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Algebraic Geometry for High School Students
Posted:
Sep 17, 1993 5:39 PM
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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."
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