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The Four Vertex Theorem and Its Converse

Award: Chauvenet Prize

Year of Award: 2011

Publication Information: Notices Amer. Math. Soc, 54(2007), no. 2, 192-207.

Summary: 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. 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.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.

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About the Authors (From Prizes and Awards, Joint Mathematics Meetings 2012)

Dennis DeTurck received his Ph.D. from 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. A 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.

MSC Codes: 
Dennis DeTurck (University of Pennsylvania) and Herman Gluck (University of Pennsylvania) and Daniel Pomerleano (University of California Berkeley) and Shea Vela-Vick (Columbia University)
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Publication Date: 
Thursday, January 24, 2013
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A discussion of the proofs of the four vertex theorem in global differential geometry and its converse