December 10, 2003

High-Powered Mathematician
Prefers Low-Tech Approach

Changfeng Gui, professor of mathematics, was honored earlier this year with one of the highest prizes in mathematics and was called "one of the leading researchers of his generation in the field of nonlinear partial differential equations."

Professor Changfeng Gui has received the PIMS Research Prize, one of the highest awards in mathematics. Photo by Dollie Harvey

Yet a visitor to his office in the Gant Science Complex is immediately struck by a barrage of yellow sticky notes, attached at knee-level to the back of his desk. Is this evidence of an attempt to cope with the flow of ideas by dashing off notes as new notions occur?

The box of crayons on top of the desk answers the question: Gui's sons, ages 5 and 7, have been visiting their father at work. The older son, Thomas, a second-grader, likes to solve math problems. The younger one, James, likes to imitate. Hence the yellow notes.

Gui also adopts a low-tech approach to his work. He prefers to think out at his desk problems that cannot be tackled even by the most powerful computers. His computer, shoved back in a corner, is something he uses for communications, he says.

His approach has been highly successful. Earlier this year, he was awarded the PIMS Research Prize, a prestigious award from the Pacific Institute for Mathematical Sciences, and was cited for exceptional work on the most difficult problems of mathematics.

Gui's main interest is nonlinear partial differential equations that can describe rates of change, such as population changes. Equations of this type are used by engineers, biologists, economists, and others. They can be used in weather prediction, the design of the shape of an airplane, or stock option calculations.

If heating a room with a radiator is described by a partial differential equation, adding to the problem a thermostat set at 70 degrees contributes a more complex, nonlinear element, Gui explains.

While the details of what he does may be difficult for a non-mathematic ian to grasp, Gui is not fazed by the most basic questions.

"I think most people can learn math better," he says, "if they're willing to learn and break the mental barrier."

For that reason, he likes teaching the less advanced calculus classes - the 112-114 series - for which students give him excellent teaching evaluations. "I'm more patient with them - I work more closely with them, and ask them to do the problem in class as a group," he says.

Solving problems in a group corresponds to the way Gui learned math while growing up in China. There, students live and study together in close quarters, usually eight to a small room.

Gui did not encounter partial differential equations until he was an undergraduate student at Beijing University. He had many interests in math (his mother was a math teacher), but he wasn't at first interested in the field that would become his specialty.

He earned a master's degree specializing in algebra at Bejing, then came to the United States to study for a Ph.D. at the University of Minnesota, which has a strong program in partial differential equation research. There he began to work on the Lane-Emden equation, named for two physicists, and that became the basis of his thesis.

Eventually, his work led him to research on the Gierer-Meinhardt system, two interacting equations that are used to study pattern formation in biology, such as changes in the color patterns of fish. The math he studies is more theoretical than computational, Gui says, but it can be applied by scientists and others.

One of his research achievements was to prove, in 1998, with collaborator Nassif Ghoussoub of the University of British Columbia, the de Giorgi Conjecture, which makes it possible to understand in depth what happens in material phase transitions, such as when water turns to ice.

The conjecture - the term for a problem that can be stated but not solved without new techniques - was raised in 1978 by an Italian mathematician, Ennio de Giorgi. No one had solved it until Gui and Ghoussoub published their paper 20 years later.

The conclusion, a neat classification of all solutions to an important nonlinear equation, took him five years to work out, Gui says. In essence, what he proved for the case of two dimensions was what de Giorgi hoped - that the obvious solution is the only one possible. Gui continues to work on proving the conjecture for more than two dimensions.

He also has been cited for work he is doing with a math department colleague, Richard Bass, on the Gibbons conjecture, work that was called "groundbreaking, important, and quite difficult" by PIMS in describing Gui's contributions to mathematics research.

Gui came to UConn in 1999 from the University of British Columbia, where he was an associate professor. The same year, he won the AndrŽ-Aisensdadt Prize, awarded to top young mathematical researchers in Canada. He says he was attracted to UConn because of the strength of the math department's partial differential equation group. The group includes Bass, and Professors Yung Choi, Israel Koltracht, and Joseph McKenna, as well as Gui.

Partial differential equation, or pde, research is a very active, "hot" area of mathematics in general right now, says Charles Vinsonhaler, professor and former math department head. Problems that previously were not possible to solve, or which took a long time to solve, now can be looked at on the computer, he says.

Or, theoretically, at a desk covered with yellow stickies.