Shawn Walker

First Position

Dissertation

Advisor:

Abstract: The combinatorics of regular cell complexes, such as convex polytopes and simplicial complexes, have long been of interest to mathematicians.

We consider the problem of relations between the face counts in simplicial complexes, specifically the class of colored simplicial complexes. We develop two different techniques for studying these face counts.

The first technique, known as shifting, attempts to ascribe a simplified form to each simplicial complex. Given a simplicial complex $\Delta$, we form a new simplicial complex, called the compression of $\Delta$. The compression of $\Delta$ maintains the same face counts as $\Delta$, but much of its other structure, both combinatorial and otherwise, is removed. We demonstrate the use of this technique in proving the Kruskal-Katona theorem and develop an analogous shift technique that preserves some of the fine structure contained in colored simplicial complexes.

The second technique, which we investigate, is that of convex or linear analysis. We apply techniques of convex analysis to an appropriate invariant to determine a number of interesting inequalities that relate the face counts of a balanced simplicial complex.

Last modified: August 10, 2005