Convinced that Marden’s Theorem deserved wider recognition, Dan Kalman ’74 spent the past few years examining gaps in the theorem and developing a straight-forward, understandable proof. The resulting paper, “An Elementary Proof of Marden’s Theorem,” was awarded the 2009 Lester R. Ford Award by the Mathematical Association of America (MAA).

Kalman said that during college, Marden's Theoreom struck him as “the most amazing combination of ideas.” It is a mathematical theorem that relates the roots of a polynomial p(x) and those of its derivative p´(x) in a complex plane.

The winning paper was published in The American Mathematical Monthly (vol. 115, no. 4, April 2008). The theorem describes a thoroughly unexpected geometric connection between the roots of a cubic polynomial p with complex coefficients and the roots of the polynomial’s derivative p’. To wit, if the roots of p are noncollinear points A, B, and C in the complex plane, the roots of p’ are the foci of the unique ellipse inscribed in triangle ABC and tangent to the sides at their midpoints.

“The extraordinary lucidity of Kalman’s argument combines with his historical scholarship on the theorem’s origins and the charm of this surprising result to make this article a model of mathematical exposition,” remarked the MAA.

A related paper, “The Most Marvelous Theorem in Mathematics,” with dynamic graphics, extended historical remarks, and detailed mathematical background, is available online. The online edition uses specially-designed MAA software to turn the article into an interactive document, with hypertext that allows readers access to additional details and insights. The hypertext also provides enough supplemental background information on the paper’s elementary mathematics topics that anyone with a command of first-year calculus can work through whatever details are unfamiliar or interesting.

The Lester R. Ford Awards are made to authors of expository articles published in The American Mathematical Monthly. They are named for Lester R. Ford Sr., a distinguished mathematician, editor of The American Mathematical Monthly, 1942-46, and past president of the Mathematical Association of America.

Kalman has been a member of the mathematics faculty at American University, in Washington, D.C. since 1993. Prior to that, he worked for eight years in the aerospace industry and taught at the University of Wisconsin, Green Bay. During the 1996-1997 academic year, he served as an associate executive director of the MAA.

Kalman has a B.S. from HMC and a Ph.D. from University of Wisconsin, Madison. A frequent contributor to all of the MAA journals, he has served on the editorial boards of both MAA book series and journals. He is the author of two MAA books, Elementary Mathematical Models and Uncommon Mathematical Excursions.

Information courtesy of Mary Schellinger, American University, and the MAA