Math 765 - Computational Commutative Algebra

✍ Scribed by Nathan Fieldsteel

• Year: 2019
• Tongue: English
• Leaves: 58
• Edition: version 2019-04-30
• Category: Library

✦ Table of Contents

January 9 - Monomial Orders and Multivariate Polynomial Division
Getting Started
Multivariate Polynomial Division
January 11 – Groebner Bases and Buchberger's Algorithm
Groebner Bases
Buchberger's Algorithm
January 14 – An Introduction to Macaulay2
Sample Macaulay2 Session
January 16 – Introduction to Affine Algebraic Geometry
Affine Varieties
Groebner Bases and Elimination Ideals
January 18 – First Macaulay2 Hands-On Session
Problem 1
Problem 2
Problem 3
January 23 - R-modules
January 25 - More on Graded Modules, Minimal Free Resolutions, Betti Numbers
January 28 - Betti Diagrams, The Koszul Complex for R/m, Proof of Hilbert Syzygy Theorem
January 30 - Projective Geometry, Hilbert Functions
February 1 - Worksheet 2
February 4
The Hilbert Function, Series, and Polynomial
February 6 - Algorithms for minimal free resolutions
More general terms orders:
February 8
Schreyer Resolutions and Schreyer Frames
February 11 - Example of Computing a Schreyer Frame
February 13 - Example of Computing a Schreyer Resolution
February 15 - An Algorithm for Computing a Schreyer Resolution
February 18
February 20 - Worksheet Day
February 22 - More on Hilbert Functions, Points in Projective Space
Geometric Information from the Hilbert Function
February 25 - Points in Projective Space, Regularity, Resolutions, Examples
Imposed Conditions
February 27 - Some background towards (stating and) proving the theorem
March 1 - Worksheet Day - Writing your own Macaulay2 functions.
March 4 - Regularity, Depth, and Local Cohomology
Local Cohomology
March 6
March 8
March 18
Completing the Proof from Last Time:
Applications of the theorem:
March 20
March 22: Loading Packages, Depth in Macaulay2
Problem 1
Problem 2
Week of March 25 - 29: Simplicial Complexes Worksheet
Problem 1
Problem 2
Problem 3
Problem 4
Week of April 1: Alexander Duality
Problem 1
Problem 2
Problem 3