This course provides a hands-on introduction to applied algebraic geometry, emphasizing the interplay between commutative and homological algebra, geometry, and combinatorics. The syllabus includes a variety of tools and methods that are valuable for computational applications involving the transformation of complex objects into a sequence of simpler components. The geometric setting will usually be projective space, graded rings and modules. Key topics covered include the dimension and Hilbert polynomials of a variety, free resolutions and regular sequences, Gröbner bases, syzygies, the Stanley-Reisner ring of a simplicial complex, and multivariate polynomial splines. Through practical examples and exercises using the computer algebra system Macaulay2, students will gain hands-on experience with computational algebra tools and the application of algebraic techniques to solve computational problems.
Cox, D. A., Little, J. B., O'Shea, D. (1998). Using Algebraic Geometry (Vol. 185). Springer.
Schenck, H. (2003). Computational Algebraic Geometry (London Mathematical Society Student Texts). Cambridge: Cambridge University Press.
Schenck, H. (2022). Algebraic Foundations for Applied Topology and Data Analysis. Springer International Publishing.