Math 715, Commutative Algebra and Polytopes.
For a polytope P of dimension d+1,
the f-vector (f_0, ..., f_d) counts the
number of i-dimensional faces in the polytope.
For example, the f-vector of a 3-dimensional
octahedron (the 3-cross-polytope) is
(6, 12, 8).
Already there are many basic questions one can ask about the f-vector,
such as:
-
For fixed dimension d and fixed number of vertices,
how small can the
entries of the f-vector be?
-
For fixed dimension d and fixed number of vertices,
how large can the
entries of the f-vector be?
-
Given a vector of entries, is it an f-vector 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 Kruskal-Katona,
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 cd-index encodes the f-vector and flag data of a polytope
and how it can be used to prove further results.
COURSE OUTLINE
-
Introduction to convex polytopes
-
Kruskal-Katona Theorem
-
Upper and Lower Bound Theorems
-
A Friendly Introduction to Commutative Algebra
-
The Stanley-Reisner Ring
-
Reisner's Topological Criterion
-
Upper Bound Theorem for Spheres
-
Flag Vectors, Coalgebras and the cd-index
TEXTBOOK:
Richard P. Stanley,
Combinatorics and Commutative Algebra,
second edition,
Birkhauser,
Boston, 1996.
http://www.ms.uky.edu/~readdy/715/Comm_Algebra_Polytopes/