This is a junior-senior level course in algebraic geometry, including the necessary background material in abstract algebra. Algebraic geometry is the study of polynomial equations and their solution sets. For example, the equation x2 + y2 - 1 = 0 determines a circle in the (x,y) plane. When more variables and equations are involved, the solution set may be a complicated set consisting of points, curves, surfaces, and objects of higher dimension. Any set which can be defined by polynomial equations is called an algebraic variety.
Given a variety, one can consider all of the polynomials which are zero on the set. This collection of polynomials is called the ideal of the variety. The ideal is a purely algebraic object while the variety is a geometric one. The interplay between the algebra and the geometry is what makes the subject so interesting. Computer algorithms for analyzing ideals have been developed in the last several decades. Using these, one can study ideals and varieties far too complicated to be worked out by hand.
Necessary background knowledge for the course includes single-variable and multi-variable calculus and linear algebra. The necessary abstract algebra is developed along the way. This consists mainly of topics from commutative algebra, especially the theory of polynomial rings. Reading, understanding and writing proofs is an important part of the course. This class is among those that can be counted toward a math major's algebra requirement (part of "Column X") as described here .
Topics often include the Euclidean algorithm, examples of real and complex curves and surfaces, Hilbert basis theorem, Groebner bases and the Groebner basis algorithm, the Nullstellensatz, elimination theory, the ideal-variety correspondence and miscellaneous examples and applications.
To suggest some flavor of Math 5385, a student might learn ...