, and I can see maybe distance R in any direction, and I say, hm, I wonder how many trees there are? And of course you can do kind of a rough estimate.
Now I’m going to switch analogies and I’ll be working in flooring. I’m going to be tiling a floor. So if you think about the space between the trees as a tile and say that has area 1, you look out a distance R and say, well, the area of the region that you can see is about πR2, it’s the area of the circle, and each of these tiles has size 1, so maybe you might guess that there are roughly πR2 trees. That’s what’s called the Gauss circle problem or the lattice point counting problem. And the fact that that is actually increasingly accurate as your range of vision gets bigger and bigger, as R gets bigger and bigger, is a beautiful theorem with an elementary proof, which we could talk about later, but what I want to talk about is when you’re looking out, turning around in this spot, you can’t see every tree.
JA: For instance, there’s a tree just to the right of you. You can see that tree, but there’s a tree just to the right of that tree that you can’t because it’s blocked by the first tree that you see. There’s a tree at 45 degrees that would have the coordinate (1,1), and that blocks all the other trees with coordinates (2,2) or (3,3). It blocks all the other trees in that line. We call the trees that we can see, the visible trees, we call those primitive lattice points. It’s a really nice exercise to see that if you label it by how many steps to the right and how many steps forward it is, call that that the integer coordinates (m,n), or maybe since we’re on the radio and can’t write, we’ll call it (m,k), so the sounds don’t get too confusing.
JA: A point (m,k) is visible if the greatest common divisor of the numbers m and k is 1. That’s an elementary exercise because, well maybe we’ll just talk a little bit about it, if you had m and k and they didn’t have greatest common divisor 1, you could divide them by their greatest common divisor and you’d get a tree that blocks (m,k) from where you’re sitting.
JA: We call these lattice points, they’re called visible points, or sometimes they’re called primitive points, and a much trickier question is how many primitive points are there in the ball of radius R, or in any kind of increasingly large sequence of sets. And this was actually computed, I believe for the first time, by Euler
KK: Probably. Sure, why not?
JA: Yeah, Euler, I think Cauchy also noticed this. These are names, anything you get at the beginning of analysis or number theory, these names are going to show up.
JA: And miraculously enough, we agreed that in the ball of radius R, the total number of trees was roughly the area of the ball, πR2. Now if you look at the proportion of these that are primitive, it’s actually 6/π2.
JA: So the total number of primitive lattice points is actually 6/π2 times πR2. And now, listeners of this podcast might remember some of their sequences and series from calc 1, or 2, or 3, and you might remember seeing, probably not proving, but seeing, that if you add up the following series: 1 plus 1/4 plus 1/9 plus 1/16 plus 1/25, and so on, and you can actually do this, you can write a little Python script to do this. You’ll get closer and closer to π2/6. Now it’s amazing, now there is of course this principle that there aren’t enough small numbers in mathematics, which is why you have all these coincidences, but this isn’t a coincidence. That π2/6 and our 6/π2 are in a very real mathematical sense the same object. So that’s my favorite mathematical theorem. So when you count all lattice points, you get π showing up in the numerator. When you count primitive ones, you get π showing up in the denominator.
KK: So the primitive ones, that must be related to the fact that if you pick two random integers, the probability that they’re relatively prime is this number, 6/π2.
JA: These are essentially equivalent statements exactly. What we’re saying is, look in the ball of radius R. Take two integers sort of randomly in between, so that m2+n2 is less than R squared, what’s the proportion of primitive ones is exactly the probability that they’re relatively prime. That’s a beautiful reformulation of this theorem.
KK: Exactly. And asymptotically, as you go off to infinity, that’s 6/π2.
JA: Yeah, and what’s fun is, if a listener does like to do a little Python programming, in this case, infinity doesn’t even have to be so big. You can see 6/π2 happening relatively quickly. Even at R=100, you’re not far off.
EL: Well the squares get smaller so fast. You’re just adding up something quite small in not too long.
JA: That’s right. That’s my favorite mathematical theorem for many reasons. For one, this number, 6/π2, it shows up in so many places. What I do is at the intersection of many fields of mathematics. I’m interested in how objects change. I’m interested in counting things, and I’m interested in the geometry of things. And all of these things come into play when you’re thinking about this theorem and thinking about various incarnations of this theorem.
EL: Yeah, I was a little surprised when you told us this was going to be your theorem because I was thinking it was going to be some kind of ergodic theorem for flows or something because the stuff I know about your field is more what my spouse does, which is more related to dynamical systems. I actually think of myself as a dynamicist-in-law.
JA: That’s right. The family of dynamicists actually views you as a favorite in-law, Evelyn. You publicize us very nicely. You write about things like billiards with a slit, which is something that we’ve been telling the world about, but until you did.
EL: And that was a birthday gift for my spouse. He had been wanting me to write about that, and I just thought it was so technical, I don’t feel like it. Finally, it’s a really cool space, but it’s just a lot to actually go in and write about that. But yeah, I was surprised to see something I think of as more number theory related show up here. That number 6/π2, or π2/6, whichever way you see it, it’s one of those things where the first time you see it, you wonder why would you ever square π? It comes as an area thing, so something else is usually being squared when you see it. Strange thing.
JA: So now what I’m going to say is maybe a little bit more about why I picked it. For me, that number π2/6 is actually the volume of a moduli space of abelian differentials.
EL: Of course!
JA: Of course it is. It’s what’s called a Siegel-Veech constant, or a Siegel constant. Can I say just a couple words about why I love π2/6 so much?
EL: Of course.
JA: Let’s say that instead of planting your trees in a square grid, you have a timber company where they wanted to shoot an ad where they shot over the forest and they wanted it to look cool, and instead of doing a square grid, they decided to do a grid with parallelograms. Still the trees are planted in a regular grid, but now you have a parallelogram. So in mathematical terms, instead of taking the lattice generated by (1,0) and (0,1), you just take two vectors in the plane. As long as they’re linearly independent, you can generate a lattice. You can still talk about primitive vectors, which are the ones you can see from (0,0). There are some that are going to be blocked and some that aren’t going to be blocked. In fact, it’s a nice formulation. If you think of your vectors as (a,c) and (b,d), then what you’re essentially doing is taking the matrix (ab,cd)[ed. note: this is a square array of numbers where the numbers a and b are in the top row and c and d are in the bottom row] and applying it to the integer grid. You’re transforming your squares into parallelograms.
JA: And a vector in your new lattice is primitive if it’s the image of a primitive vector from the integer lattice.
EL: Yeah, so there’s this linear relationship. You can easily take what you know about the regular integer lattice and send it over to whatever cool commercial tree lattice you have.
JA: That’s right. Whatever parallelogram tiling of the plane you want. What’s interesting is even with this change, the proportion of primitive guys is still 6/π2. The limiting proportion. That’s maybe not so surprising given what I just said. But here’s something that is a little bit more surprising. Since we care about proportions of primitive guys, we really don’t care if we were to inflate our parallelograms or deflate them. If they were area 17 or area 1, this proportion wouldn’t change. So let’s just look at area 1 guys, just to nail one class down. This is the notion of an equivalence class essentially. You can look at all possible area 1 lattices. This is something mathematicians love to do. You have an object, and you realize that it comes as part of a family of objects. So we started with this square grid. We realized it sits inside this family of parallelogram grids. And then we want to package all of these grids into its own object. And this procedure is usually called building a moduli space, or sometimes a parameter space of objects. Here the moduli space is really simple. You just have your matrices, and if you want it to be area 1, the determinant of the matrix has to be 1. In mathematical terms, this is called SL(2,R), the special linear group with real coefficients. There’s a joke somewhere that Serge Lang was dedicating a book to his friend R, and so he inscribed it “SL2R,” but that’s a truly terrible joke that I’m sorry, you should definitely delete from your podcast.
KK: No, that’s staying in.
EL: You’re on the record with this.
JA: Great. That’s sort of all possible deformations, but then you realize that if you hit the integer lattice with integer matrices, you just get it back. Basically the space of all lattices you can basically think of as 2 by 2 matrices with real entries and determinant 1 up to 2x2 matrices with integer entries. What this allows you to do is allows you to give a notion of a random lattice. There’s a probability measure you can put on this space that tells you what it means to choose one of these lattices at random. Basically what this means is you pick your first vector at random, and then you pick your second vector at random as uniformly as possible from the ones that make determinant 1 with it. That’s actually accurate. That’s actually a technically accurate statement.
Now what that means is you can talk about the average behavior of a lattice. You can say, look, I have all of these lattices, I can average. And now what’s amazing is you can fix your R. R could be 1. R could be 100. R could be a million. And now you can look at the number of primitive points divided by the number of total points in the lattice. You average that, or let me put it a slightly different way: you average the number of primitive points and divide by the average number of total points.
JA: That’s 6/π2.
EL: So is that…
JA: That’s not an asymptotic. That’s, if you average, if you integrate over the space of lattices, you integrate and you look at the number of primitive points, you divide by the average number of total points, it’s 6/π2.That’s no matter the shape of the region you’re looking in. It doesn’t have to be a ball, it can be anything. That’s an honest-to-God, dead-on statement that’s not asymptotic.
EL: So is that basically saying that the integer lattice behaves like the average lattice?
JA: It’s saying at the very large scale, every lattice behaves like the average lattice. Basically there’s this function on the space of lattices that’s becoming closer and closer to constant. If you take the sequence of functions which is proportion of primitive vectors, that’s becoming closer and closer to constant. At each scale when you average it, it averages out nicely. There might be some fluctuations at any given scale, and what it’s saying is if you look at larger and larger scales, these fluctuations are getting smaller and smaller. In fact, you can kind of make this precise, if you’re in probability, what we’ve been talking about is basically computing a mean or an expectation. You can try and compute a variance of the number of primitive points in a ball. And that’s actually something my student Sam Fairchild and I are working on right now. There are methods that people have thought about, and there’s in fact a paper by a mathematician named Rogers in the 1950s who wrote about 15 different papers called Mean Values on the Space of Lattices, all of which contain a phenomenal number of really interesting ideas. But he got the dimension 2 case slightly wrong. We’re in the process of fixing that right now and understanding how to compute the variance. It turns out that what we do goes back to work of Wolfgang Schmidt, and we’re kind of assembling that in a little bit more modern language and pushing it a little further.
I do want to mention one more name, which is, I mentioned it very briefly already. I said this is what is called a Siegel-Veech constant. Siegel was the one who computed many of these averages. He was a German mathematician who was famous for his work on a field called the geometry of numbers. It’s about the geometry of grids. Inspired by Siegel, a mathematician named William Veech, who was one of Evelyn’s teachers at Rice, started to think about how to generalize this problem to what are called higher-genus surfaces, how to average certain things over slightly more complicated spaces of geometric objects. I particularly wanted to mention Bill Veech because he passed away somewhat unexpectedly.
EL: A year ago or so?
JA: Yeah, a little bit less than a year ago. He was somebody who was a big inspiration to a lot of people in this field, who really had just an enormous number of brilliant ideas, and I still think we’re still kind of exploring those ideas.
EL: Yeah, and a very humble person too, at least in the interactions I had with him, and very approachable considering what enormous work he did.
JA: That’s right. He was deeply modest and an incredibly approachable person. I remember the first time I went to Rice. I was a graduate student, and he had read things I had written. This was huge deal for me, to know that, I didn’t think anybody was reading things I’d written. And not to make this, I guess we started off with remembering Steve, and we’re remembering Bill.
There’s one more person who I think is very important to remember in this context, somebody who took Siegel’s ideas about averaging things over spaces and really pushed them to an extent that’s just incredible, and the number 6/π2 shows up in the introduction to one of the papers that came out of her thesis. This was Maryam Mirzakhani, who also we lost at a very, very young age. She was a person who, like Veech, had incredibly deep contributions that I think we’re going to continue to mine for ideas, and she’s going to continue having a really incredible legacy, who was also very encouraging to colleagues, contemporaries, and young people. If you’re interested in 6/π2 and how it connects to not just lattices in the plane but other surfaces, her thesis resulted in three papers, one in Inventiones, one in the Annals, and one in the Journal of the American Math Society, which might be the three top journals in the field.
JA: For the record, for instance, I think of myself as a pretty good research mathematician, and I have a total over 12 years of zero in any of those three journals.
KK: Right there with you.
JA: The introduction to this paper, she studies simple closed curves on the punctured torus, which are very closely linked to integer lattice points. She shows how 6/π2 also shows up as what’s called a Weil-Peterson volume, or rather π2/6 shows up as what’s called a Weil-Peterson volume of the moduli space. Again, a way of packaging lots of spaces together.
EL: We’ll link to that, I’m sure we can find links for that for the show notes so people can read a little more about that if they want.
JA: Yeah. I think even there are very nice survey papers that have come out recently that describe some of the links there. These are sort of the big things I wanted to hit on with this theorem. What I love about it is it’s a thread that shows up in number theory, as you pointed out. It’s a thread that shows up in geometry. It’s a thread that shows up in dynamical systems. You can use dynamics to actually do this counting problem.
JA: Yeah, so there’s a way of doing dynamics on this object where we package everything together to get the 6/π2. It’s not the most efficient, not the most direct proof, but it’s a proof that generalizes in really interesting ways. For me, a theorem in mathematics is really beautiful if you can see it from many different perspectives, and this one to me starts so many stories. It starts a story where if you think of a lattice, you can think about going to higher-dimensional lattices. Or you can think of it as a surface, where you take the parallelogram or the square and glue opposite sides and get a torus, or you can start doing more holes, that’s higher genus. It’s rare that all of these different generalizations will hold really fruitful and beautiful mathematics, but in this case I think it does.
KK: So hey, another part of this podcast is that we ask our guest to pair their theorem with something. So what have you chosen to pair your theorem with?
JA: So there’s a grape called, I’m just going to look it up so I make sure I get everything right about it. It’s called primitivo. So it’s an Italian grape. It’s closely related to zinfandel, which I kind of like also because I want primitive, and of course I want the integers in there, so I’ve got a Z. Primitivos are also an excellent value wine, so that makes me very happy. It’s an Italian wine. Both primitivo and zinfandel are apparently descended from a Croatian grape, and so what I like about it is it’s something connected, it connects in a lot of different ways to a lot of different things. Now I don’t know how trustworthy this site is, it’s a site called winegeeks.com. Apparently primitivo can trace its ancestry from the ancient Phoenicians in the province of Apulia, the heel of Italy’s boot. I’m a big fan of the Phoenicians because they were these cosmopolitan seafarers who founded one of my favorite cities in the world, Marseille, actually Marseille might be the first place I learned about this theorem, so there you go.
EL: Another connection.
JA: Yeah. And it’s apparently the wine that was served at the last supper.
EL: I’m sure that’s very reliable.
JA: I’m sure.
EL: Good information about vintages of those.
JA: I would pair it with a primitivo wine because of the connections, these visible points are also called primitive points by mathematicians, so therefore I’m going to pair it with a primitivo wine. Another possible option, if you can’t get your hands on that, is to pair it with a spontaneously fermented, or primitive beer.
EL: Oh yeah.
JA: I’m a big fan of spontaneously fermented beers. I like lambics, I like other things.
EL: Two choices. If you’re more of a wine person or more of a beer person, you’ve got your pairing picked out. I’m glad you’re so considerate to make sure we’ve got options there.
JA: Or I might drink too much, that’s the other possibility.
KK: No, not possible.
EL: Well it’s 9:30 where you are, so I’m hoping you’re not about to go out and have one of these to start your day. Maybe at the end of the day.
JA: I think I’ll go with my usual cappuccino to start my day.
KK: Well this has been great fun. I learned a lot today.
EL: Yeah. Thanks for being on. You had mentioned that you wanted to make sure our listeners know about the website for the Washington math lab, which is where you do some outreach and some student training.
JA: That’s right. The website is wxml.math.washington.edu. It’s the Washington Experimental Math Lab. WXML is also a Christian radio station in Ohio. We are not affiliated with the Christian radio station in Ohio. If anybody listens to that, please don’t sue us. So what I said at the top of the podcast, we’re very interested in trying to create as large as possible a community of people who are creating their own mathematics. To that end, we have student research projects where undergraduate students work together with faculty and graduate students and collaborative teams to do exploratory and experimental mathematics, teams have done projects ranging from creating sounds associated to number theory sequences to updating and maintaining OEIS and Wikipedia pages about mathematical concepts to doing research modeling stock prices, modeling rare events in protein folding, to right now one of my teams is working on counting pairs and triples and quadruples of primitive integer vectors and trying to understand how those behave. So that’s one side of it. The other side is we do a lot of, like Evelyn said, public engagement. We run teacher’s circles for middle schools and elementary schools throughout the Seattle area and the northwest, and we do a lot of fabrication with 3d printing teaching tools. Right now I’m teaching calculus 3, so we’re printing Riemann sums, 3d Riemann sums as we do integration in two variables. The reason I’m spending so much time plugging this is if you’re in a university and this sounds intriguing to you, we have a lab starter kit on our webpage which gives you information on how you might want to start a lab. All labs look different, but at this point we just had our Geometry Labs United conference this summer. There are labs at Maryland, at the University of Illinois Urbana-Champaign, at the University of Illinois in Chicago, at George Mason University, at University of Texas Rio Grande Valley, Kansas State. There’s one starting at Oklahoma State, at the University of Kentucky. So the lab movement is on the march, and if you’re interested in joining that, please go to our website, check out our lab starter kit, and please feel free to contact us about what are some good ways to get started on this track.
EL: All right. Thanks for being on the show.
JA: Thanks so much for the opportunity. I really appreciate it, and I’m a big fan of the podcast. I loved the episode with Eriko Hironaka. I thought that was just amazing.
KK: Thanks. We liked that one too.
JA: Take care, guys.