Low Dimensional Topology and Real Projective Geometry

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

The major theme of this project is to develop further the theory of real projective and related geometric structures on manifolds. This is an area where little work has been done and there are many interesting questions. In particular questions of existence, moduli spaces and their compactifications. A second area is to explore the existence of various kinds of surface in 3-manifolds, in particular the virtual Haken question and its relatives. The PI and Tillmann are developing a theory of transversally oriented normal surfaces which we expect to have significant applications. Another project is to develop the theory of Voronoi decompositions (nearest-point neighbor sets for a discretization of a space) beyond the well know setting of Euclidean space to general Riemannian manifolds. Such decompositions are frequently used in computational situations to approximate a continuum by a finite grid. These techniques will enable this to be done in a much more general framework than Euclidean space. Geometry and topology are playing an increasingly important role in many areas of mathematics and science. In physics ever more sophisticated mathematical ideas from these areas are being used to develop string theory and related areas. In computer science the effective use of graphics requires sophisticated geometry. In biology the construction of evolutionary trees uses techniques from geometry. Projective geometry was born in the Renaissance from the work of the mathematician and artist Gerard Desargues who wanted to accurately depict what he saw on a flat canvas. This led in the nineteenth century to a great mathematical theory. After a period of less activity in the twentieth century projective geometry is again reappearing in a variety of unlikely places. It perhaps offers the possibility for a unified understanding of three dimensional spaces. This is an area with many new problems suitable for training new PhDs.