====== Thomas Zaslavsky (Binghamton) ======

====== Polygon Space ======

===== Abstract for the Combinatorics and Number Theory Seminar 2003 April 7 =====

A //polygon// is a Hamiltonian circuit of the complete graph on n vertices. If we assign real-number ``lengths'' to the edges, each polygon has a length (that is, a real number), which induces a linear quasiordering of the set of all polygons. We call such a quasiordering //realizable//.

Now suppose the ``lengths'' really are lengths. That is, we pick n points in Euclidean space E<sup>d</sup>, (P<sub>i</sub>) = (P<sub>1</sub>, P<sub>2</sub>, ..., P<sub>n</sub>), and define the length of edge ij to be the distance d(P<sub>i</sub>, P<sub>j</sub>). There are some obvious questions. Which realizable quasiorderings are realizable by points in E<sup>d</sup>? One could allow some of the points to coincide, or not; these give different answers. Given points (P<sub>i</sub>) , inducing a certain realizable quasiordering, which other realizable quasiorderings are realizable by points (Q<sub>i</sub>) arbitrarily near (P<sub>i</sub>)? I will discuss these questions.

This will be a very informal talk with at most bits of hints of proof.