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.

References: 

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.