Low Dimensional Topology and Real Projective Geometry

本课题的主要主题是进一步发展流形上的实射影及相关几何结构的理论。这是一个工作很少的领域，有许多有趣的问题。特别是关于存在性、模空间及其紧化的问题。第二个领域是探索3流形中各种曲面的存在性，特别是虚拟Haken问题及其相关问题。PI和Tillmann正在发展一种横向定向法线面的理论，我们期望它有重要的应用。另一个项目是发展Voronoi分解理论（空间离散化的最近邻集），超越众所周知的欧几里德空间到一般黎曼流形的设置。这种分解经常用于计算中，通过有限网格近似连续体。这些技术将使它能够在比欧几里得空间更一般的框架中完成。几何和拓扑学在数学和科学的许多领域中发挥着越来越重要的作用。在物理学中，来自这些领域的更为复杂的数学思想正被用来发展弦理论和相关领域。在计算机科学中，图形的有效使用需要复杂的几何学。在生物学中，进化树的构建使用了几何学的技术。射影几何诞生于文艺复兴时期，由数学家兼艺术家杰勒德·德萨格（Gerard Desargues）提出，他想要在平面画布上准确地描绘他所看到的东西。这导致了19世纪一个伟大的数学理论的诞生。在二十世纪经历了一段不那么活跃的时期之后，射影几何再次出现在各种不太可能出现的地方。它也许提供了对三维空间统一理解的可能性。这是一个有许多新问题的领域，适合培养新的博士。

The major theme of this project is to develop further the theory of real projective and related geometric structures on manifolds. This is an area where little work has been done and there are many interesting questions. In particular questions of existence, moduli spaces and their compactifications. A second area is to explore the existence of various kinds of surface in 3-manifolds, in particular the virtual Haken question and its relatives. The PI and Tillmann are developing a theory of transversally oriented normal surfaces which we expect to have significant applications. Another project is to develop the theory of Voronoi decompositions (nearest-point neighbor sets for a discretization of a space) beyond the well know setting of Euclidean space to general Riemannian manifolds. Such decompositions are frequently used in computational situations to approximate a continuum by a finite grid. These techniques will enable this to be done in a much more general framework than Euclidean space.Geometry and topology are playing an increasingly important role in many areas of mathematics and science. In physics ever more sophisticated mathematical ideas from these areas are being used to develop string theory and related areas. In computer science the effective use of graphics requires sophisticated geometry. In biology the construction of evolutionary trees uses techniques from geometry. Projective geometry was born in the Renaissance from the work of the mathematician and artist Gerard Desargues who wanted to accurately depict what he saw on a flat canvas. This led in the nineteenth century to a great mathematical theory. After a period of less activity in the twentieth century projective geometry is again reappearing in a variety of unlikely places. It perhaps offers the possibility for a unified understanding of three dimensional spaces. This is an area with many new problems suitable for training new PhDs.