1. “Purposeful messiness”



John Urschel comes out of a rainy Cambridge night into the gleaming, very of-the-moment vegetarian restaurant where we’re meeting—his suggestion. He’s wearing a charcoal-gray sweater, his beard is perfectly groomed, there is not a drop of water on his glasses. He’s a very big guy but he doesn’t look as big as he is. He’s here to talk about math. He is exactly on time.



“Very rigorous, very structured,” Urschel tells me, when I ask him to describe his personal habits. “Nothing drives me crazy more than being late.”



At which point he pauses and rethinks. “I think I am messy,” he says. “But it’s a very purposeful messiness. My desk might be messy but everything’s where it’s supposed to be. I don’t like people messing with my mess.”



For those of you who don’t spend a lot of time around our tribe, Urschel is being an extremely typical mathematician here. This isn’t a matter of his work habits or his physical presentation. As far as those things go, there is no typical; the range of mathematicians is pretty much coterminous with the range of human beings. When I was starting out in math I wore John Lennon glasses and a lot of corduroy shorts. I know mathematicians who look and dress like Paris runway models, and mathematicians who look and dress like grindcore drummers, and mathematicians who look and dress like wind-battered sea captains. John Urschel looks and dresses like a successful pro athlete who has moved on to a string of high-profile media gigs, which, in part, he is.



John Urschel teaches a STEM lesson at Dundalk High School in Baltimore. Steve Ruark/AP Images for Texas Instruments

What mathematicians do have in common, what is in a way our foundational habit, is the tiny move Urschel executes between the first thing he says and the second.



We say things, or write them down, in order to see if they feel true. We say it and then we think about it. And whatever’s slightly not right, we circle back and revise, making more precise, shaving off the parts that don’t pass inspection. It’s our process in math and it leaks out into the way we talk about our desks and our lives.



Urschel is a Ph.D. student at MIT, working on applied math, a field for which “purposeful messiness” is actually not too bad a description. Urschel doesn’t spend his time in the purely abstract reaches of abstract number systems or exotic geometries that could never exist in the physical world. In applied math, the structured rigidity of theory is always just barely containing the messiness of the world we live in. For Urschel, a big part of that world is computation. A pure mathematician like me might ask, in an airy way, which problems have answers. Urschel is more likely to ask which problems have answers that an actual computer can figure out without using all the planet’s electricity or running until the galaxies crap out. Among his specialties is optimization, which, in non-technical terms, means “the mathematics of doing things better and cheaper with machines.” As you might imagine, it’s hot right now.



Urschel orders a chickpea dish. I get the meatless meatballs. I start eating but after a few seconds I realize Urschel doesn’t have any food. I apologize for starting early and put down my fork. “No,” he says, “I already finished mine.”



He really is pretty big.

2. “It was a struggle that I really loved”



Things weren’t always so organized. “We would be late for things all the time when I was a kid,” he tells me. Late for football practice. Late for school—so often, he says, that the school stopped keeping track. His life was complicated: Urschel grew up in Buffalo, living primarily with his mom, Venita Parker, a former nurse who went back to school to become a lawyer during Urschel’s early childhood. Urschel’s father was a cardiothoracic surgeon and a former college football player, who for much of Urschel’s youth was a section chief at Beth Israel Deaconess Medical Center in Boston.



Early on, Urschel’s grandmother took care of him. After she died, when he was in elementary school, Urschel would go with his mother to night court. While his mom argued cases, he’d sit in the back of the courtroom and read The Big Book Of Tell Me Why, a kids’ compendium of science questions. “How does lightning work? How does this work? How does that work?” Urschel says. “It was big enough where I never got bored of it.”



It wasn’t obvious that Urschel was going to be a mathematician. He didn’t compete in the Math Olympiad or precociously one-up his teachers. He was the kind of kid who often slides through the school system unnoticed; his grades were all perfect, so there was no problem, so no one paid attention.



Football was another story. He had the kind of strength and size—listed at six-foot-three, 275 pounds by the end of high school, dominating opposing linemen on both sides of the ball—that was impossible to overlook.

“I have, or had, more math talent than football talent,” Urschel tells me. “I fully believe this.” But the world is on the lookout for football players. His coaches at Canisius, an all-boys Jesuit high school, continually encouraged him to think big, bigger, biggest. If he worked at it, he could play in college. He could play in the Big Ten. There was a chance he could play in the NFL. From his math teachers? Zero.



John Urschel (64) battles with Ohio defensive lineman Carl Jones (89) during the first quarter of a football game in 2012. AP Photo/Gene J. Puskar, File

That changed at Penn State, where Urschel enrolled in 2009 to study math and play football. (He was offered a full scholarship to Stanford, but turned it down; he had already committed to PSU.) If you believe in the ideal of the student-athlete—if you think it’s something other than a cynical narrative-launder of a billion-dollar entertainment industry masquerading as a wing of higher education—you are going to like the story of John Urschel in college a lot. Urschel was taking second-semester calculus with Vadim Kaloshin, a young Russian professor Penn State had recently pried away from Caltech. One day, about a month into the term, Kaloshin got a note from the athletic department about Urschel, a standard team check-in to make sure Urschel’s grades were OK.



They were better than OK. Urschel had 100 percent in everything.



But there was one more issue. Urschel was a Big Ten football player, with the schedule that entailed. He came to Kaloshin with a form to fill out; he needed permission to miss some Friday classes in order to travel with the team.



This is the moment where the possible tracks of Urschel’s life split and curve off from each other. Kaloshin might have just signed the form. Professors sign a lot of forms. Urschel was doing well in the course, and there was no chance he couldn’t keep up with the work. It was the simplest way.



But Kaloshin had a different idea. Urschel was doing so well in calculus—maybe he was ready for something harder. “Do you want to read a mathematics book?” Kaloshin asked him.



The book was a text in Kaloshin’s research specialty, chaos theory and dynamical systems. The subject, in some form, is as old as Newton. The universe operates according to known physical laws. So, in principle, if you know the precise location and velocity of all the chunks of matter in the solar system, you can predict the course of their motion—their dynamics—over the whole future course of time. But that “in principle” is hiding a lot. Any measurement we can make in the real world is inaccurate to some extent; and it turns out even tiny inaccuracies in our knowledge of the present blow up into total confusion about the future. That sounds like chaos, all right, but there’s math to it. We’ve developed ways to understand when and how and how badly that blow-up comes to be; we can be precise about imprecision!



Kaloshin told Urschel to read the first eight chapters of the book and then come back to him. That seemed like enough material to keep Urschel productively occupied for the last few months of the semester.



It took him two weeks.



Dynamical systems was an inspired choice for Urschel—and not only because it concerns massive bodies exerting force on one another. It starts out simple and textbooky. As Newton already knew, a system with only two planets behaves completely predictably; the planets orbit each other in perfect ellipses for all time. But when you make the seemingly modest change from two planets to three, you are suddenly face to face with one of the deepest mathematical challenges humans have encountered: the three-body problem. Three planets in motion can do just about anything! The planets can rotate placidly, or two can conspire to slingshot the third into space, or they can dance forever in a figure eight. No one alive understands the space of possibilities.



Mathematicians like me often teach math as if we know everything. The truth, which we reveal to our students too rarely, is that we know almost nothing. One step outside the lighted circle of the understood, and everything is unmapped wilderness. A lot of people find this unsettling. For Urschel, it was thrilling. “It was so hard,” he told me. “And it was a struggle that I really loved.”



The world thinks mathematicians are people for whom math is easy. That’s wrong. Sure, some kids, like Urschel, have little trouble with school math. But everyone who starts down the road to creating really new mathematics finds out what Urschel did: It’s a struggle. A prickly, sometimes lonely struggle whose rewards are uncertain and a long time coming. Mathematicians are the people who love that struggle.



Finding your true love is a challenge. And it’s sobering to think how easily Urschel could have missed it. If he hadn’t been an elite athlete, nobody would have been monitoring his academic progress month by month. He might have gone through college as he had high school, quietly excelling and thereby never drawing any special notice.

If he hadn’t been an elite athlete, nobody would have been monitoring his academic progress month by month. He might have gone through college as he had high school, quietly excelling and thereby never drawing any special notice.

We don’t have a system for making mathematicians, the way we do for football. Every high school in America has a football team, but not every high school has a math team. And math team, in any case, doesn’t capture math whole; it’s a game about math, not the thing itself.



Finding your way to mathematics is still old-fashioned and analog. It relies on the chance encounter between two people, or between a person and a book, or between a person and an idea. If you know mathematicians, you know some who seem to have been on the track since childhood, and others who are only in the business by a stroke of luck. Persi Diaconis was a kid magician who became friends with the mathematical popularizer Martin Gardner on the card trick invention circuit. Joan Birman was bored in by-the-numbers calculus in college, became an engineer, then found at her first job building microwave frequency meters that one of those boring problems was exactly what she needed to get the meter to exhibit the right response curve. Solomon Lefschetz had both his hands blown off in an industrial accident and had to find a career he could undertake without them. Rodrigo Bañuelos was a teenager working in a car wash, with only one year of formal education, when he met Juan Francisco Lara, a Ph.D. student at UCLA who got him into Pasadena City College, where he took eighth-grade algebra for the first time; now he’s a math professor at Purdue. Laura DeMarco was planning to be a math teacher until her sophomore year of college, when a law professor tipped her off that professors do research. She’d had no idea. Even the math professors? Yes, said her probability prof, and that’s where she started. June Huh, an unpublished poet trying to transition into science journalism, sat in on Heisuke Hironaka’s algebraic geometry class with the idea of writing a profile; instead, Huh dumped journalism and became an algebraic geometer.



We don’t know how to scale this process. There aren’t enough Vadim Koloshins to fan out to every high school and college, and the high schools and colleges that have that kind of mentor are disproportionately those that serve wealthy kids with educated parents. We are probably missing a lot of John Urschels.



Kaloshin suggested a research problem to Urschel, who dove into it with full force. At Tampa for the Outback Bowl, he spent every non-football moment reading math and emailing back and forth with his mentor, trying to nail down the finer points of instability in a three-body problem involving the Sun, Jupiter, and an asteroid. That problem eventually became his first research paper, published in Celestial Mechanics and Dynamical Astronomy. (It’s probably in that back issue on your night table you keep meaning to get to.) “It was quite astonishing,” Kaloshin told me, that someone who’d only had a semester of college math could carry out this level of original work.



Urschel came to Penn State a football player who was good at math. From that moment, he was a mathematician who was good at football.



3. “I’m awful at doing things I don’t want to do.”



The list of strengths on John Urschel’s NFL scouting report leads off with: “Highly intelligent—will be successful with or without football.”



He laughs when I read it to him. “That’s a good thing to say for me in life, it’s not a good thing you want on your scouting report.”



It seems crazy to downgrade a player for having the potential to succeed outside professional football. Playing in the NFL is a dream for millions. Only a few have a scrap of a chance. It’s not something you do because you have no other choice! For Urschel, though, math was the dream, and football was a job. A very well-compensated job televised around the world, but a job.



John Urschel runs a drill at the NFL football scouting combine in 2014. AP Photo/Michael Conroy

The Baltimore Ravens, who signed Urschel in 2014, did their best to honor his mathematical dream. He enrolled in MIT’s Ph.D. program, and the team issued press releases touting his off-season coursework. For Christmas, quarterback Joe Flacco gave Urschel a box of Hagoromo Fulltouch chalk, a discontinued Japanese brand prized by mathematicians around the world.



But the NFL couldn’t offer what he’d had as a student-athlete at Penn State, where learning and playing were both officially part of his life. Urschel allows that he knows lots of players whose educational life was treated as a distraction, and that what rises into public awareness about academic misconduct in big-time college sports is “a small percentage” of the titanic cheatberg underneath the water. He’s not naïve about college football. But he believes in it.



“There’s something really enjoyable about college football that I really didn’t get out of pro football,” he says. “There’s something to be said about being this team, this unit, you go to school together, you live together, you hang out all day, you eat together… When you go out and you play on Saturdays, you’re representing your teammates, and you’re also representing the university.”



That communal feeling just wasn’t there in Baltimore. “In the NFL, you’re playing for a paycheck,” he says. “It feels somewhat transient … The moment I’m not producing to a level that they like, they fire me.”



And there was always the question about his brain. He had a lot planned for it. Did it make sense for him to slam it, dozens of times every Sunday, against other bodies of approximately the same outlandish size and strength as Urschel’s own? After 49ers rookie linebacker Chris Borland unexpectedly retired after the 2014-15 season, citing concerns about his long-term neural soundness, Urschel wrote an article called “Why I Still Play Football” for the Players’ Tribune:



“I play because I love the game,” Urschel wrote. “I love hitting people… This is a feeling I’m (for lack of a better word) addicted to, and I’m hard-pressed to find anywhere else. My teammates, friends and family can attest to this: When I go too long without physical contact I’m not a pleasant person to be around.”



So where did that feeling go? How does an ex-football player, who no longer bashes his head against anything more tangible than a partial differential equation, satisfy the jones to hit?



“Wanna know the craziest thing?” he tells me. “It’s gone. It’s gone. There’s something to be said for getting older.” In the athletic prime of his 20s, Urschel says, he’s gone mild. “I was such an aggressive, pent-up kid. Even in college I had this aggression about me… But every year, just a little less, a little less.” He doesn’t love hitting like he used to. “Not that I dislike hitting,” he quickly points out.



But it’s true: he doesn’t seem like a guy who wants to crush you. He seems like a kindly young sweater dad. Which is in fact what he’s just become. Urschel met his fiancée, the writer Louisa Thomas, when she was writing a profile of him for Grantland. She lived in Los Angeles, and he was in Baltimore, but they started dating almost immediately. The profile got shelved. They moved in together after seeing each other just four times. Their first child, a daughter, was born in December.



“I’m not going to be an achievement parent,” he tells me. “I’m gonna give her every opportunity, and make sure she has more opportunities than I did, but at the same time I’m also not going to push on anything… I don’t have any preconceived notions of what she’s going to do, what she’s going to be, or what she’s going to accomplish.”



(Internet Research Department Report: Urschel and Thomas’s Amazon baby wish list included Eric Carle’s My Very First Book of Numbers, a game called “Math For Love Prime Climb,” a kids’ chess set, and Spectrum math workbooks from Pre-K all the way through eighth grade.)



The more I talk to Urschel, the less strange it seems that he’d walk away from big money and national fame in the NFL. It feels like an almost obvious choice. He just stopped liking football as much as he liked math. “This is one of my worst attributes,” he tells me. “I’m awful at doing things I don’t want to do.”



4. “What do I have in my toolbox?”



What Urschel wants to do, lately, is think about graph theory. Bear with me, sports fans!



A “graph,” in modern math language, isn’t an Excel chart or a high-school blackboard sketch; it’s more like what in plain English we might call a network. It’s a collection of objects and a collection of connections between the objects. This might sound abstract—but that’s a good thing. The objects might be social media accounts, and the connections links; or the objects might be strains of flu, and the connections genetic relationships; or the objects might be neurons and the connections synapses. The abstraction of a “graph” allows us to use the same mathematical language and techniques to talk about all these entities, just as Newton’s theory of universal gravitation revealed fruit falling to earth and moons orbiting Jupiter as different tellings of the same mathematical story.



John Urschel lectures at the National Museum of Mathematics (image via YouTube)

In the 19th Century, applied mathematicians studied the kind of problems Urschel worked on in college: bodies in motion through space. In the 21st, we think just as much about information in motion through networks. The forces at play are stranger and less deterministic than good old gravity. And we’re still waiting for that all-explaining apple to fall.



One thing graph theorists do is try to understand large-scale features of a network, given fine-grained information about the objects and the connections between them. If you have a list of every website on earth, and all the links between them, can you correctly figure out which sites are the most important and useful? (A breakthrough on this problem, back in the 1990s, was the launch point for an algorithm called BackRub, eventually renamed Google.) If you can measure the neurons in the brain and the connections between them, can you tell the difference between a healthy brain and a sick one—or one that’s not sick yet, but will be?



Or you can go the other way: if you can observe the outward behavior of a network, can you make good inferences about its internal structure? That’s what Urschel’s most recent work is about. Suppose you’re an Internet merchant and you have a database of every customer and what they bought. From this information, you might like to understand the relationships between items. Maybe there are certain pairs of items which tend to repel each other; a customer who buys one tends not to buy the other. You, the merchant, would want understand which pairs of items are connected to each other in this way; in other words, you want to know the graph that describes the items and their interrelations. Urschel is part of a team at MIT working on just this kind of problem—can you detect the underlying graph structure? How accurately? How fast?



Urschel works closely with MIT professors who specialize in optimization and machine learning, the mathematical fields driving progress in the new economy. But he’s not interested in leaving academia for the richer and faster world of tech. “I like having the freedom to think about whatever interests me. And the moment something stops interesting me, I leave it.” Applied math in its contemporary form fits Urschel’s temperament exactly. The world conveys problems in endless supply. The problems are connected with each other but each has its own personality and demands new insights. There are few better jobs to have if you don’t like being bored.



“Some people, they say, ‘Hey, I’ve got this hammer, this is my tool,’ and every single problem they see they go around trying to hammer on it… But then there’s mathematicians who go around looking for problems, and they say, ‘What do I have in my toolbox, do I have anything that works? No?’ Then they go to the hardware store, start looking for something someone’s made, and if no, they go to the shop and they start making a tool of their own. I’ve encountered both types of mathematicians and I like the latter type much more.”

5. “There’s not many of me”



I haven’t mentioned yet that John Urschel is African-American. Urschel hasn’t mentioned it yet either. He rarely does. Urschel gave a speech at the annual meeting of the National Council of Teachers of Mathematics last year. After the talk, an audience member started asking a question:



“Demographically, you’re going to represent less than 1 percent of the people who receive Ph.D’s…”



Urschel breaks in: “Because I’m a football player, or….?” Big laughs from the crowd.



“I don’t like talking about it,” Urschel tells me, when I ask him about moving from a majority-Black workplace to a field where Black people remain scarce. (The most recent stats put the figure for black Ph.D.’s at about 1.5 percent.)



Then he talks about it. For a guy who doesn’t like talking about it, he has kind of a lot to say. So I just listen while he ranges on a while.



“I actually don’t like thinking about it too much and I don’t like acknowledging it,” he says. “Yes. I realize this. I’m black. I’m a mathematician. There’s not many of me…”



“I’m very aware of the fact that if you look up the most famous black mathematicians in history, the majority of them are famous for being black… The first black person to get a Ph.D. in math, the first black person to get a Ph.D. in math from Yale, the first black woman to get a Ph.D. in math… These are important people, and these are important things. You need pioneering people. These are people who have pioneered so people like me don’t have to be the first black person to get a Ph.D. from MIT in math, which I’m certain I’m not. I don’t know. I’m pretty certain I’m not. I’ve got to not be. Though I don’t actually know.”



He’s not the first. But the list isn’t long. The most famous graduate is probably Arlie Petters, a mathematical physicist, who got his Ph.D. from MIT in 1991 and is now a professor and dean at Duke. The earliest black mathematician I could find with a Ph.D. from MIT is Clement McCalla, who finished there in 1973 and had a long career as an applied mathematician at AT&T. (Update: after this story was published, Clement McCalla wrote to say that Earl Brown had earned a math Ph.D. from MIT even earlier, in 1968.) Overall, of the 1,921 Ph.D.s awarded in math by US universities in 2016, only 29 were earned by African-Americans.



Urschel says he wants to be a mathematician whose existence and visibility sends a positive message to students who don’t look like the majority of their professors. Playing in the NFL isn’t the most traditional way for a mathematician to get visibility, but it’s worked; according to the site TrueSciPhi, Urschel is the number four mathematician on Twitter, with just over 50,000 followers. So a lot of people who don’t ordinarily follow math are seeing jokes about the Banach-Tarski theorem and photos of his baby cuddling a copy of Introduction to Algorithms. His latest public project is a YouTube series for Marvel on the science of superpowers. Urschel has become an ambassador for the profession, which badly needs one. Most Americans don’t know a mathematician is something you can even still be, let alone that a mathematician might be a young guy with a sharp wit on social media who could pick you up with one hand and throw you across the room.



Mathematicians are famously prone to “impostor syndrome,” that nagging sense that you don’t really belong, that the friendly faces all around are just humoring you, that all your hard work would be a triviality for the true champions of the field. Urschel sees it in the students he teaches at MIT: “Self doubt—am I good enough?—just constantly too much stress.”



There’s no such thing as impostor syndrome in football, he tells me. “Everyone in college football thinks they’re NFL-bound,” he tells me. “If you have that personality of ‘what am I doing here?’, you’ve gotten weeded out a while ago.”



Urschel gives you the sense he’s a guy who may never in his life have asked “What am I doing here?” Or maybe football pounded that question out of him. He conveys a sense of being exactly where he’s supposed to be, so unusual for someone his age that I thought at first he had to be faking it. Over the course of our talk I come to see that it’s real.



Math, like football, is competitive. But, as Urschel told the Notices of the American Mathematical Society, the real competition isn’t with other mathematicians: it’s “competitive with the problem, where it’s me vs. the unknown.” Our ignorance about the universe is on one side of the ball; the whole worldwide network of mathematicians is on the other, pressing forward as a unit. Back on a college campus, back in the math department, Urschel is part of a team again, one that’s been playing for thousands of years and shows no signs of disbanding soon.



“I think I’m actually leading my best life,” he says. “I get up in the morning and I ask myself, what would I rather be doing? And the answer is nothing. There’s not a single thing in the world I could say I’d rather be doing. I want for nothing. Which is a beautiful, beautiful thing.”

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Thanks to Alex Fine and the students at the Baltimore Academy of Illustration for working on art for the featured image for this post. More illustrations here.