Computational Commutative Algebra
This is a graduate level course on computational commutative algebra. We are going to work on finitely generated graded modules over polynomial rings. The topics include how to compute their combinatorial and homological invariants, e.g. Groebner bases, Hilbert series and betti diagrams. During the course, we are going to work through a lot of examples together via computer algebra system Macaulay2.
Lecturer
Date
3rd March ~ 29th May, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Friday | 15:20 - 16:55 | A3-3-201 | ZOOM 06 | 537 192 5549 | BIMSA |
Prerequisite
Basic knowledge on rings, ideals, varieties and their correspondence should be sufficient. I am going to review them in the first lecture anyway.
Syllabus
1. Preliminaries: projective spaces, graded rings and modules, chain complexes and homologies.
2. Free resolutions, Betti diagrams and Hilbert functions.
3. Monomial ideals and initial ideals
4. Groebner bases
5. The geometry of syzygies
6. Other related topics
2. Free resolutions, Betti diagrams and Hilbert functions.
3. Monomial ideals and initial ideals
4. Groebner bases
5. The geometry of syzygies
6. Other related topics
Reference
1) I. Peeva: Graded Syzygies
2) B. Hasset: Introduction to Algebraic Geometry
3) V. Ene and J. Herzog: Groebner basis in Commutative Algebra
4) D. Eisenbud: Commutative Algebra with a View toward algebraic geometry
5) E. Miller and B. Sturmfels: Combinatorial Commutative Algebra
2) B. Hasset: Introduction to Algebraic Geometry
3) V. Ene and J. Herzog: Groebner basis in Commutative Algebra
4) D. Eisenbud: Commutative Algebra with a View toward algebraic geometry
5) E. Miller and B. Sturmfels: Combinatorial Commutative Algebra
Audience
Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Beihui Yuan gained her Ph.D. degree from Cornell University in 2021. She has joined BIMSA in 2023. Her current research interests include application of commutative algebra in pure and applied mathematics problems.