Graduate Student Profile - Craig Citro
Mathematics

If you’re looking for someone who understands Fermat’s Last Theorem, Fuchsian groups, or the Teichmüller flow, you might stop by the Rock Wall in the Wooden Center. Rock-climbing is a favorite hobby of Craig Citro, his fellow graduate students in mathematics, and their professors. "I’d guess that one in four people in the department climbs regularly and the other three at least have tried it," he says. "It's like problem solving with your body."

Rock climbing has a lot in common with work on the very abstract and obscure issues in advanced mathematics, Craig says. You might try a particular climb "twenty or thirty times and not be able to do it," he says, "then you'll shift the way your foot is positioned by 10 degrees and all of a sudden the climb is the easiest thing in the world."

Similarly, "I spend 29 days in a row beating my head against a wall trying to understand something in Number Theory and making zero or negative progress," he explains, "and then one day a bunch of things become clear and I make a lot of progress all at once. That one day is enough fun that I'm willing to come back and struggle again the next month."

While you may find people at the Wooden Wall who understand the issues in advanced mathematics, don't count on them being able to explain what they do. At least so far, although Number Theory is the oldest branch of mathematics, with 2,000 years of history, no one's written Number Theory for Dummies. Craig's girlfriend knows "the math words I use a lot," he says, "but she doesn't really know what I do," which is studying p-adic L-functions. "I'm trying to prove something about their value at a certain point," he says. An L-function is an analytic object. "A strong theme in modern mathematics is trying to draw connections between large, well-developed areas, for example, in Number Theory, connecting some algebraic object with an analytical object—say, an elliptical curve and a modular form.

In 19th-century Europe, mathematicians studied the symmetry of groups of geometric objects, developing a lot of information that didn't seem particularly useful at the time. Then, a century later, "quantum chemists had a meaningful reason to understand this because the symmetry of tetrahedrons or other shapes relates to what happens when different atoms come together to form molecules," Craig says. Some of today's mathematicians are looking at symmetry groups of polynomials in algebraic equations, and if the usefulness of the information is unclear at the moment, that doesn't mean there's no practical reason for pursuing this knowledge.

At the beginning of the 20th century, a German mathematician named David Hilbert challenged his colleagues to solve 23 mathematics problems, and nearly all were "put to bed in one way or another," Craig says. At the beginning of the 21st century, The Clay Mathematics Institute set out what it called Seven Millennium Problems, adding a new and very 21st-century spin. Some hard-working—and lucky—mathematician will win $1 million for the solution of each problem. As two of the problems are related to the behavior of L-functions, Craig says, "It's clear that L-functions are the right objects to be studying at the moment."

Craig's path to L-functions began in a typical manner: From the beginning of his school years, he always excelled in mathematics. In high school, a calculus teacher named Pete Lederberg gave him an elementary book on Number Theory—one he'd used himself in college. Craig "thought I wanted to do math, but I didn't know what that meant." He did, however, understand what video games were, and he and his friends were adept not only at playing but also at creating them. As a result, he went off to Georgia Tech as a computer science major and stuck with the subject after a transfer to Indiana University.

After a while, however, he decided that the topics in computer science "weren't of great interest to me," and when he cast about for another major, "math seemed a natural candidate." Changing majors with little more than a year to go, Craig had to rush to meet the requisites, and he arrived at UCLA under prepared. To do advanced mathematics requires a solid background, and "I came in with essentially none," he says. "I had to spend a lot of time catching up and filling in the gaps of my undergraduate education."

During that period, he took a reading course with Haruzo Hida, a highly regarded mathematician in Number Theory, who calls Craig "the most multitalented student I ever had," combining excellence as a mathematician, computer skills, and "outstanding communication and teaching ability. I expect him to become a superb teacher/researcher." Professor Hida became Craig's thesis adviser. For a dissertation in mathematics, "you settle on something that you'd like to believe is true, and your goal is to come as close to proving that as you possibly can," Craig says. The idea is to write a thesis that, with a little tweaking, becomes your first paper as a professional mathematician.

Craig's tools in this quest basically come down to pencil and paper and perhaps some coffee. An old joke says that mathematicians are machines that turn coffee into theorems, and Craig notes that if they're not at the Wooden Center Rock Wall, mathematicians can often be found in the café at Kerckhoff Hall. In other words, he can do his research just about anywhere, and his girlfriend knows that dinner may be briefly interrupted while he has an inspiration. "I keep a piece of paper in front of me and scribble something down," he says.

Toothpaste may also have something to do with his progress. "Sometimes I'll be working on something for several hours, and I'll finally decide about 4 a.m. that it's time to throw in the towel for the night," he says. "Then I'll be brushing my teeth, and it will hit me — this is what I was missing, and I go back to work." At least that happens once every 30 days or so.

Published in Fall 2007, Graduate Quarterly