We can informally describe algebraic geometry as the geometry of curves and surfaces defined by polynomial equations. In this course the emphasis is on concrete computations with polynomials using computer packages and on the interplay between algebra and geometry. Abstract algebra will be presented as needed; the only algebra prerequisite is sophomore level linear algebra. The other required prerequisite is multivariable calculus -- Math 2263 or Math 2374 or equivalent.
More specifically, the course will be based on the book Ideals, Varieties and Algorithms,by Cox, Little, and O'Shea. In one semester we can cover the first four chapters. If we study a fairly minimal number of proofs, we'll also have time for part of either chapter 5 or chapter 8. The course should be accessible to students with a reasonably solid background from sophomore level mathematics. We'll start by introducing some basic stuff about affine varieties -- particularly curves in R² and surfaces and curves in R³. We'll mention briefly the parallel concepts in C² and C³. This leads to discussion of the algebraic concept of polynomial ideals. We will use suitable graphics software to help with visualization of implicitly defined curves. From time to time the class will meet in a math department computer lab, to learn how to use this and other software. Previous experience with Matlab, Maple, or Mathematica can be helpful, but is not required.
Our software tools also can be used for visualizing surfaces in R³. We'll have only a limited amount to time to work with that topic, but at least a brief introduction will be given.
The method of Groebner bases provides an effective method to determine whether a polynomial is an element of a given ideal. This can be viewed as the multivariable analogue of long division of polynomials in one variable. Among other things, it leads to an algorithm for solving systems of polynomial equations. This algorithm can be viewed as a far-reaching generalization of the method of Gaussian elimination from linear algebra.
Computations will be done by hand at first, but then we will learn how to use a computer algebra package -- Maple -- to study more interesting examples. We will then apply Groebner basis methods to study other topics, such as projections of varieties, finding the implicit equations of a variety given in parametric form, and finding the singular points of a variety.
After developing some proficiency with computational methods, we will apply them to develop some introductory parts of the theory of algebraic geometry. At the end of the semester, we'll have a short introduction to either projective algebraic geometry [chapter 8] or polynomial and rational functions on a variety [chapter 5].
In summary: at the beginning we will assume knowledge only of sophomore level linear algebra and polynomials in several variables, including partial derivatives. Treating the subject in a fairly concrete way and moving forward systematically from this starting point, we will reach a level where students can understand some of the theory in algebraic geometry. Appropriate instruction about the computer tools will be provided -- including online help pages developed for this course, starting in the Fall Semester of 2000. For a more introductory description of the Matlab scripts that we use for plotting implicitly defined curves, see the Summer 2000 REU project report by Rory Mulvaney. (In PDF format ... )
Info for math majors: Math 5385 is on the Algebra List for the
Upper Division
Math Requirements, actually being listed in the Column X portion of the
list.
For further information: Please
send me an e-mail, or call me
at the phone number listed below,
or
visit
the class homepage (
http://www.math.umn.edu/~roberts/math5385/).
Prof. Joel Roberts
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA
Office: 351 Vincent Hall
Phone: (612) 625-1076
Dept. FAX: (612) 626-2017
e-mail:
roberts@math.umn.edu
http://www.math.umn.edu/~roberts