Computational Commutative Algebra
This is a graduate level course on computational commutative algebra. The central theme in this course is the study of (finite) free resolutions, which provides a lot of invariants for graded modules. We are going to learn tools to study and computes free resolutions, as well as using free resolution as a tool to study geometry of projective varieties. During the course, we are going to work through a lot of examples together. The audience are strongly encouraged to generate their own examples and test their ideas with computer algebra systems, e.g. CoCoA, Singular and Macaulay2.
Lecturer
Date
27th February ~ 16th May, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 09:45 - 11:50 | A3-4-312 | ZOOM 01 | 928 682 9093 | BIMSA |
Prerequisite
A first course in commutative algebra and algebraic geometry. Basic knowledge on rings, ideals and varieties should be sufficient.
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) H. Schenck: Computational Algebraic Geometry
3) D. Eisenbud: The Geometry of Syzygies
4) D. Eisenbud: Commutative Algebra with a View toward algebraic geometry
5) E. Miller and B. Sturmfels: Combinatorial Commutative Algebra
2) H. Schenck: Computational Algebraic Geometry
3) D. Eisenbud: The Geometry of Syzygies
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.