Low Dimensional Topology and Geometry

这项建议主要涉及几何和拓扑的几个方面。其中一个项目建议研究射影(及相关)结构和组合三角剖分。第二个项目是将辫子的代数理论与具有给定辫子边界的表面(无论是3维还是4维)的分类联系起来。特别地，通过辫子群来探索4-流形的规范理论与这类曲面的复杂性之间的联系。第三个项目是研究三维流形的虚Haken问题，Voronoi分解在数学和计算机科学中具有重要的意义。人们使用有限点集来将欧几里得空间细分为凸单元，方法是为每个点分配由比给定集合中的任何其他点更接近给定点的点组成的空间的子集。这种分解在计算中经常被用来用有限网格来逼近连续统。这一理论在欧几里得(平坦)空间中是众所周知的，我们正在将其推广到黎曼(曲线)几何的情况

This proposal primarily concerns several aspects of geometry and topology. One of the projects proposes to study projective (and related) structures and combinatorial triangulations. A second project is to relate the algebraic theory of braids to the classification of surfaces (in either 3 or 4 dimensions) with boundary a given braid. In particular, to explore the connection between the gauge theory of 4-manifolds and the complexity of such surfaces via the braid group. A third project is to study the virtual Haken question for 3-manifolds which are bundles or semi-bundles.Voronoi decompositions are of great importance in mathematics and computer science. One uses a finite set of points to subdivide Euclidean space into convex cells by assigning to each point the subset of space consisting of points closer to the given point than to any other point in the given set. Such decompositions are frequently used in computational situations to approximate a continuum by a finite grid. This theory is well known in Euclidean (flat) space, and we are extending it to the case of Riemannian (curved) geometry