### Graph bisections meet lines of scrimmage

You don’t usually read about spectral graph theory and professional football in the same sentence. Meet *John Urschel*.

Urschel was born in Winnipeg and moved with his family to Williamsville, New York when he was four years old. His father worked near Niagara-On-The-Lake, Ontario (my hometown, no joke), and he attended high school in Buffalo. He played football in college as part of the Penn State Nittany Lions. The NFL drafted him in 2014, and now at the age of 24 he plays as a guard in the Baltimore Ravens.

According to my brother-in-law Denis Gluck, and avid football fan who played the sport at university, Urschel is a *right guard* whose job is block for and to protect the quarterback (see RG in the figure below; C would be the quarterback position).

While at Penn State, Urschel completed his undergraduate degree in Mathematics, boasting a 4.0 GPA. In late January 2016, the 300 pound, 6 ft. 3 football player announced on Twitter that he was beginning his PhD at M.I.T.

Urschel is not the first NFL player who entered a mathematics doctoral program. Frank Ryan of the Cleveland Browns became the first pro football player in the league to hold a PhD in mathematics in 1964. Others in the NFL have pursued their PhD and other advanced degrees at various stages of their career. For example, Blaine Nye of the Dallas Cowboys received a PhD in Finance in 1981 after his retirement from the NFL in 1976.

Urschel is a published academic equipped with a Google Scholar page. He has a paper in *Journal of Computational Mathematics* named “A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians,” and an accepted article in the journal *Linear and Multilinear Algebra* on spectral approximations of graph bisections. That’s an impressive accomplishment for any new doctoral student, let alone one in the NFL.

### Aside on spectral graph theory

John Urschel’s published mathematical work focus on *spectral graph theory*, which uses uses algebraic techniques to study networks. You can think of a graph as a binary matrix, with a 1 in its i-j entry if there is an edge joining vertices i and j. Otherwise, the i-j entry is 0. See the figure below for an example with five vertices.

If you have taken a first-year linear algebra course at university, then you likely learned of *eigenvalues*. Using the adjacency matrix, we can naturally consider eigenvalues of graphs. The eigenvalues of a graph reveal many of its properties, ranging from how to draw a graph or how to partition it. They play a role in the famous PageRank algorithm used in the Google search engine.

Graph theorists don’t just study the adjacency matrix, but other related matrices like Laplacians. There is a rich literature (both books and papers) on Laplacians, which I won’t enter into here. Permit me to mention that my good friend and legendary mathematician Fan Chung Graham literally wrote the book on spectral graph theory.

### Love what you do

I mentioned to Fan once that she must work so hard to have achieved all her mathematical success. She responded simply: “I love what I do.” That is so true: when your work is your passion, you don’t view it as work.

When I heard about Urschel’s academic journey, I thought it was such an inspiring story, especially for the upcoming generation studying mathematics. Urschel’s successes cross many endeavors and help break down stereotypes of mathematicians. He clearly loves mathematics, has a talent for it, and that usually translates into success in graduate studies. By the way, to round out his interests, he is an avid chess player, and enjoys hitting people on the field!

Another bit of wisdom here is that you can excel in more than one thing in life, and if you work hard, you can achieve your goals. In 2015, I wrote a novel while training for a marathon (and finishing with a PB) all while publishing in mathematics journals and supervising my graduate students. *You define your own limits in life.*

Urschel references work-life balance on Twitter, and I love this tweet especially. The blank page is so tantalizing to mathematicians.

### Safety first

Concussions in professional sports such as football and hockey are a source of considerable controversy lately, with much media attention and even a movie *Concussion* starring Will Smith. Aware of the controversy, and in response to an NFL colleague quitting over concerns for his health, Urschel wrote a thoughtful essay on the topic called *Why I Still Play Football.*

I hope Mr. Urschel takes care of health the best he can when on the field. His mind is something he should strive to preserve. The average NFL career lasts 3.5 years, although some have stayed for 20+ years. Mathematical productivity can flourish for decades if you remain healthy and stay current.

### Super Bowl of the mind

I wish all the best to Mr. Urschel for the successful completion of his doctoral work in mathematics. As I describe in my blog aimed at newcomer PhD students, if he finishes his PhD, I guarantee it will be one of greatest accomplishments of his life. Think of it as a Super Bowl of the mind.

Higher mathematics requires the right attitude. Progress on your research can be slow, but it so gratifying when it finally comes. As with most intellectual pursuits, it’s not about the money. In John Urschel’s own words:

“I have a bright career ahead of me in mathematics. Beyond that, I have the means to make a good living and provide for my family, without playing football. I have no desire to try to accumulate $10 million in the bank; I already have more money in my bank account than I know what to do with. I drive a used hatchback Nissan Versa and live on less than $25k a year. It’s not because I’m frugal or trying to save for some big purchase, it’s because the things I love the most in this world (reading math, doing research, playing chess) are very, very inexpensive.”

Anthony Bonato