Problems in Commutative Algebra. Semester: Fall Instructor: Chris Brav Course description: The seminar will illustrate the general theory of algebraic geom- etry and commutative algebra by focusing on the special case of varieties over a field, using concrete techniques, particularly Groebner bases, resultants, and elimination theory. Prerequisites: Some familiarity with rings, ideals, and modules. Curriculum: Division algorithm for polynomials in many variables. Groebner bases. Hilbert's Basissatz Ideal membership problem. Computing solutions of polynomial systems of equations. Computing the closure of the image of a map of affine varieties (elimination theory). Resultants and elimination theory. Hilbert's Nullstellensatz Irreducible components. Dimension. Projective elimination theory. Textbooks: Cox, Little, O'Shea. Ideals, Varieties, and Algorithms. Springer. Hassett. Introduction to Algebraic Geometry. Cambridge Univ. Press.