Instructor: Chris Francisco

Description: This course will provide an introduction to using the computer algebra system Macaulay 2 in commutative algebra and algebraic geometry and cover introductory material on Hilbert functions and free resolutions. My goal is to provide the basics by lecturing, give students an opportunity to explore some of these topics with the computer and make conjectures, and then discuss the results in class. The exact material we cover will depend on the background and interests of the class, but possible topics include Groebner bases, stable ideals, Gorenstein ideals, bounds on Betti numbers and the growth of Hilbert functions, generic initial ideals, connections to combinatorics via Stanley-Reisner theory, Hilbert functions and resolutions of special ideals arising in algebraic geometry, and rings of invariants.

I plan to discuss a number of open questions and encourage students to explore special cases of them. I will probably ask students to do a (small) project that uses Macaulay 2 to investigate something in their research or that they're learning about.

Prerequisites: A graduate course in algebra, including basic knowledge of modules and exact sequences, is enough. I'll help everyone get set up on Macaulay 2; no special computer knowledge is necessary. Feel free to talk to me if you're concerned about whether you have sufficient background.

Textbook: None required, but we will take some topics from Cox, Little, and O'Shea's "Ideals, Varieties, and Algorithms" and Hal Schenck's "Computational Algebraic Geometry."