The Joy of x

John Urschel: From NFL Player to Mathematician

John Urschel, who retired from playing professional football with the Baltimore Ravens to become a mathematician, talks to host Steven Strogatz about the fascinations of graph theory that lured him away from the NFL.

John Urschel is a doctoral student in mathematics at the Massachusetts Institute of Technology who has already published several well-received papers. What’s even more remarkable is that before pursuing his Ph.D., he played professional football for three seasons with the Baltimore Ravens. Urschel talks to host Steven Strogatz about juggling two demanding careers, his decision to trade in football for math, and the kinds of math problems that fascinate him. This episode was produced by Camille Petersen. Read more at Production and original music by Story Mechanics.

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John Urschel: I was like the most nervous I think I’ve ever been in my life.

Steven Strogatz: Oh.

Urschel: Because, okay, you compare this to let’s say a football game, or you compare this to like the NFL combine where, you know, where your performance really affects whether or not you make it into the NFL.

Strogatz : Mm-hmm.

Strogatz [narration]: From Quanta Magazine, this is “The Joy of x.” I’m Steve Strogatz. In this episode, John Urschel.


Urschel: My performance on this exam really affected whether or not like I would continue on the path that was like my life dream, my life’s dream, yeah.

Strogatz: Yeah, from the — from the little kid running around in the bookstore.

Urschel: Yeah, my life’s dream was not to be a professional football player.

Strogatz: That’s so interesting.

Strogatz: Meet my friend, John Urschel. He’s had at least two lives, so far. First as a professional football player for the Baltimore Ravens, where he was an offensive lineman. He played center and guard, but it was only for about three years before he decided to follow his real passion, which was to become a — a mathematician. And again, he’s at the highest echelon of, of that sport or that game except, in this case, he’s training at the Massachusetts Institute of Technology, where he’s a graduate student in the math program. Part of the fun for me of getting to know him is seeing a scientist at the beginning of the life cycle rather than at the end. This is someone who’s just launching what I, I’m sure is gonna be a brilliant career.

Urschel: I’ve always found application to be a sort of, like, a nice cherry but never — never — it’s never motivated me.

Strogatz: It’s not the whole sundae for you [LAUGHS].

Urschel: No, it’s like application will never motivate me to do a problem.

Strogatz: It’s interesting, ’cause a lot of your stuff, well, feels like machine learning, linear algebra, you know, graph theory. They have all kinds of applications, but you seem to like to dig out the math that makes the machine work.

Urschel: Yes.

Strogatz: Or who — how would you put it? What, what is that —?

Urschel: I think that’s absolutely right. I like to somehow, you know, I — I’ve found these areas to be quite interesting and I like understanding how things work.

Strogatz: Mm-hmm.

Urschel: I like understanding why. I’m a why person.

Strogatz: [LAUGHS]

Strogatz : I don’t think we should primarily conceive of John as a football player. I really think of him as a puzzle solver and he seems to have been fascinated by puzzles since childhood.

Strogatz: I like the picture of you as, that I have in my head, but you need to correct me if I’m not getting it right, as a little boy looking at puzzle books that his mother gave him.

Urschel: Yeah.

Strogatz: So what does that picture really look like with more color in it?

Urschel: Yes, with more color: You can imagine my mom would take me to the local Barnes & Noble’s and, you know, I would run around all excited.

Strogatz: [LAUGHS]

Urschel: And I would, like, go and I would look at all the different puzzles and different puzzle books, and she would tell me, you know, “You can choose one,” you know. And she would show me different — I mean, okay, when we started, at first, she would just buy me a couple, and — but then, you know, once I, once I liked them, then she would let me go to the book — the book shop and pick them out.

Strogatz: What was the appeal of puzzles for you?

Urschel: It was fun.

Strogatz: [LAUGHS] Okay.

Urschel: Yeah, I, I have no… I, it’s not something complex. It’s not something complicated. It’s that it was challenging and the satisfaction of solving a puzzle and also sort of the enjoyment of, sort of, wrestling with it.

Strogatz: How did this compare with your experience in school? Because in principle, school has puzzles, and things to learn, and challenges.

Urschel: Yes. In reality, school was not the most enjoyable thing for me, in part because, one, I was — I was slightly antisocial.

Strogatz: Hm.

Urschel: I was. I mean, uh, there’s this, there’s this story my mom loves to tell over and over again.

Strogatz: Hm.

Urschel: I’m in first grade and the teacher — my first-grade teacher and the principal call my mother, and they call her in to let her know that they think I am mentally challenged, that I have, you know, a mental disability, and that I need to be held back to take the first grade over again and be put in a first-grade class with mentally-challenged — other mentally-challenged children. And my mother is, you know, as any parent would be, is extremely, extremely upset. And not only is she extremely upset, thankfully, she’s extremely defiant, because she is certain that I do not have a mental disability and that I’m actually quite bright. And also she sort of, herself, perceived this as somewhat racial in a, you know, in a way. And so she had them test me to test to see whether or not I, you know, in fact had these issues. And I scored extremely well on, you know, all fronts. And in particular in mathematics, I was completely off the charts —

Strogatz: [LAUGHS]

Urschel: — to the extent that they recommended to my mother, they said, “We are extremely sorry. We recommend he takes third grade next. He doesn’t need second grade.”

Strogatz: Oh, wow, so they knew that you weren’t in the right grade but they just had it backwards [LAUGHS].

Urschel: Yeah, and, well, and they also gave good reasons for why they thought I was, you know, mentally challenged, because they said, “Well, during playtime, he doesn’t play with anyone else.”

Strogatz: Yeah.

Urschel: “He’s just by himself.”

Strogatz: Mm-hmm.

Urschel: “He doesn’t pay attention in class. When we ask him questions, he doesn’t answer, and, and yeah. He’s never — he’s never looking where he’s supposed to be and, he doesn’t talk, and, you know, when we ask him questions, he seems to never know what’s going on, or never know the answer, or never want to answer.”

Strogatz: Mm-hmm. But so — so you say your mother thought there was this racial dimension to it.

Urschel: Yes, so this was in a, like, a suburb of Buffalo, and I think there were no black kids in school except for me, and not only… So she thought that, “Okay, well, John, he’s little. He’s, you know, got a single parent, and you know, they sort of … and he’s not paying attention.” Rather than giving me the benefit of the doubt and questioning “Is he not paying attention because he’s bored or because he doesn’t understand?” She thought that they sort of immediately went to, sort of, the end that they thought seemed likely, given what I looked like and what my home situation was.

Strogatz: Yeah, but you say you don’t see the story that way.

Urschel: I sort of see it as — as a story that really says more about what I was like as a little kid —

Strogatz: [LAUGHS]

Urschel: — than like what the school system was like because, you know, clearly, I was very, you know, I, you know, I — I was somewhat of a problem child.

Strogatz: Yeah, it sounds like you’re — like you can empathize with the school people.

Urschel: Yes, of course. I was clearly a problem child that was … you know, I was not the typical child and not in a, not in a good way.

Strogatz: I mean, it seems like we should start talking about … you have this interesting chapter of your life that goes on for quite a while with, with sports and football, especially.

Urschel: Yeah, yeah, no. I do. What, what — what do you want to know?

Strogatz: What should we — what should we know? So you started in getting interested in high school. I suppose that’s maybe your growth spurt around — you start getting very big.

Urschel: I actually started getting interested in middle school.

Strogatz: In middle school.

Urschel: And so it’s interesting because the reason why I got into sports was because of conformity.

Strogatz: Mm-hmm.

Urschel: I, you know, I had been bullied, and I changed school systems, and I decided, “You know what? I am not getting bullied anymore. I am going to fit in and how am I going to fit in? Conformity.”

Strogatz: That’s amazing. That’s such a formative thing in your life, this whole bullying period.

Urschel: Yeah. No. Yeah, I decided, “This is not happening anymore. This is not good. I don’t like it. It’s been going on for far too long. What I’m going to do is, I’m just gonna look and see what are the popular kids doing?”

Strogatz: Yeah.

Urschel: “And whatever they’re doing, I’m going to do that exact thing.”

Strogatz: Huh.

Urschel: So I got there and what do all the popular kids do? Well, they play lacrosse and they like to play street hockey.

Strogatz: Yeah.

Urschel: These are my two new favorite sports.

Strogatz: Okay [LAUGHS]. There you go.

Urschel: I’m now a lacrosse player and I am quite a good street hockey goalie.

Strogatz: So you — you take on these sports that are the — the popular kids’ sports, and —

Urschel: Yes, and guess what?

Strogatz: What?

Urschel: I become like part of the popular kid friend group.

Strogatz: Yeah.

Urschel: Yeah. Everything’s fine now, so I get into sports. I’m on the lacrosse team. I even do, like, travel box lacrosse, which is lacrosse played in a, a hockey rink.

Strogatz: Oh, okay.

Urschel: Yes, and, yeah, so now all of a sudden, I play sports.

Strogatz: Mm-hmm.

Urschel: And right around this time that I’m playing sports, my father simultaneously moves back near me. So for a decent, you know, for some chunk of my life, he was living in Boston because he was the — he was the chief of surgery at Beth Israel Deaconess, Harvard’s hospital, and he moves back. And when he moves back, he decides that I’m overweight, and I’m out of shape, and that he needs to fix this, yeah. And so he decides, “I’m going to fix this.”

Strogatz: Mm-hmm.

Urschel: And so what he does is, he all of a sudden becomes extremely involved in my life. And he picks me up from school every single day after school, and he takes me to the gym, and we’re lifting weights and we’re running up — running up and down stairs, and we’re doing laps on the, you know, on the track.

Strogatz: Was this one, like, an especially happy time for you? You’re getting to hang out with your dad so much.

Urschel: Yeah, it was an extremely happy time. I like really — I really loved it and these were some of my, like, favorite memories.

Strogatz: I bet, yeah.

Urschel: Like, yeah, and, uh, I — I got —

Strogatz: And he, himself, had been a — an athlete.

Urschel: Yeah, he was a college football player. He had a chance to play professionally but he decided to, sort of, like, finish medical school and stuff.

Strogatz: There’s a little gap here in going from your high school days playing lacrosse and stuff to you in college, ’cause that’s a big decision, right? Where are you gonna go to college.

Urschel: Yes.

Strogatz: At some point, football has to become part of our — our discussion here.

Urschel: Yeah, so I was playing football in high school. The coaches immediately after the first day of practice, they say, “Oh, this one. We need to — we need to do something with him.”

Strogatz: Yeah?

Urschel: You know, they could tell, you know… I had horrible technique. I, you know, didn’t know, you know, anything about how to play football but I was really good at hitting people.

Strogatz: [LAUGHS] Okay, there you —

Urschel: So after the first day of practice, he’s no longer on the freshman team. He’s on the junior varsity team.

Strogatz: After the break, John goes to the Ph.D. playoffs.


Strogatz: Flesh this out a little. How can you be a pro football player and in training to be a pro mathematician?

Urschel: Don’t do it. This is my advice for anyone.

Strogatz: [LAUGHS] Anyone in the NFL thinking of —

Urschel: Do not do it. Do not do it. I have to say that, first of all, I’m so — I’m so grateful to be at MIT and this is an amazing place for me. Don’t do it. Don’t do it. And this, like, as much as I hate to admit it, this really came — this really played a part in my decision to retire when I did, I think, because, you know, I think a part of me really did not want to do that again.

Strogatz: What are you talking about?

Urschel: Okay, so what I’m talking about is being in the full — being sort of like, oh, being a player in the NFL, a full-time job, a very serious full-time job, and you get down time, though, as any person does. Okay, it’s not some crazy amount of hours, but then I’m doing coursework.

Strogatz: Like for people who don’t know much about football, every — you — you play on Sunday. You got to get better by — yeah, tell us.

Urschel: I’ll tell you the exact schedule.

Strogatz: Yeah, tell us, yeah, just to give us a, a picture.

Urschel: Perfect. So I’ll tell you my exact schedule.

Strogatz: Okay, yeah.

Urschel: So — so the Ravens are slightly different than most teams in the league based on when they give us our off day. Most teams have their off day on Tuesday. We did ours on Monday.

Strogatz: Mm-hmm.

Urschel: So we play on Sunday. Let’s say we have a 1:00 game. We get done around 5:00 or so. I’m out by 5:00. I go home.

Strogatz: You’re all now sore.

Urschel: Sore, beat up.

Strogatz: Yeah.

Urschel: And 5:00 PM until 11:00 AM on Tuesday — so 5:00 PM Sunday, 11:00 am on Tuesday, I am not a football player. I am a full-time Ph.D. student at MIT and I am trying to do all my assignments as quickly as possible.

Strogatz: I thought you were gonna say either that you were asleep or you’re in a like some kind of hot tub or something.

Urschel: No, I — I need to get all my work done.

Strogatz: You need to get all your work done.

Urschel: Yeah, so at the time, what I was doing —

Strogatz: That’s your only window.

Urschel: This is my only window, and so I just absolutely — and then, okay, so then Tuesday is like, uh, half day of work. Wednesday is a full day of football work, very long day. Thursday is a long day. Friday is more of like a three-quarters day, and then Saturday is your travel day and you start to —

Strogatz: Which could mean going across the whole country or whatever, yeah.

Urschel: Yeah exactly. And so that whole period, I’m fully football, because well, it’s a serious job and I need to really pay attention. I need to watch film. I need to do things, otherwise, I’m not gonna have a job.

Strogatz: Mm-hmm.

Urschel: And I want to let you know that this absolutely is too much.


And then, you know, I finish the — I finish the season. We don’t make the playoffs, which… Okay, of course I’m sad about, but the one plus of not making the playoffs is I now have a month to study for my quals.

Strogatz: [LAUGHS] Right. Maybe you should tell us what that means. What — what are the quals?

Urschel: Yeah, so, uh, qualifying exams. It’s a set of exams you take that test your proficiency in certain areas of math, and you need to pass these things in order for you to sort of move along to… Say that, okay, this person knows enough math that we feel comfortable sort of pushing him along the Ph.D. process.

Strogatz: Mm-hmm. Now so let put an asterisk on it.

Urschel: Yes.

Strogatz: From the professor’s point of view, and you probably know this.

Urschel: Yes. Tell me from the professor’s point of view.

Strogatz: So from the professor’s point of view, this is a good chance for us to get rid of students who don’t belong.

Urschel: Yes, yes. I wasn’t going to say this.

Strogatz: Well, this is a hurdle.

Urschel: Yes. It is a serious hurdle.

Strogatz: This is a very serious hurdle, and sometimes students are admitted who are risky in some way, either their — their preparation is a little thin but they have some kind of potential, or, you know, you can think of various reasons you’d take a chance on someone.

Urschel: So let me say that I’m currently pointing at myself right now.

Strogatz: Okay, yes [LAUGHS].

Urschel: When you say like, you know —

Strogatz: I was being a little abstract in my description, I bet.

Urschel: Yes, but no, no, it’s true. So this was a — this was a very serious moment for me to say I belong here, because first of all, I did my undergrad at Penn State. This is hardly an institution that is, you know, famous for pumping out brilliant undergrad mathematicians.

Strogatz: That’s right. Well, I mean it does have a strong math department, as it turns out.

Urschel: It has a strong math department, but —

Strogatz: MIT doesn’t consider it a peer institution.

Urschel: Exactly. MIT does not consider it a peer institution.

Strogatz: Nor does Penn State. I mean [LAUGHS] —

Urschel: Yeah, nor does Penn State. Let’s be real, and I love Penn State, but, okay, I — I can’t think of the last mathematician, you know, undergrad, you know, major, math major at Penn State who has, you know, done a Ph.D. at, you know, let’s say a top tier institution. You know what I mean? It’s just —

Strogatz: It’s not the standard route.

Urschel: It’s not the standard route, you know. First of all, it’s not the standard route. Second of all, I’m this professional football player who’s been out of school, you know,  so it’s not like I’m in school and then applying. I’m out of school.

Strogatz: Mm-hmm. No, you — you’re definitely an unconventional student.

Urschel: Yeah, I’m completely unconventional. They accept me halfway, like, through the year, so I start in January.

Strogatz: Hm. That’s also very unconventional, yeah.

Urschel: Very unconventional. And so all of this put together, I’m an unconventional, you know, student there, so I felt like, you know, I — I need to do well here. Otherwise, this could spell serious trouble.

Strogatz: Mm-hmm.

Urschel: I was very aware of this.

Strogatz: Mm-hmm.

Urschel: I’m, you know, I’m no fool, and so I lock myself into like my office with a chalkboard and with all these notes to try to prepare for these three courses that my qualifying exam is on. I am doing this, like, 14 hours a day. I am killing myself because I am behind and I am just — like, I have to do this.

Strogatz: Yeah.

Urschel: And then I also have to prepare for these other two courses.

Strogatz: Sure. So are the qualifying exams done as an oral exam or a written exam?

Urschel: Yes, so I — I wish they were written. They — they were oral.

Strogatz: Yeah.

Urschel: They were oral exams.

Strogatz: So antisocial John Urschel is standing in front of his teachers.

Urschel: In a very, very stressful situation.

Strogatz: By himself.

Urschel: By myself.

Strogatz: With what picture? What do we picture? Three — three dudes?

Urschel: Three dudes, and by the end, the consensus they have is that, yes, I’ve passed. I know the material, but, uh, they did say that I did seem really nervous early on [LAUGHS]. They said like — like, “You know the material, but clearly you’re — you’re very nervous early on,” is what they said, which is fair, because I was incredibly nervous. I was like the most nervous I think I’ve ever been in my life. Once I got introduced to higher level mathematics, I knew that my dream in life was to be a math professor, and not just be a math professor, but to be one who, like, actually makes, you know, at least one nice discovery, at least one.

Strogatz: [LAUGHS]

Urschel: And has a chance to, you know, continually learn, work on hard problems… Even if I fail and I, you know, I fail again and again and again. That’s fine. Just being able to work on them. Interacting with sort of other people, young students. Really, I just want to, you know, think about things, think about tough things, even if I don’t, you know, make any progress. I mean this goes back to me as a kid. The — the enjoyment is sort of the struggle.

Strogatz: Yeah. Yeah, yeah. It’s a very, very consistent theme running through your life, it seems. Interesting.

Urschel: Yeah, so this is like my — this is my, I mean, my dream is to be a — be a professor.

Strogatz: This seems to be one of these happy dreams that — that’s gonna turn out that way. So far so good.

Urschel: I hope so. Yeah, I hope so.

Strogatz: Yeah, I mean it’s — it’s maybe you should tell us a little about, um, you know, so you passed the — the qualifying exam. All right, you’re nervous, but they say you passed, and then you got to work and you’ve been doing now for some number of years.

Urschel: I finished three and a half years now, and so I’m starting the end of my fourth year this fall.

Strogatz: And so this is it. I mean, normally when I talk to my scientists in this show, they’re people who are sometimes in their 60s. So you’re one of our first, I think you are, our first guest who’s at an early stage in the professional developmental life cycle. The making of a scientist or a mathematician. What word would you use to describe what you’re trying to become?

Urschel: Yeah, mathematician.

Strogatz: Simply mathematician.

Urschel: Yeah, mathematician, of course.

Strogatz: That’s the word for you. No adjectives on it?

Urschel: No, mathematician.

Strogatz: Okay, so for you, pure or applied, which some people make a big deal about?

Urschel: I don’t, I don’t really distinguish these things so much.

Strogatz: Yeah. Okay, so tell us a little, then, about what kinds of questions you like and what you like about them?

Urschel: Yes, so the questions that I often like answering are questions of the form either — either there’s a well-known area or a well-known technique that is not well- understood, and I like understanding theoretical guarantees of perhaps a certain technique or theoretical properties of a certain problem.

Strogatz: But when you speak of a theoretical guarantee, tell us what that would mean.

Urschel: Suppose there’s a machine learning problem.

Strogatz: Mm-hmm.

Urschel: What I mean by a machine learning problem is, you have some framework where you, say, you get some samples and now you’re asked to, let’s say, learn something back, maybe a distribution. And you want to ask questions of the form. How well can you learn back your distribution given your samples?

Strogatz: Mm-hmm.

Urschel: And, okay, often this is asked from a heuristic point of view, and you look at techniques to do this — whether it be sort of gradient descent-type techniques or some variant of that — but you try to solve these things in practice. Whereas in this framework I’m typically interested in, understanding when this is an easy problem to solve, when it’s a hard problem to solve, and why.

Strogatz: Mm-hmm.

Urschel: I’m interested in looking at techniques that maybe people already use and understanding under what circumstances is this guaranteed to work.

Strogatz: Yeah. Yeah, yeah. It’s a — it’s an interesting niche. It’s almost like I’m trying to think of what an analogy would be for this. It’s as if there are tools that people have. Like you could literally picture a drill, you know, or a hammer, or a saw.

Urschel: Yeah, and I want to know “what does this hammer work on?” How well can we be guaranteed that it works? People are already hammering away with it.

Strogatz: So they know it’s an effective hammer.

Urschel: Yes, but I want to know “can we quantify how effective it is?”

Strogatz: Yeah. Where would it break down?

Urschel: Yes, exactly. This is one of my big interests. And then other interests I have are sort of answering theoretical questions about, uh, about things that — that we use, perhaps theoretical questions where it’s not obvious how useful they are. Like for instance, I care about theoretical questions regarding your ability to, say, solve an eigenvalue problem or solve a linear system, and theoretical from the point of, you know, asymptotic running time. Or I’m concerned about your ability to, say, approximate a certain signal you have. I actually think about these things in a very abstract setting in the sense that, yes. I understand where these things come from, but it’s enough for me to know where these things come from, people care about them, and for me, it is a purely mathematical problem which, in fact, is even simpler in my mind when I just think about it as a mathematical problem.

Strogatz: So am I gunking it up by being too concrete with this audio example?

Urschel: No, you’re not gunking it up because this is — I think that this is something that helps a lot of people to understand things. But to me — here’s the way I think about it. And so here’s the abstract version of the problem. Suppose, okay, for instance, we have a, a cube, a unit cube in d dimensions.

Strogatz: Okay [LAUGHS].

Urschel: Okay?

Strogatz: We can do it in three dimensions for our — for anyone who wants to picture it, an ordinary, three-dimensional cube.

Urschel: Three dimensions, d equals three, but —

Strogatz: But it’s —

Strogatz: You’re getting a flavor of what John likes to do.

Urschel: Yes.

Strogatz: He wants to do it in any number of dimensions.

Urschel: Yes. And now what we’re going to do is, we’re going to break up this unit cube into n pieces.

Strogatz: Mm-hmm. Cool.

Urschel: Okay? And in fact, rather than break it up into n pieces, I’ll do something even more intuitive for our listeners.

Strogatz: Yeah?

Urschel: We’re going to just pick n points in the cube. Okay? We pick n points.

Strogatz: Got it.

Urschel: Rather than breaking it up into n pieces, I think this is more intuitive.

Strogatz: Okay, cool.

Urschel: We pick n points in the cube and we’re gonna try to pick these n points so that we minimize the following thing.

Strogatz: Mm-hmm.

Urschel: We want to minimize the average square distance from any point we randomly — from any other point randomly chosen in the cube, and the closest point of the n points that we just listed.

Strogatz: I hear you. Right. Can I — I always like to be concrete.

Urschel: Yes.

Strogatz: Tell me if this is close.

Urschel: Yes.

Strogatz: If you thought of this cube as this big block of three-dimensional space, which is like here I am on the surface of the Earth, but I also have access to a helicopter so I can go up into the third dimension.

Urschel: Yes.

Strogatz: And maybe imagine some futuristic thing where people are living all over the place in this three-dimensional cube.

Urschel: And you can imagine something like hospitals.

Strogatz: That’s exactly what I was gonna say.

Urschel: This is a good example.

Strogatz: Where should you put the hospitals?

Urschel: Yeah, where do you put the hospitals?

Strogatz: So everyone’s close to a hospital no matter where they live.

Urschel: Right, and the way you measure closeness is based off square distance.

Strogatz: Yeah.

Urschel: Yeah, which turns out to be a good way to measure these things, yeah.

Strogatz: So good. I’m glad I was on the right track.

Urschel: That’s an example I use a lot. Hospitals, yeah.

Strogatz: Well, because it’s a real thing, right? People in the field of operations research try to think about where should you put the ambulances, where should you put the hospitals, given where the people live.

Urschel: Yeah, I mean this is a — you can think of this problem as a specific — a specific type of example of like an optimal transport.

Strogatz: It’s beautiful. So this is the mathematical natural generalization of this very concrete hospital problem.

Urschel: Yes.

Strogatz: Cool.

Urschel: Yes.

Strogatz: Cool. So this is, you know, and this is such a typical math move that you take a — a thing that comes from the real world and now you make it more beautiful, and more abstract, and maybe more general. It will apply to many more things than the hospital problem.

Urschel: Yes, yes.


Strogatz : When we come back. How to get a traveling salesman to all 48 state capitals in the contiguous US in the most efficient, shortest possible distance. Also, what does a classic movie have to do with fixing math education today?

Strogatz : John and I talked about an old problem that’s a classic chestnut in math. It goes by the name of the Traveling Salesman problem. In a nutshell, it’s — it’s the question of if you wanted to get a salesman to all 48 state capitals in the continental U.S. in the shortest possible distance, how would you do it?

You know, it probably doesn’t sound like a very pressing problem. There aren’t too many traveling salesmen left anymore, but it embodies much bigger principles. It has big implications for everything from military operations, air traffic control, medicine, essentially anything where you’ve got countless possibilities and you’re looking for the best or the cheapest or the fastest way to do something. Especially, you know, in some problems that are so big like this, there’s just no hope of listing every possible option, so you need a more efficient way to solve the problem. John worked on a subspecies of this kind of problem, goes by the name of the Asymmetric Traveling Salesman problem.

Urschel: Asymmetric is, just imagine the amount of time it takes for you to get from Point A to Point B is different than the amount of time it would take you to get from Point B to Point A.

Strogatz: Okay, got it. So if we thought of all the capitals of all the states in the United States, there would be 50 cities, and, uh, actually it’s a little confusing with Hawaii being not —

Urschel: Yeah, let’s get rid of Hawaii and Alaska, yeah.

Strogatz: Okay, let’s do the continental U.S.

Urschel: Yeah, continental U.S.

Strogatz: So those 48 states, so — so if you had to go from one capital to the next and you want to minimize the total distance that you travel —

Urschel: You want to visit all 48 capitals.

Strogatz: Yeah, nobody visits twice. You don’t go to any place twice. Oh, you could. It’s usually not a good idea to go backwards.

Urschel: Sometimes it is.

Strogatz: Can it be?

Urschel: Well, it’s important that you’re allowed to.

Strogatz: Okay, so you’re allowed to visit any city as much as you want —

Urschel: Yeah, but you want to visit all of them.

Strogatz: But the question is total — visit them all in the shortest amount of distance.

Urschel: And we can say total distance.

Strogatz: But the asymmetric problem you say is like the distance from —

Urschel: Albany to, uh —

Strogatz: Harrisburg.

Urschel: — is different than the distance from Harrisburg to Albany.

Strogatz: It could be ’cause there’s so much construction on one of the roads.

Urschel: Yeah, exactly. So this would be the asymmetric setting, and now you want to try to, you know, visit these 48 capitals in as little distance as possible, and now, so first of all, this is known to be something called NP-hard, which means, well, I’m not even going to attempt to say what it means.

What it means is that what the — the amount of time that we think it would take to solve this problem exactly is very large compared to the number of places we have to visit. So what people work on is they try to come up with ways to quickly find a route that may not be the best route, but it guaranteed to be within some factor of the best route.

Strogatz: Oh, yes. I see.

Urschel: I spent, like, well over a year looking at this and I made no progress. And that happens, and that’s okay, except it’s not okay when you’re a Ph.D. student.

Strogatz: [LAUGHS] Well, that’s true. Learning how to back out sometimes, or I want to say quit, really — I mean strategic quitting is a valuable skill.

Urschel: Yeah, so we completely backed out and, you know, I said, okay. I’ve really spent a chunk of time trying to work toward something that I’m clearly not making progress on. I need to be realistic. I need to start working on something where I feel like I can make some steps towards something. I can have some publications. I need to catch up. I need to start being very active ’cause this is important, and this is, sort of, has a big effect on my life, and so this is — this is what I’ve been doing. Because it’s not good, you know, as a Ph.D. student, to have, you know, you know, to be working on something for a year, year and a half, and have nothing to show for it.

Strogatz: No. You’ve gotta put points on the board.

Urschel: Yeah, so I, you know, for a year and a half, I put no points on the board, and so the past, like, recent — you know, recent time, I’ve really been working hard. And so I put out, you know, I put out a paper like a few months ago involving sort of some — some questions about, sort of, your ability to draw graphs in the plane in sort of a nice sense. And, you know, I’m — I’m working on this quantization stuff, and so I’m making progress.

Strogatz: Sounds good.

Urschel: And I — and I feel really good about it.

Strogatz: Sounds really good, yeah, and I see you got a big smile there [LAUGHS].

Urschel: Yeah, no, I’m really — I’m really happy about the stuff I’m doing and I — I enjoy it a lot, so I really enjoy it. It’s like — it’s — it’s really cool stuff.

Strogatz: Fantastic.

Urschel: And it’s a flavor that I really didn’t — I didn’t really, I didn’t study at all before. Because — okay, the way I came to this. — this is gonna sound crazy, but, okay, and this is something I should have said. This whole quantization problem… Listeners probably, you know… Unless you work in information theory, you wouldn’t have heard of this, but this quantization problem: When you replace a continuous signal with, let’s say, a discrete set of points, this is something called the K-means problem.

Strogatz: Oh, yeah. That’s a famous thing.

Urschel: Which is a famous thing in machine learning, and, you know, if you’re a data science person, everyone knows this. My sister-in-law’s boyfriend, you know, he works in, like, politics. He knows this algorithm, and so … which was —

Strogatz: Because of gerrymandering or something?

Urschel: Because it’s a common technique in just, like, data science. Any time you have some data and you just want to break it into parts, it turns out the K-means algorithm is the single most popular technique to do that.

Strogatz: I hear people mentioning it all the time now.

Urschel: Yeah, so, so yes. This is K-means.

Strogatz: So you have a natural audience, it sounds like. I mean, you’re a step or two away from K-means.

Urschel: Yeah, so of course, you can imagine, based off of my previous work, that I came to this through K-means, though it has taken a very abstract flavor.

Strogatz : Another part of John’s work is about changing the way that students learn math. He thinks there’s a lot missing in the way that we’re doing math education.

Urschel: It’s important, I think, to understand the context of what you’re learning in terms of just knowing a tiny bit about when was it discovered. Why was it discovered? What was its use? Because I feel like, you know, as kids, or at least when I was a kid, I felt like this math that I was learning, you know, although like I enjoyed it, you know, it just came out of this, like, black box.

Strogatz: Sure.

Urschel: Whereas, you know, when you’re learning addition or multiplication, you’re learning all these things, to think that people came up with these things so long ago so that they could solve, like, problems where they have, you know, six people working and they have 11 loaves of bread. How much does each person get? Or you have two groups of people and 50 loaves of bread, but one group of people working is more than the other group. I mean these are famous examples that we’ve found … like, I think it’s like hieroglyphs of — of these —

Strogatz: Absolutely. Ancient cultures were thinking about stuff like this.

Urschel: Yeah. I mean this is what people were using numbers for.

Strogatz: Yeah, and collecting taxes, and surveying, and measuring land. I mean that’s in our — in the word “geometry” itself is land measurement. You know, this was a problem of figuring out the area so that the king would know how much grain to collect and tax.

Urschel: Yeah, I mean, and as strange as it sounds, it’s like not a single thing, in my opinion, that a kid learns in mathematics in high school curriculum is sort of anything that isn’t born out of a very practical use that was needed in the history of our world.

Strogatz: Mm-hmm.

Urschel: And I think kids don’t understand that. But I will say that in physics class or in other classes, you do get a flavor of the people who discover certain things and you do get a flavor of, you know, why. You do get a flavor of this, or at least I got a flavor of this.

Strogatz: Some fields are better with that. Yeah, the physicists tend to be better about that than — than in math. When you go to classrooms nowadays, what kinds of things do you try to do with the students?

Urschel: I mean, okay, you know me. I love puzzles.

Strogatz: Okay.

Urschel: So yeah, you know, I get in a classroom and I love — I love doing puzzles, and also my audience of choice is always an AP Calculus course.

Strogatz: Yeah.

Urschel: I mean, this is just, it feels like this is — it feels like I can do the most in this sort of high school classroom. And I have a sort of competitive advantage over the next person who might — they might want to bring in to talk about math in that, okay, I feel like there’s lots of people who can talk about like math or talk about the importance of science at the level of like a sixth-grader or a seventh-grader, but not so many people can talk to, let’s say, a room of people who just took Calc BC who are going to college, who think they might want to study certain scientific disciplines, that have knowledge of mathematics, where I can give them good advice.

Strogatz: Right.

Urschel: And so I am a resource. I answer questions, but also, I give them a little bit of a lecture. It can be, you know, going through puzzles, like sort of interesting problems, like sometimes, you know, I’ll — I’ll show them, like, the Riemann rearrangement theorem.

Strogatz: Oh, that’s a great one. So maybe we should try to give the heart of what that’s about. I — I mean I could try, but you could try.

Urschel: Okay. I can try. So I want you to imagine the following sum. I’m just gonna add up these numbers. I’m gonna add up 1, subtract 1/2, add 1/3, subtract 1/4, add 1/5, and subtract 1/6, and 1/7, subtract 1/8, and 1/9, and so on and so forth, and you can imagine where I’m going with this.

Strogatz: So this is — this is all the fractions 1 over something.

Urschel: One over something.

Strogatz: 1 over 1, 1 over 2, 1 over 3, but with alternative plus and minus signs between them.

Urschel: It’s a plus if the denominator is odd. It’s a minus if the denominator is even.

Strogatz: Okay.

Urschel: And I want you to add these things in the order that I just added them.

Strogatz: Yeah.

Urschel: Okay? And if you add all these things together, you get a fixed number. Oh, I should know this. I think it’s like —

Strogatz: Log 2.

Urschel: Yeah, exactly.

Strogatz: Right? Natural logarithm of 2, so it’ll be about .69.

Urschel: Yes, exactly.

Strogatz: Yeah. I — I think so. I mean I think if I remember right, I think so.

Urschel: No, that’s right. Good memory. Good.

Strogatz: [LAUGHS]

Urschel: Now what I’m going to do is I’m just gonna change the order that I’m gonna add the numbers.

Strogatz: Yeah.

Urschel: So instead, I’m just gonna let’s say add 1, subtract 1/2, subtract 1/4, add 1/3, subtract 1/6, subtract 1/8.

Strogatz: Okay, so some tricky thing where you’re still using the same numbers, but in some new order.

Urschel: Same numbers, add 1/5, subtract 1/10, subtract 1/12.

Strogatz: You’re not putting plus signs in front of things that used to have negatives.

Urschel: No.

Strogatz: You didn’t do any of that.

Urschel: I didn’t do anything of that.

Strogatz: It’s the same numbers.

Urschel: The exact same numbers and take 1 minus 1/2, minus 1/4. What’s 1 minus 1/2? It’s 1/2, so now I have 1/2 minus 1/4.

Strogatz: Right.

Urschel: I take 1/3 minus 1/6, minus 1/8.

Strogatz: Yeah.

Urschel: What’s 1/3 minus 1/6?

Strogatz: Uh, another 1/6.

Urschel: A sixth, so I have 1/6 minus 1/8.

Strogatz: Yes.

Urschel: And then I go on, and on, and on.

Strogatz: Oh, boy. I see what’s happening.

Urschel: And now I have 1/2 minus 1/4, plus 1/6, minus 1/8, plus 1/10, minus 1/12, plus 1/14, minus 1/16, but look. I have a common denominator of 2. So I had better pull out 1/2, but when I pull out 1/2, what do I get? I get 1 minus 1/2, plus 1/3, minus 1/4, plus 1/5, minus 1/6, plus 1/7, minus 1/8, and somehow, I got half of what I started with. I started with the log of 2.

Strogatz: Yeah, and now you got 1/2 log 2.

Urschel: And now I have 1/2 log 2.

Strogatz: So, okay. I realize that might be hard to have listened to, but what John just did was he took a big — an infinite set of numbers that added up in what in the jargon of calculus would be a convergent series, and it converged to a certain number, .69.

Urschel: Yes.

Strogatz: He then took the same numbers and just added them in a different order, and now he got an answer that’s half as big.

Urschel: Yes.

Strogatz: It’s only 1/2 of .69.

Urschel: And the beautiful result of the Riemann rearrangement theorem says that what really — what really determines the sum is sort of the ratio of these things that are being taken out. And in fact, you can produce any number you want. Suppose there’s always a ratio of taking things out from the positive bin and the negative bin such that the sum of this infinite series will be any number you want. You could — Steve could tell me a number right now.

Strogatz: Sure. I want it to add up to 42.

Urschel: Forty-two, and there is a way for me to reorder the sum that I just told you such that it equals 42.

Strogatz: It’s unbelievable.

Urschel: It’s a beautiful result.

Strogatz: It’s the most amazing — I mean, it’s — it was a total shock when it was first discovered by this great genius, Riemann, that we’re talking about, and it underscores how — how counterintuitive infinity is.

Urschel: I have to admit, I love when my intuition gets corrected.

Strogatz: Yeah.

Urschel: Yeah, and I have to say this is one thing that I’ve been enjoying about, sort of, what I’m doing my thesis on. My tuition gets corrected all the time.


We thought we were going to get back to talking about sort of underrepresented groups in mathematics and we never really got back to that.

Strogatz: We haven’t, no. It was just a generic you in the classroom.

Urschel: Yeah.

Strogatz: Um, let’s — I — I gotta talk to John about this.

Urschel: Okay, let’s talk about it.

Strogatz: So let’s talk about it [LAUGHS].

Urschel: Yes. No. It’s — first of all, it’s — it’s an issue. It’s a real issue that you look at, let’s say, top math departments because we’re mathematicians, and you don’t see women, and you don’t see African Americans when we’re talking about, let’s say, the American mathematician situation.

Strogatz: Certainly not in proportion to the representation in the population.

Urschel: Yes.

Strogatz: Not even close.

Urschel: Not even close, and this — this tells you something, and I’ll say what it says to me. So first of all, I am going to sort of discard the following hypothesis as ridiculous. The hypothesis that all of the babies in this country that have, you know, innate, great innate ability for mathematics and are little genius babies are all being born male, white, and into strong households, namely with, you know, strong socioeconomic backgrounds. And this is absolutely ridiculous, right? Of course.

Strogatz: I think so, too.

Urschel: And so — sort of, okay, I’m being a little sort of like loose here, but in light of that, we’re sort of left with the conclusion that there’s clearly something going on between when these children are being born all the way through their upbringing where somehow we have really, really brilliant, smart minds being lost and not being served.

I think that this is — this is an issue and I think it is an issue we should care about. And I do talk about it as a need for our community because I — I do want to stress that it’s not just an issue for those children. It’s an issue for us as, as well, and so that’s why I’m phrasing it like this because I think it’s an issue we should care about.

Strogatz: Suddenly, I’m flashing back to the old movie, Stand and Deliver. I wonder if it’s too — too long ago for you, because you’re a young man still.

Urschel: Too — what year was this?

Strogatz: Yeah, okay, well, Stand and Deliver, I don’t remember what year it is, but it’s the story of a — a high school teacher named Jaime Escalante in East Los Angeles who was — he was an aerospace engineer, if I’m remembering right, or some kind of engineer, but Mr. Escalante taught students that everybody else had written off as, you know, these are kids that have no potential, or are troublemakers, or — and he taught them the course you’re talking about, Advanced Placement Calculus in high school, and there were colleagues, at least according to the movie — and I think in real life, ’cause I know some of his students who have gone on to become professors, you know — that, that Mr. Escalante’s expecting too much out of these students. “They can’t even learn Algebra. How are they gonna learn BC Calculus? You know you’re setting these kids up for disappointment.” And he says, “That’s not true. The kids will rise to the level that the teacher expects.”

Urschel: Mm-hmm.

Strogatz: And I always found that very — I mean I can get choked up with that line. I think that’s such a… I, I have found that to be true, that if you expect a lot, kids deliver a lot. I mean your own story seems to have born that out, so maybe that’s a — an appropriate thing for us to reflect on. Let’s ask more of each other and expect a lot out of everybody. Everybody has a lot of potential that’s getting untapped.

Urschel: No, I think that’s —

Strogatz: I mean that’s your point, right?

Urschel: Yeah, I think, you know, that’s perfectly fair. That’s exactly my point.

Strogatz: Well, let’s do our own part [LAUGHS] to help with this. And meanwhile, you’ve still got all your potential left untapped here, not totally, but you’re on your way.

Urschel: Yes, yes. I’m on my way and I’m really — I’m really enjoying the process.

Strogatz: So am I hearing right? Your Ph.D. is not a done deal yet. You still haven’t graduated.

Urschel: No. I still haven’t graduated.

Strogatz: So let’s look forward to that.

Urschel: Yes.

Strogatz: And your upcoming postdoctoral years, and hopefully I see you as a colleague very soon in the professor ranks.

Urschel: Yes.

Strogatz: I’ll — I will teach you the secret handshake later.

Urschel: Okay, perfect, perfect.

Strogatz: [LAUGHS]


Next time on the Joy of x, cosmologist Janna Levin and I wax poetic about black holes.

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