Math 8680: Topics in combinatorics
Combinatorial commutative algebra

Fall 2003

Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256, Telephone (with voice mail): 625-6682, E-mail: 
Classes: Mon-Wed-Fri 2:30-3:20pm, Vincent Hall 206
(but I will attempt to change the time after the first class
to allow attendance at Math 8660 Topics in Probability). 
Office hours: Wednesdays at 3:35 PM;
one can always arrange to meet me by appointment 
Course content: When studying commutative rings, there are several invariants one wishes to calculate, some of them homological. In general, these invariants may be hard to compute. However, for particularly nice classes of rings (such as quotients of polynomial rings by monomial ideals and semigroup rings), they can be more tractable, and more fun, due to connections with combinatorics and combinatorial topology.

This course will be about these kinds of rings, and their connections with combinatorics. Some of the topics we hope to address are:

  • Introduction to Groebner bases
  • Stanley-Reisner rings and simplicial complexes
  • Monomial ideals and their homological properties
  • Semigroup rings and polyhedral cones
  • Counting integer points in polytopes (Ehrhart polynomials)
  • If time permits, we may discuss some of these topics:
    -local cohomology
    -Grassmannian and flag varieties
Prerequisites: Graduate algebra (or some understanding of what polynomial rings, ideals and quotient rings are). Some exposure to simplicial homology and/or homological ideas (e.g. what an exact sequence is) would be helpful, but not absolutely required.  
Source materials
There is a text on the subject, in preparation:
Combinatorial commutative algebra, by E. Miller and B. Sturmfels.
A version is currently (Aug. 19, 2003) at the web-site of Prof. Ezra Miller.
I will eventually have Alpha Print distribute a version, once the course enrollment has stabilized.

Here are some other related and potentially useful texts, all on reserve in the Math Library:
Ideals, varieties and algorithms, by D. Cox, J. Little, and D. O'Shea
Commutative algebra with a view toward algebraic geometry, by D. Eisenbud
Combinatorics and commutative algebra, by R. Stanley
Groebner bases and convex polytopes, by B. Sturmfels
Cohen-Macaulay rings, by W. Bruns and J. Herzog

Course requirements and grading There will be no homeworks or exams. Registered students are expected to attend regularly, and must either
- hand in solutions to 5 homework problems from the text by the last day of class, or
- give a talk before the 12th week of the semester on a paper that I approve.
Students choosing to give a talk will be expected to meet with me at least once after I have approved the choice of subject, but before they give the talk, to discuss its content.
Back to Reiner's Homepage.