Starting Tuesday, University of Minnesota Professor Vic Reiner will teach an introductory course on algebraic geometry for ten to twelve high school students. Rather than attempt to develop deep theory, Reiner has chosen a computational approach to the subject. The students will become familiar with the mathematics software package MAPLE, as well as learning some of the algorithms that MAPLE uses. In order to have the proper computational facilities, the class will meet at the Geometry Center.
Reiner is a Combinatorist, so I asked him how he had decided on algebraic geometry as the topic of his course. He explained that he chose the topic because of a well written new textbook on the subject. The book, "Ideals, varieties and algorithms: an introduction to computational algebraic geometry and commutative algebra," by David Cox, John Little, and Donal O'Shea, was just published in 1992. It combines three courses, developed independently by the authors. Reiner first saw the book at an NSF Regional Geometry Institute at Amherst. He likes it particularly well for this course because it manages to develop the basic concepts of algebraic geometry with a minimum prerequisite knowledge. The only necessary previous mathematics background is linear algebra. The book discusses the subjects that one cares about in algebraic geometry, namely manipulation of polynomials and basic ideas of varieties, but it does not get into deep theoretical material; the authors choose instead to show practical algorithms. As quickly as the second chapter, the book introduces Groebner bases, the underlying idea in numerical algorithms for finding roots of polynomials.
Here are a few of the highlights that Reiner hopes to cover in the class. To begin they will learn some basic concepts. For example, they have not had any complex analysis. However, using a heuristic topological argument involving winding numbers, Reiner plans to justify the fundamental theorem of algebra, thereby motivating use of complex rather than real spaces. He will also discuss basic ideas of fields.
The students will learn to numerically solve a system of polynomial equations with MAPLE and graph the answer with graphics packages from the Geometry Center. In order to remove the mystique as to how MAPLE works, Reiner will break down the algorithms into small steps. He also plans to discuss Pythagorean triples and Fermat's Last Theorem.
Reiner's course is part of a larger program called the University of Minnesota Talented Youth Mathematics Program, abbreviated UMTYMP. UMTYMP includes an accelerated high school math curriculum, followed by a calculus sequence. The first three years of calculus are in depth versions of the same classes usually taught to calculus students, although taught differently than at a college, as UMTYMP pioneers experimental teaching methods and curricula. The fourth year of calculus is actually a topics class, the subject of which varies from year to year. Last year the class covered probablility and combinatorics. Next year will be differential equations and dynamical systems. This will be the first year of the algebraic geometry class. Reiner says, "These students are very good. The students could handle this class at Mt. Holyoke, Amherst, and Holy Cross, the places where the courses were first taught by Cox, Little, and O'Shea; I'm sure these high school students will have no trouble."