Math 714, Topics in Combinatorics
Combinatorics, Commutative Algebra and
Topology of Simplicial Complexes
(C-CATS-C)
Spring semester 2010
For a polytope P of dimension n, the f-vector
(f_0, ..., f_(n-1)) counts the number of i-dimensional
faces in the polytope.
For example, the f-vector of an
icosahedron
is (12, 30, 20).
Already there are many basic questions one can ask about the f-vector,
such as:
-
For fixed dimension and fixed number of vertices, how small can
the entries of the f-vector be?
-
For fixed dimension 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
flag data of a polytope and how it can be used to prove
further results about polytopes and subdivisions of manifolds.
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
- Discrete Morse theory
- Flag Vectors, the cd-index and manifolds
TEXTBOOK:
Richard P. Stanley, Combinatorics and Commutative Algebra, second edition,
Birkhäuser, 1996.
ANNOUNCEMENT:
pdf
WEBPAGE:
http://www.math.uky.edu/~jrge/714/
I look forward to seeing you in January 2010.
jrge@ms.uky.edu