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The Joy of Solving Equations

If one were asked to use just one word to describe Bernd Sturmfels' lecture at the MAA Carriage House on May 17, "colorful" might come to mind. Sturmfels, a mathematician at the University of California, Berkeley, delighted a full house with his illuminating discussion of Gröbner bases, using color-coded examples filled with complex brown polynomials and much simpler blue polynomials.

Born in Germany, Sturmfels received his Ph.D. from the University of Washington in 1987 and has held positions at Cornell University, New York University, as well as at Berkeley.  He has received numerous honors, including the MAA's Lester R. Ford Award for expository writing (1999) and designation as a George Pólya Lecturer.

Sturmfels described a Gröbner basis as a set of multivariate polynomials that has desirable algorithmic properties.  He emphasized that "Gröbner bases give a method of transforming a feasible solution using local moves into a global optimum." He noted that Gröbner bases are fundamental in algebra and that they have many applications, including optimization, coding, robotics, and statistics.

A part of his discussion centered on the idea of what to do with the change in your pocket. "You should take the change in your pocket and optimize that portfolio!" he declared. Using Gröbner bases, Sturmfels showed the crowd a "simple" way to figure out the smallest number of coins that it takes to make $1.17. He noted that a Gröbner basis could be used to optimize any type of currency, including galleons, sickles, and knuts, the money used in the fictional Harry Potter books.

Sturmfels' entertaining presentation drew laughter from the crowd a number of times. He described, for example, how the procedure came to be known as a Gröbner basis. He explained that a Gröbner basis can be computed by a method that was introduced in Bruno Buchberger's 1965 dissertation. "And, like any good student, Buchberger named the method after his advisor, Wolfgang Gröbner," Sturmfels noted. On the other hand, he explained, the algorithm used for computing Gröbner bases now carries Buchberger's name.

The evening came to a close with some questions from the audience and a hearty round of applause for Sturmfels and his remarkably lucid presentation of Gröbner bases and their many applications. And the audience left with a better appreciation of pocket change.—R. Miller


An interview with Bernd Sturmfels appeared in the January 2008 issue of MAA FOCUS.

 

This MAA Distinguished Lecture was funded by the National Security Agency.

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