On triangulating a square

Geometry Topology Seminar
Monday, February 27, 2012 - 14:05
1 hour (actually 50 minutes)
Skiles 006
Emory University
I will discuss the following geometric problem. If you are given an abstract 2-dimensional simplicial complex that is homeomorphic to a disk, and you want to (piecewise linearly) embed the complex in the plane so that the boundary is a geometric square, then what are the possibilities for the areas of the triangles? It turns out that for any such simplicial complex there is a polynomial relation that must be satisfied by the areas. I will report on joint work with Jamie Pommersheim in which we attempt to understand various features of this polynomial, such as the degree. One thing we do not know, for instance, if this degree is expressible in terms of other known integer invariants of the simplicial complex (or of the underlying planar graph).