FAYETTEVILLE, Ark. - When geophysicists seek to find oil fields, they sometimes turn to mathematicians like Dmitry Khavinson for help.

Khavinson, who recently received renewed funding from the National Science Foundation for his research, studies complex analysis, a way of mathematically describing many processes in the physical world. Complex analysis, often used in the fields of hydrodynamics and engineering, involves finding the optimal solution to problems set in domains so large that the bigger portion is out of reach. Engineers and scientists bring these problems to mathematicians, who then attempt to derive mathematical problems—and solutions—to describe the scientific processes that take place.

Take the oil fields, for instance. In theory, the geophysicists could place electrodes in the earth and put a current between them to determine the resistance of the underground area. They can measure the signal and check the results to determine where an oil field may lie.

But this method will not always succeed, and mathematical models can show this variance in success and failure.

"Mathematically it boils down to finding the global behavior," Khavinson said. In the equations formed, Khavinson looks for singularities, or places where the solution "goes bad." These singularities describe where a process like finding an oil field might hit a snag.

Khavinson will examine important boundary functions, special functions, best approximations and zero values for solutions of different equations with applications in physics and applied mathematics.

Complex analysis, like that used in Khavinson’s work, also can be used to describe the movement of a bubble in a fluid between two plates. Equations generated through complex analysis can describe the shape of the bubble at any given time—a challenging mathematical problem, given that such a bubble can develop glove-like "fingers" and change its shape in other ways.

"That type of problem you can spend your life on," Khavinson said.

The three-year NSF grant will fund investigations with applications in mathematical physics, applied mathematics and numerical analysis and will involve both undergraduate and graduate students in research and outreach.