Computational Algebraic Geometry

Math 7970

Summer 2024


Meeting days/times and credit: Lectures Monday and Wednesday, 930-1045. Five weeks, first class July 1. Room: Parker Hall 246. Office hour after class Monday and Wednesday 11-12. Credit variable, grade determined by HW, make a decent effort and you'll get an A.

Course Description: Algebraic Geometry is a beautiful and active area of modern mathematics. Recent advances in computing and algorithms have made it possible to calculate many abstract invariants that were previously out of reach. We will cover one chapter per lecture, with student run problem sessions on Tuesday and Thursday. If you spend a couple hours doing the homework (there are generally only 5-6 problems per chapter), you'll get the most benefit out of the class.

Schedule by week:
Lecture 1: Basics of Commutative Algebra
Lecture 2: Projective Space and Graded Objects
Lecture 3: Free Resolutions and Regular Sequences
Lecture 4: Gröbner Bases and the Buchberger Algorithm
Lecture 5: Combinatorics, Topology and the Stanley–Reisner Ring
Lecture 6: Categories and Functors: Localization, Hom, and Tensor Product
Lecture 7: Geometry of Points and the Hilbert Function
Lecture 8: Snake Lemma, Derived Functors, Tor and Ext
Lecture 9: Curves, Sheaves, and Cohomology
Lecture 10: Projective Dimension, Cohen–Macaulay Modules, Upper Bound Theorem

Grading: Your grade will be determined by homework scores; you'll get out of the class what you put in.

Academic Integrity: I encourage group work on the homework problems. This does not include copying each others solutions.

Textbook: Computational Algebraic Geometry. However, for impecunious students, there is an online draft of the published version.

Updated 7/1/24 (hks).