Math 715, Commutative Algebra and Polytopes.
For a polytope P of dimension d+1,
the fvector (f_0, ..., f_d) counts the
number of idimensional faces in the polytope.
For example, the fvector of a 3dimensional
octahedron (the 3crosspolytope) is
(6, 12, 8).
Already there are many basic questions one can ask about the fvector,
such as:

For fixed dimension d and fixed number of vertices,
how small can the
entries of the fvector be?

For fixed dimension d and fixed number of vertices,
how large can the
entries of the fvector be?

Given a vector of entries, is it an fvector of
a simplicial polytope
(or more generally, of a simplicial complex)?
The proofs of the first two questions, known as the Lower and Upper Bound
Theorems, are very geometric.
Already the third result, due to KruskalKatona,
suggests some of the algebraic tools later developed
to answer deeper questions about
polytopes.
We will discuss these three questions
during the first third of the course.
The middle third will serve as an introduction
to
commutative algebra techniques for studying polytopes.
During the last part of the course,
we will show how a noncommutative polynomial called
the cdindex encodes the fvector and flag data of a polytope
and how it can be used to prove further results.
COURSE OUTLINE

Introduction to convex polytopes

KruskalKatona Theorem

Upper and Lower Bound Theorems

A Friendly Introduction to Commutative Algebra

The StanleyReisner Ring

Reisner's Topological Criterion

Upper Bound Theorem for Spheres

Flag Vectors, Coalgebras and the cdindex
TEXTBOOK:
Richard P. Stanley,
Combinatorics and Commutative Algebra,
second edition,
Birkhauser,
Boston, 1996.
http://www.ms.uky.edu/~readdy/715/Comm_Algebra_Polytopes/