COMPLEXITY AND RIGIDITY IN LOW DIMENSIONAL GEOMETRY

低维几何的复杂性和刚性

• 批准号: 1311844
• 负责人: Yair Minsky
• 金额: $ 34.4万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1311844&HistoricalAwards=false
• 关键词: COMPLEXITY; RIGIDITY; LOW; DIMENSIONAL; GEOMETRY

The PI plans to investigate aspects of the interaction of topology and geometry in 2 and 3 dimensions, focusing in particular on the relations between topological and geometric complexity. A primary feature in this investigation is a certain localization of complexity that occurs in maps of surfaces, and has deep relations to the theory of 3-manifolds and their hyperbolic structure, as well as to the geometry of the Teichmuller and moduli space of conformal structures on a surface. The structure of the complex of curves on a surface plays a role in all these settings, which the PI plans to continue investigating and refining. Particular goals include genus-independent versions of existing theorems on the complex of curves, or versions where genus dependence is explicit; quantitative control of Thurston's skinning map; and better understanding of other settings where gluing maps determine the structure of hyperbolic 3-manifolds. Particular attention will be paid to the compressible-boundary case. In Teichmuller theory, the PI will investigate the behavior of Weil-Petersson geodesic flow and its relation to combinatorial invariants arising from the complex of curves.Geometry and topology play a fundamental role in mathematics. Geometric spaces arise in many settings, either as explicit objects we visualize directly, or via abstractions such as configuration spaces of systems or parameter spaces. The topology of a space is the way it is combinatorially put together without regard to geometry, but often these topological descriptions suffice to determine the geometry uniquely. This phenomenon is called rigidity and plays a central role. In low dimensions there is a confluence of many different aspects of mathematics which interact with rigidity, including analysis, complex analysis and dynamics. Moreover, the visualizability and accessibility of the low-dimensional setting brings many subtle phenomena into focus and serves as a testing ground for new mathematical ideas. Within this setting we propose to investigate a number of quantitative aspects of rigidity, particularly ways in which complexity in the topological description translates to features of the geometry.

PI计划研究在2和3维度中拓扑与几何形状相互作用的各个方面，尤其集中在拓扑复杂性和几何复杂性之间的关系上。这项研究的主要特征是在表面地图中存在某种复杂性的定位，并且与3个manifolds及其双曲线结构的理论以及Teichmuller和Moduli在表面上的结构的几何形状有着深厚的关系。表面上曲线复合体的结构在所有这些环境中都起作用，PI计划继续研究和完善。特定的目标包括与曲线复合物或属依赖属的版本有关的现有定理的独立属版本。瑟斯顿皮肤图的定量控制；并更好地理解胶合图确定双曲线3个manifolds的结构的其他设置。特别关注可压缩的案例。在Teichmuller理论中，PI将研究Weil-Petersson测量流的行为及其与曲线复合物引起的组合不变性的关系。几何和拓扑在数学中起着基本作用。几何空间在许多设置中都出现，无论是我们直接可视化的显式对象，还是通过抽象（例如系统或参数空间的配置空间）。空间的拓扑结构是将其组合在一起而不考虑几何形状的方式，但通常这些拓扑描述足以确定几何形状。这种现象称为刚性，起着核心作用。在低维中，数学的许多不同方面存在与刚性相互作用的汇合，包括分析，复杂分析和动态。此外，低维设置的可视化性和可访问性将许多微妙的现象引入了重点，并作为新数学思想的测试基础。在这种情况下，我们建议研究刚性的许多定量方面，尤其是拓扑描述中复杂性转化为几何特征的方式。