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

拟议的研究是在该地区的几何和动力学。目的是了解动力系统轨道的渐近行为。一个具体的问题涉及双曲三流形中的曲面叠层。而不是研究递归现象发生在一个紧凑的空间agonist方法。双曲空间的几何学有一个明确的视觉边界，代表了许多可能的发散到无穷大的方式。此外，周围的双曲几何形状影响所考虑的层合板的叶片的几何形状。基本的问题是，然后研究这些几何形状如何与他们的方法visualboundaries，并检查的影响，ambienthperbolic几何形状施加在叶子上。 在这个程序中，考虑叶片具有更强的几何或分析性质或两者兼有的叠层也是很自然的，例如当它们满足最小曲面微分方程时。这种假设允许使用分析和概率工具，并且可以获得更精确的信息。叶是极小曲面的层积对于理解其他三流形的几何也是相关的。它们似乎在三维流形的拓扑双曲化猜想中起着重要的作用，因为已知非双曲三维流形具有极小曲面的层。 这个建议也包括了半单群作用的刚性领域的问题。主要的焦点是在所谓的格罗莫夫的中心定理，一个主要的工具，在理解对称的几何结构的流形。动力系统被用来模拟许多领域的过程，例如天气、物理或化学过程，以及生物体及其形态的进化。它们也被用于建模过程，这些过程是根据一些规则从有限的数据中演变而来的，这些规则要么是事先指定的，要么是随机的，就像大脑中的神经网络或计算机中的数字处理器系统一样。本研究的目的是了解某些动力系统的定性结构和渐近行为。该计划的主题之一是研究二维系统在三维空间中演化的行为，特别是周围空间特征的相互作用及其轨迹的几何和渐近行为。这可以通过多种方式来实现：通过纯粹的几何方法，通过研究某些定义其轨道的微分方程，或者通过概率方法。了解这些动力系统和空间的基本特征，同时具有内在的几何兴趣和美感，可能与偏微分方程、固态物理、晶体和准晶体结构及其缺陷、物理力学、计算和算法等领域有关。

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.