COMPLEXITY AND RIGIDITY IN LOW DIMENSIONAL GEOMETRY

PI计划在2维和3维中研究拓扑和几何的相互作用，特别是关注拓扑和几何复杂性之间的关系。这个研究的一个主要特征是曲面映射中出现的复杂性的某种局部化，它与三维流形理论及其双曲结构，以及曲面上保形结构的TeichMuller和模空间的几何有很深的关系。曲面上曲线复合体的结构在所有这些设置中都起着作用，PI计划继续调查和完善这些设置。具体目标包括现有关于曲线复形的定理的亏格独立版本，或亏格依赖是显式的版本；瑟斯顿蒙皮映射的定量控制；以及更好地理解粘合映射决定双曲三维流形结构的其他背景。将特别注意可压缩边界情况。在Teichmuller理论中，PI将研究Weil-Petersson测地线流的行为及其与曲线复数产生的组合不变量的关系。几何和拓扑在数学中起着基础作用。几何空间出现在许多环境中，要么是作为我们直接可视化的显性对象，要么是通过抽象，如系统的配置空间或参数空间。空间的拓扑是将其组合在一起的方式，而不考虑几何，但通常这些拓扑描述足以唯一地确定几何。这种现象被称为僵化，并发挥着核心作用。在低维度中，与刚性相互作用的数学的许多不同方面汇合在一起，包括分析、复杂分析和动力学。此外，低维环境的可视化和可访问性使许多微妙的现象成为焦点，并成为新的数学思想的试验场。在这一背景下，我们建议研究刚性的一些量化方面，特别是拓扑描述中的复杂性转化为几何特征的方式。

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.