PHILADELPHIA — A team of mathematicians from the University of Pennsylvania has been named winners of the 2012 Chauvenet Prize, given by the Mathematical Association of America to the author or authors of an outstanding expository article on a mathematical topic. The prize was awarded at the Joint Mathematics Meeting.
Dennis DeTurck, Herman Gluck, Daniel Pomerleano and David Shea Vela-Vick authored the paper, "The Four Vertex Theorem and Its Converse," which was published in Notices of the American Mathematical Society in 2007.
DeTurck is dean of the College of Arts and Sciences, the Robert A. Fox Leadership Professor in the School of Arts and Sciences and a professor of mathematics. Gluck is also a professor of mathematics at Penn. Pomerleano received his bachelor's degree from Penn in 2007 and is completing his graduate studies at the University of California, Berkeley and Vela-Vick received his Ph.D. from Penn in 2009 and is a National Science Foundation postdoctoral fellow at Columbia University.
The award announcement praised the paper as a "carefully crafted survey [with] enough mathematical details to give the reader a sense of the proofs, but not so many to obscure the big picture."
"As far as we know, this is the first time anyone has won the Chauvenet for work they've done while they were students," DeTurck said; Pomerleano contributed to the paper as an undergraduate and Vela-Vick as a graduate student.
The team wrote the paper with a broad audience in mind, aiming to appeal to mathematicians with varied backgrounds. The subject was the four vertex theorem, which deals with the curvature of curves in the plane. It states that a simple closed curve in a plane, other than a circle, must have at least four "vertices," or points where the curvature has a local maximum or local minimum.
Maximum curvature, DeTurck said, can be imagined as the point on a road at which a driver has to turn the steering wheel most sharply to stay on track. Minimum curvature is where the smallest turn of the wheel is needed.
In its original form, the concept was proved by Indian mathematician Syamadas Mukhopadhyaya more than 100 years ago. Others with an interest in geometry have since built upon the theorem, and in 1998, when prominent Swedish mathematician Bjorn Dahlberg died, he left on his desk a manuscript proving that the converse of the theorem was also true.
To compose the paper, DeTurck, Gluck, Pomerleano and Vela-Vick worked to distill the proofs down to their essence.
"The goal was we wanted to say this in as clear a way as we possibly could," DeTurck said. "It's just a matter of presenting it to each other over and over again until you realize the crux of the idea."
All four authors have previously been honored for their work in mathematics.
DeTurck has won various awards including the SAS Ira Abrams Award, the Lindback Award and the Mathematical Association of America's Haimo Award for Distinguished Teaching. Gluck has been honored with a National Academy of Sciences-National Research Council Fellowship at Berkeley and the Institute for Advanced Study, an Alfred P. Sloan Research Fellowship at Harvard University, a NATO Senior Fellowship in Science at Zurich and Amsterdam, a Guggenheim Fellowship at Penn and the University of Bonn and with the Lindback Award and Dean's Award for mentoring undergraduates at Penn.
Pomerleano has won the Waldemar J. Trijitzinsky Memorial Award of the American Mathematical Society. Vela-Vick has received the Dean's Award for Distinguished Teaching by a Graduate Student at Penn and an NSF Postdoctoral Fellowship at Columbia.