RUI: Surfaces and their horizons, geometric structures, and pseudogroups

The University Corporation, Northridge, Northridge CA

Investigators

Abstract

The proposed research is in the area of geometry and dynamics. The objective is to understand the asymptotic behavior of orbits of dynamical systems. One specific problem deals with surface laminations in hyperbolic three-manifolds. Rather than studying recurrence phenomena taking place in a compact space a global approach is taken. The geometry of hyperbolic space has a well defined visual boundary representing the many possible ways of diverging to infinity. Moreover, the ambient hyperbolic geometry affects the geometry of the leaves of the lamination under consideration. The basic problem is then to study how these geometries relate with respect to their approach to the visual boundaries, and to examine the influence that the ambient hyperbolic geometry exerts on the leaves. Within this program it is also natural to consider laminations whose leaves have stronger geometric or analytic properties, or both, as for example when they satisfy the minimal surface differential equation. Such hypothesis allows for the use of analytical and probabilistic tools, and more precise information can be obtained. Laminations whose leaves are minimal surfaces are also relevant to understand the geometry of other three-manifolds as well. They appear to play an important role in the topological hyperbolization conjecture for three-manifolds, as is known that non-hyperbolic three manifolds have a lamination by minimal surfaces. This proposal also includes problems in the area of rigidity of actions of semisimple groups. The main focus is in the so-called Gromov's centralizer theorem, a major tool in understanding the symmetries of geometric structures on manifolds. Other questions relating to the structure of pseudogroups of transformations are also proposed. Dynamical systems are used to model processes in many areas, for example the weather, physical or chemical processes, and the evolution of living organisms and their morphology. They are also used for modeling processes which evolve from a finite amount of data according to some set of rules, either specified before hand or of a random nature, as neural networks in the brain or systems of digital processors in a computer. The objective of the proposed research is to understand the qualitative structure and asymptotic behavior of certain dynamical systems. One of the topics in this proposal is to study the behavior of two-dimensional systems evolving in a three-dimensional space, and specifically the interaction of the features of the surrounding space and the geometry and asymptotic behavior of their trajectories. This can be approached in a variety of ways: by purely geometric means, by studying certain differential equations that define their orbits, or via a probabilistic approach. Understanding basic features of these dynamical systems and these spaces, while having intrinsic geometric interest and beauty, could be relevant, for example, in areas like partial differential equations, solid state physics, structure of crystals and quasi-crystals and their defects, statistical mechanics, computation and algorithms.