Outline of Sottile's course (18 June 2007) ==================================================================== Week 1: Introduction to algebraic geometry and Groebner bases _________________________________________________________________________________ 23 July Examples Definition: - Polynomials - Affine varieties Examples (make a Maple worksheet) - zeroes of a univariate polynomial - plane algebraic curves - quadric hypersurfaces Mention parametric representations and define hypersurfaces - intersection of two plane curves - intersection of two quadrics - hyperboloid intersect a sphere Examples of non-varieties Ideal-Variety Correspondence Subsets of C[x_1,...,x_n] <====> Subsets of A^n Ideals of C[x_1,...,x_n] <====> Subsets of A^n Ideals of C[x_1,...,x_n] <====> Subvarieties of A^n Radical Ideals in C[x] <====> Subvarieties of A^n State and prove the Nullstellensatz from statement about maximal ideals of C[x_1, ... ,x_n]. _________________________________________________________________________________ 24 July Groebner Bases Motivation: The ideal membership problem. - Its solution in 1 variable. Warm-up: Monomial ideals - d=1 (relate to ideal membership in 1-variable) - d=2 staircase picture - Dickson's Lemma Main idea: Term orders - Give examples Groebner bases Corollary Hilbert Basis Theorem. Ideal membership test. Buchberger Algorithm _________________________________________________________________________________ 25 July Resultants and Bezout's theorem. _________________________________________________________________________________ 26 July Solving systems of POlynomial equations _________________________________________________________________________________ 27 July Structure of algebraic varieties Zariski topology Irreducible decomposition Dimension Functions, regular and rational Coordinate ring, local ring, smooth and Singular points ==================================================================== Week 2: Toric varieties and deformation techniques 30 July Projective varieties 31 July Toric Ideals 1 August Hilbert functions and degree 2 August Toric varieties Sparse polynomial systems Kouchnirenko's theorem 3 August Bernstein's theorem and Numerical homotopy