Experimental Low-Dimensional Geometry and Topology

首席研究员使用计算机可视化工具来帮助研究低维微分几何和代数几何以及拓扑中的许多问题。特别是，这些问题涉及到各种三维几何的等距离散群的基本定义域、各种三维空间形式中多边形的模空间以及低维半单李群紧化族的直接构造。开发了用于探索三维双曲空间等距循环群无穷远球面上的基本域的交互式软件，并将其推广到双曲空间、其他群和其他几何的内部。研究了多边形的模空间，以确定其自身几何的各个方面，并且由于它们与穿孔黎曼曲面上矢量束的模空间以及一些辛和几何不变理论商的关系。双曲盘的等距群的某些代数紧化的模空间也生成了象，其他代数紧化的模空间以及紧化本身也生成了象。这个项目的一个重要组成部分是本科生的直接参与：虽然数学具有不同层次的抽象，但计算机任务对认真的学生来说是可以访问的，并为他们提供了一个亲身参与当前数学研究的不同寻常的机会。在今天的许多商业电脑游戏中可以看到奇妙的交互式三维图形，这是近年来在计算机软件的复杂性和计算机硬件的速度和能力方面发生的惊人进步的结果。但从更大的意义上说，它们可以归因于基于硬科学的工程知识的增加，最终是基于深刻的数学理解。因此，有些令人惊讶的是，这些计算和图形工具在纯数学研究中很少使用。首席研究员使用交互式计算机可视化在各种不同领域进行数学实验，这些领域都有一些与二维，三维或四维几何相关的组件；由于这是纯数学，实验证据必须在纯抽象的环境中得到证明。这个项目的一个重要部分是本科生的直接参与：在这里，利用许多本科生现在拥有的非常高水平的计算机能力来创建驱动数学探索的软件是可能的，同时给这些本科生一个极其难得的机会来直接体验当前的数学研究。

Poritz9806408The principal investigator employs computer visualization tools to aid in the study of a number of questions in low-dimensional differential and algebraic geometry and topology. In particular, these questions have to do with fundamental domains for discrete groups of isometries of various three-dimensional geometries, moduli spaces of polygons in each of the three-dimensional space forms, and direct constructions of families of compactifications of semisimple Lie groups of low dimension. Interactive software has been developed to explore fundamental domains on the sphere at infinity for cyclic groups of isometries of three-dimensional hyperbolic space and this is extended to the interior of hyperbolic space, other groups and other geometries. Moduli spaces of polygons are investigated to determine aspects of their own geometry and because of their relationship to moduli spaces of vector bundles over punctured Riemann surfaces and some symplectic and geometric-invariant theory quotients. Images have also been made of moduli spaces of certain algebraic compactifications of the group of isometries of the hyperbolic disk and further images are generated of moduli spaces of other algebraic compactifications as well as of the compactifications themselves. An important component of this project is the direct involvement of undergraduate students: while the mathematics is of various levels of abstraction, the computer tasks are accessible to serious students and provide them with an unusual opportunity for hands-on involvement in current mathematical research.The fantastic interactive three-dimensional graphics that can be seen in many of today's commercial computer games are the result of the phenomenal advances that have occurred in recent years in the sophistication of computer software and the speed and power of computer hardware. But in a larger sense they can be attributed to the increase of engineering knowledge based on hard science and, ultimately, on deep mathematical understanding. It is thus somewhat surprising that these computational and graphics tools have been used very little in pure mathematical research. The principal investigator uses interactive computer visualization to perform mathematical experiments in a variety of different fields that all have some component relating to two-, three- or four-dimensional geometry; as this is pure mathematics, experimental evidence must then be justified in a purely abstract setting. An important part of this project is the direct involvement of undergraduate students: here again it is possible to take advantage of the very high level computer abilities that many undergraduates now possess to create the software that drives the mathematical explorations, while at the same time giving these undergraduates the extremely rare opportunity to have immediate experience of current mathematical research.