Combinatorics in Hyperbolic Geometry and Exporting Teichmuller Theory

首席研究员计划一个研究项目，通过与其他几个数学学科的联系和相互作用，研究几何中的一个基本重要课题——泰奇穆勒理论。几何和拓扑学领域的核心是对表面的研究，例如像行星这样的球形固体的表面，或者甜甜圈的表面，或者更复杂的三维固体的表面。在Teichmuller理论中，一个关键的目标是理解一个固定类型的曲面所能容纳的所有可能的几何形式。例如，一个天体物理学家可能有兴趣对一个假想行星的所有可能的地形进行编目。可能性是无数的：可能有丘陵或山地地形，高地，不同海拔的山谷，等等。神奇的是，Teichmuller理论提供了一种方法，可以将所有可能的地形打包成一个可以研究的几何对象。Teichmuller理论及其数学亲戚的实际应用比比皆是，从计算机可视化和图形设计，到进化生物学和遗传学，以及两者之间的许多其他重要学科。在更多的技术细节中，首席研究员提出了一个由三部分组成的计划，用于研究Teichmuller和双曲几何，映射类群，以及有限生成群和度量空间的更一般的类。首先，PI计划开发动态和组合工具来研究Teichmuller空间、映射类群和双曲3-流形，并使用这些工具来分析组合学和动力学交叉的基本问题，如格点计数问题，以及从组合的角度研究双曲3-流形。例如，PI计划证明曲面S的曲线复形与S上的双曲3流形纤维的几何形状之间的关系，以一种对底层曲面S的拓扑敏感的方式。接下来，PI将把这些工具推广并扩展到其他群和空间，例如自由群的外自同构群和外空间；例如，PI将在Teichmuller空间上的Weil-Petersson度量的启发下，开始对图的模空间上的度量进行研究。最后，PI计划在更广泛的群和空间类别中提出计数和其他类型的动态问题的类比，并在各自的上下文中使用广义工具来解决这些问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The principal investigator plans a research program for studying Teichmuller theory, a fundamentally important topic in geometry, by way of its connections and interactions with several other mathematical disciplines. Central to the fields of geometry and topology is the study of surfaces, for example the surface of a spherical solid like a planet, or the surface of a donut or of a more complicated three dimensional solid. A key objective in Teichmuller theory is to understand all possible geometric forms that a fixed type of surface can admit. For example, an astrophysicist might be interested in cataloging all possible topographies of a hypothetical planet. The possibilities are countless: there could be hilly or mountainous terrains, highlands, valleys of varying elevations, and so forth. Miraculously, Teichmuller theory gives a way to package the entire plethora of possible topographies into one geometric object which can then be studied. The practical applications of Teichmuller theory and of its mathematical relatives abound, from computer visualization and graphics design, to evolutionary biology and genetics, and to many other important disciplines in between.In more technical detail, the principal investigator proposes a three part plan for studying Teichmuller and hyperbolic geometry, the mapping class group, and more general classes of finitely generated groups and metric spaces. First, the PI plans to develop dynamical and combinatorial tools for studying the Teichmuller space, the mapping class group, and hyperbolic 3-manifolds, and to use these tools to analyze fundamental questions at the intersection of combinatorics and dynamics, such as lattice point counting problems, and the study of hyperbolic 3-manifolds from a combinatorial perspective. For example, the PI plans to demonstrate relationships between the curve complex of a surface S and the geometry of a hyperbolic 3-manifold fibering over S, in such a way that is sensitive to the topology of the underlying surface S. Next, the PI will generalize and extend these tools to other groups and spaces, such as the outer automorphism group of the free group and the Outer space; for instance, the PI will initiate a study of metrics on moduli spaces of graphs that are inspired by the Weil-Petersson metric on Teichmuller space. Finally, the PI plans to pose analogs of counting and other types of dynamical problems in a wider class of groups and of spaces, and use the generalized tools to attack these questions in their respective contexts.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.