Explicit Geometry in Low Dimensions

Temple University, Philadelphia PA

Investigators

Abstract

Proposal: DMS-0072622 PI: Igor Rivin Abstract: I propose to concentrate on constructive (and, if possible, efficient) methods for finding and characterizing OPTIMAL geometric representations for combinatorial, topological, or geometric objects in low dimensions. In one dimension we will study graphs, in two dimensions, Riemann surfaces, in three dimensions -- compact manifolds. We will study direct geometric invariants (volume, curvature integrals, diameter) as well as less tractable but in many ways attractive spectral invariants (individual eigenvalues or determinants (regularized or not) of elliptic operators). In all cases, our goal is find the nicest geometric representation of a topological object. An obvious example is that of the "round" metric on the sphere. People have been lucky to discover this metric early, because the natural processes of weather work stones into approximately round shapes. Unfortunately, nature is not there to help us with more complicated mathematical objects, so we have to invent artificial processes which will work them into "canonical" shapes, so we can better understand the underlying structure.