Research
My general area of interest is discrete geometry, currently most of my research
has to do with problems involving the geometric realizability of simplicial
complexes and the combinatorial properties of abstract polytopes. Below
are links to materials connected to my research (or will be).
Nonconvex Embeddings of the Exceptional Simplicial 3-Spheres with Eight
Vertices.
This is joint work carried out while I was a student at the
University of Washington. There are two simplicial 3-spheres with eight
vertices that do not admit realizations as the boundaries for convex
4-polytopes. The question we investigate (and answer in the
affirmative) is whether or not there exist geometric embeddings of
these spheres in 4-space. It has since been suggested by Ted
Bisztriczky that such a realization probably ought to be called a
3-convex realization, a term I am happy to promote.
J. Mihalisin and G. Williams. Nonconvex embeddings of the exceptional
simplicial 3-spheres with 8 vertices. J. Comb. Th. (A), 98(1):74--86,
2002.
Petrie Polygons, etc.
A petrie polygon is classically defined to be a sequence of edges in a polyhedron
such that any two adjacent members of the sequence lie in a face, but no
three successive. If you stand a cube up on a corner, this would correspond
to the sequence of edges that make up the "equator" of the cube. There
are ways of extending this definition to higher dimensions (inductively).
I'm currently working with a related idea I've chosen to call a Petrie
Scheme, each of which is a sequence of flags (maximal chains in the poset
of faces) in a polytope that has (amongst other properties) all of their
rank 1 elements forming a Petrie polygon. The central question I am
trying to answer is for which polytopes are the Petrie schemes nice. By
this I mean have the property that each face which appears in the scheme
appears at most once. We call such a scheme "acoptic" from the greek
word for "cutting" because these schemes don't "cut" across themselves.
G. Williams. Petrie Schemes. PhD thesis, University of Washington Seattle, 2002.
G. Williams. Petrie schemes. Canad. J. Math. 57 (2005), no. 4, 844--870.
Abstract Polytopes
Abstract
polytopes are combinatorial objects that generalize the lattice
structure of polytopes. The lattice structure of a polytope ignores the
geometry and pays attention only to information about which faces of
the polytope are contained by which, which share edges or vertices with
which, and so on. A flag
in an abstract polytope is a maximal chain of the faces, loosely
speaking a sequence of the form vertex, edge, face, 3-face, ..., facet
where each element of the list is contained in the next. We say that an
abstract polytope is regular if its automorphism group (its set of symmetries) is able to swap any flag to any other flag in the polytope. Michael Hartley proved as part of his dissertation that any
abstract polytope may be represented as a quotient of a regular
abstract polytope. My own work in this area has been a collaboration
with Hartley to model some of the more familiar classes of polytopes
and try to understand the relationships between their special
properties and the structure of their quotient represations.Doing research with me
There
are many open questions connected with each of the topics listed above,
and if you drop by my office and ask, I'll gladly give you an ear full.