Title: Combinatorics of Monomial Ideals.
Course Description:
Monomial ideals are a staple of combinatorial commutative algebra. The fundamental idea is that simplicial complexes can be viewed as (squarefree) monomial ideals, in such a way that combinatorics and topology of simplicial complexes translates into algebraic notions and invariants. This area enjoys many beautiful results, where abstract algebraic tools are used to prove seemingly unrelated enumerative statements.
I will cover the basics of this theory with minimal prerequisites (Math 653, essentially only familiarity with polynomials and ideals) and learn the necessary algebra, topology and combinatorics as we go along. Familiarity with simplicial homology is a plus, but not strictly required.
Reference books:
Combinatorial Commutative Algebra, by Miller and Sturmfels
Combinatorics and Commutative Algebra, by Richard Stanley
(both are available for free download through SpringerLink, when using a university connection)