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Enumerative Algebraic Geometry of Conics

by Andrew Bashelor, Amy Ksir, and Will Traves

Award: Hasse

Year of Award: 2009

Publication Information: The American Mathematical Monthly, vol. 115, no. 8, October 2008, pp. 701–728.

Summary: Jacob Steiner asked this question in 1848: Given five conics in the plane, how many conics are tangent to all five? The authors of this article take this question and use it as a vehicle to take the readers on a tour of enumerative algebraic geometry. Beyond the answer to Steiner's problem (3,264), Bashelor, Ksir and Traves expose both the intuition and some of the complexities of the algebra involved.

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About the Authors: (From the Prizes and Awards booklet, MathFest 2009) 

Andrew Bashelor is a 2005 graduate of the United States Naval Academy from which he received his bachelor of science degree in mathematics (with Honors). During his senior year he participated with a select group of undergraduates in the Trident Scholar Program and his research over the course of the year served as the basis of this paper.

Will Traves first learned about the enumerative geometry of conics from Frank Sottile, when Will was a doctural student at the University of Toronto. He was later a Project NExT Fellow (brown dot) and now teaches at the United States Naval Academy.

Amy Ksir grew up in Laramie, WY where she had the good fortune to learn geometry from Dr. Mary Jane Cowles Wolfe. She earned her B.A. at Rice University and her Ph.D. at the University of Pennsylvania. After a postdoctoral fellowship at Stony Brook, during which time she was a Project NExT Fellow (gold dot), she joined the faculty at the United States Naval Academy. She was honored to give a series of lectures based on this article at the Program for Women in Mathematics at the Institute for Advanced Study in 2007.

Subject classification(s): Algebra and Number Theory | Geometry and Topology | Analytic Geometry | Conics
Publication Date: 
Wednesday, September 2, 2009