Metric Geometry of Groups and Surfaces

AbstractAward：DMS-0906086首席研究员：Moon DuchinThis award is funded under the American Recovery and Reinvestment Act of 2009（Public Law 111-5）.本提案中的项目围绕低维拓扑、渐近几何和群体行动的动力学等主题展开。 Teichmuller空间和映射类群的特殊关注对象，以及为该设置开发的工具往往是有希望的更广泛的研究度量几何和几何群论。 PI建议建立在最近的工作上平坦度量，它们的长度谱，以及相关的边界理论，以及合成曲率条件及其在随机游动中的应用。 未来研究的具体途径包括奇异平面空间的渐近性，从对称空间到群论的填充不等式和高发散函数;以及使用度量半空间的相交模式来研究等距的动力学。虽然相当抽象，但这项工作中的思想也可以证明是非常适用的-例如，瑟斯顿的表面流动理论对流体动力学中的混合研究有着直接而有力的应用。 然而，更直接的是，这个研究领域解决了理解复杂空间的几何结构的基本问题，无论是大的，小的，还是随时间变化的。

AbstractAward: DMS-0906086Principal Investigator: Moon DuchinThis award is funded under the American Recovery and Reinvestment Act of 2009(Public Law 111-5).The projects in this proposal are clustered around themes inlow-dimensional topology, asymptotic geometry, and the dynamicsof group actions. Teichmuller space and the mapping class groupare objects for special attention, and tools developed for thatsetting are often promising for wider study in metric geometryand geometric group theory. The PI proposes to build on recentwork on flat metrics, their length spectra, and the associatedboundary theory, as well as work on synthetic curvatureconditions and their applications to random walks. Specificavenues of future investigation include asymptotics on the spaceof singular flat surfaces; filling inequalities and higherdivergence functions adapted from symmetric spaces to grouptheory; and the use of intersection patterns of metrichalf-spaces to study the dynamics of isometries.While quite abstract, ideas in this work can also provebeautifully applicable-- for instance, Thurston's theory of flowson surfaces has direct and strong applications to the study ofmixing in fluid dynamics. More immediately, though, thisresearch area addresses foundational questions towardunderstanding geometric structure of complicated spaces in thelarge, in the small, and changing over time.