The Mathematical Association of America has selected Dennis DeTurck, Herman Gluck, Daniel Pomerleano, and David Shea Vela-Vick as the winners of the 2012 Chauvenet Prize for their article "The Four Vertex Theorem and Its Converse," Notices of the American Mathematical Society, 54 (2007), no. 2, 192-207. Full citations and biographical information for each winner is available below.
The Chauvenet Prize is awarded to the author of an outstanding expository article on a mathematical topic. First awarded in 1925, the prize is named for William Chauvenet, a professor of mathematics at the United States Naval Academy. It was established through a gift in 1925 from J. L. Coolidge, then MAA President. Winners of the Chauvenet Prize are among the most distinguished of mathematical expositors. Read more about the award.
This award will be presented during the Joint Prize Session on Thursday, January 5, 2012, at the 2012 Joint Mathematics Meetings in Boston, Massachusetts.
The four vertex theorem is a beautiful result in global differential geometry. It says that any smooth simple closed curve in the plane must have at least four "vertices"—local extrema for the curvature function. The vertices of a (noncircular) ellipse, for example, are located where it meets its major and minor axes. The four vertex theorem was proved for convex curves in 1909 by Syamadas Mukhopadhyaya and for general curves in 1912 by Adolf Kneser.
Remarkably, the converse of the four vertex theorem is also true. Any continuous real-valued function on the circle with at least two local maxima and two local minima is the curvature function for some simple closed curve in the plane. The converse was proved for positive functions in 1971 by Herman Gluck and for arbitrary functions in 1997 by Björn Dahlberg.
The authors of this excellent expository article—Dennis DeTurck, Herman Gluck, Daniel Pomerleano, and David Shea Vela-Vick—sketch Robert Osserman’s 1985 proof of the four vertex theorem and Dahlberg’s proof of the converse. The article ends with some generalizations of the four vertex theorem and biographical sketches of Mukhopadhyaya, Kneser, and Dahlberg.
This carefully-crafted survey has enough mathematical details to give the reader a sense of the proofs, but not so many to obscure the big picture. Experts and non-experts alike are sure to enjoy and understand this well-written and well-illustrated article. It is also a wonderful tribute to Björn Dahlberg, who passed away in early 1998, with his unpublished proof of the full converse still on his desk.
Dennis DeTurck received his Ph.D. from the University of Pennsylvania (Penn) in 1980, and after a stint at NYU returned to Penn's Mathematics Department. He is now the Dean of the College at Penn and lives with about 500 new friends on its campus in Riepe College House, for whom he bakes innumerable cookies. He can't wait to see what he and his twenty-something sons, Greg and Gary, will be when they're all grown up.
Herman Gluck was born in New York City, brought up in the Bronx, and attended Bronx High School of Science, where he co-captained the math and tennis teams, and found Doris, the love of his life. Then came NYU, with afternoon sojourns to Columbia and evening sessions at Courant, followed by marriage, graduate school at Princeton (where he worked with Ralph Fox in knot theory), and the birth of their son, Mark. This was followed by a postdoc fellowship year at Berkeley and IAS, four years at Harvard, and the birth of their daughter, Robi. Forty-five years later and counting Herman is at Penn, surrounded by fabulous colleagues, students and friends, as well as son-in-law Steve and granddaughters, Mandy and Kamila. Rich to bursting life includes tennis competition and voice lessons.
Daniel Pomerleano got his B.A. at the University of Pennsylvania in 2007 and is finishing up his Ph.D. at UC Berkeley studying mathematical physics. He enjoys playing chess and travelling to and living in new destinations and looks forward to the continuation of his mathematical journey.
Shea Vela-Vick was born and raised in Albuquerque, New Mexico. He attended Rice University, where he received his B.A. in mathematics in 2004. It was there that he met his future wife, Monica. From Houston, Shea moved to Philadelphia to attend graduate school at the University of Pennsylvania. In 2009, he received his Ph.D. under the supervision of John B. Etnyre. He commutes to NYC where he is completing his third year as an NSF Postdoctoral Fellow at Columbia University. By day, Shea works on low-dimensional topology/geometry. By night, he changes poopie diapers for his newborn, Austin Lucas.