Seminar Math 295: Combinatorial Commutative Algebra.
MWF 1:30-2:20 in room 228E Physics
Christian Haase 211B Physics 660-2875 haase@math.duke.edu

$\underline{x^2y^2} + xy^2 - 3 xy + y + x + 1$

Subject: ``The last decade has seen a number of exciting developments at the intersection of commutative algebra with combinatorics. New methods have evolved out of an influx of ideas from such diverse areas as polyhedral geometry, theoretical physics, representation theory, homological algebra, symplectic geometry, graph theory, integer programming, symbolic computation, and statistics. The

purpose of this [course] is to provide a selfcontained introduction to some of the resulting combinatorial techniques for dealing with polynomial rings, semigroup rings, and determinantal rings. [The course] mainly [covers] combinatorially defined ideals and their quotients, with a focus on numerical invariants and resolutions, especially under gradings more refined than the standard integer grading.''
(From [1].)

Literature: We will largely follow

[1]

Miller, Sturmfels Combinatorial Commutative Algebra Springer GTM 227 (2005).
Currently available from http://www.math.umn.edu/~ezra/cca.html

Other standard references are

[2]

Stanley Combinatorics and Commutative Algebra Birkhäuser (1996).

[3]

Bruns, Herzog Cohen-Macaulay Rings CUP (1993).

Outline: Possible topics are

• 
  Monomial ideals (squarefree, cellular resolutions, Alexander duality, generic) Chapters 1,4-6.
• 
  Binomial ideals (semigroup rings, multigradings, resolutions, Ehrhart polynomials) Chapters 7,8,11,12.
  

Format: We will not rush through the material. The emphasis is on a hands-on understanding. One lecture per week by a student about a worked out example, software presentation, or similar.
Prerequisites: Curiosity, commitment, commutative algebra basics.