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

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The proposed research is in the area of geometry anddynamics. The objective is to understand the asymptotic behaviorof orbits of dynamical systems. One specific problem deals withsurface laminations in hyperbolic three-manifolds. Rather thanstudying recurrence phenomena taking place in a compact space aglobal approach is taken. The geometry of hyperbolic space has awell defined visual boundary representing the many possible waysof diverging to infinity. Moreover, the ambient hyperbolicgeometry affects the geometry of the leaves of the laminationunder consideration. The basic problem is then to study how thesegeometries relate with respect to their approach to the visualboundaries, and to examine the influence that the ambienthyperbolic geometry exerts on the leaves. Within this program itis also natural to consider laminations whose leaves havestronger geometric or analytic properties, or both, as forexample when they satisfy the minimal surface differentialequation. Such hypothesis allows for the use of analytical andprobabilistic tools, and more precise information can beobtained. Laminations whose leaves are minimal surfaces are alsorelevant to understand the geometry of other three-manifolds aswell. They appear to play an important role in the topologicalhyperbolization conjecture for three-manifolds, as is known thatnon-hyperbolic three manifolds have a lamination by minimalsurfaces. This proposal also includes problems in the area ofrigidity of actions of semisimple groups. The main focus is inthe so-called Gromov's centralizer theorem, a major tool inunderstanding the symmetries of geometric structures onmanifolds. Other questions relating to the structure ofpseudogroups of transformations are also proposed.Dynamical systems are used to model processes in many areas, forexample the weather, physical or chemical processes, and theevolution of living organisms and their morphology. They are alsoused for modeling processes which evolve from a finite amount ofdata according to some set of rules, either specified before handor of a random nature, as neural networks in the brain or systemsof digital processors in a computer. The objective of theproposed research is to understand the qualitative structure andasymptotic behavior of certain dynamical systems. One of thetopics in this proposal is to study the behavior oftwo-dimensional systems evolving in a three-dimensional space,and specifically the interaction of the features of thesurrounding space and the geometry and asymptotic behavior oftheir trajectories. This can be approached in a variety of ways:by purely geometric means, by studying certain differentialequations that define their orbits, or via a probabilisticapproach. Understanding basic features of these dynamical systemsand these spaces, while having intrinsic geometric interest andbeauty, could be relevant, for example, in areas like partialdifferential equations, solid state physics, structure ofcrystals and quasi-crystals and their defects, statisticalmechanics, computation and algorithms.