Efficient Triangulations of Three-Manifolds

DMS-0204707威廉·H·雅各布这个项目涉及到在研究和理解三维流形中开发和使用高效的三角剖分。三维流形的三角剖分确定了三维流形中的一类曲面，称为法线曲面。一个基本的观察是，在大多数情况下，当三维流形包含一个有趣的曲面时，对于三维流形的任何三角剖分，三角剖分中的法线曲面都表现出相同的有趣特征。这为研究三维流形和解决某些决策问题提供了一种有效的方法。例如，正规面理论导致了确定两个包含内射面的三维流形何时是同胚的理论上的分解，导致了对这些相同流形的分类，并导致了确定三个球面中的一个纽结是否为非纽结。然而，即使存在有趣的法线曲面，三角剖分也可能有许多枯燥的法线曲面，这使得理论论证相当乏味，计算尝试几乎是不可能的。高效三角剖分的指导思想是减少不必要的法面，提高基于三角剖分算法的计算效率。他们在做这件事上表现出了惊人的成功。然而，更令人惊讶的是，高效三角剖分揭示了三维流形的单顶点和对偶三角剖分与新的组合结构之间的有趣关系，这些关系似乎与三维流形的拓扑密切相关。本项目将探索有效的三角剖分，特别是它们在三维流形的同胚问题/分类问题和计算复杂性问题中的应用。低维拓扑学汇集了许多数学研究领域，并提供了一个相互作用和跨数学进步的共同点。它为大多数物理现象提供了一个自然的几何模型。这一领域的研究为计算几何、拓扑学和复杂性理论做出了重要贡献。它在物理学、计算机可视化以及医学和生物建模方面都产生了影响。

DMS-0204707William H. JacoThis project involves the development and use of efficienttriangulations in the study and understanding of 3--manifolds. Atriangulation of a 3-manifold determines a class of surfaces in the3--manifold called normal surfaces. A fundamental observation is that in most cases when a 3--manifold contains an interesting surface, then forany triangulation of the 3--manifold the same interesting feature isexhibited by a normal surface in the triangulation. This has provided apowerful method for studying 3-manifolds and solving certain decisionproblems. Normal surface theory has, for example, lead to the theoreticalresolution of determining when two 3--manifolds, which contain injectivesurfaces, are homeomorphic, to the classification of these same manifolds,and to the determination of whether a knot in the three-sphere is theunknot. However, even with the presence of interesting normal surfaces, agiven triangulation may have numerous uninteresting normal surfaces, whichmake theoretical arguments quite tedious and computational attempts nearlyimpossible. The guiding principle of efficient triangulations is reducingunnecessary normal surfaces and improving the efficiency of computationsfor triangulation based algorithms. They have exhibited remarkablesuccess in doing this. However, more surprisingly, efficienttriangulations have exposed interesting relations between one-vertex andideal triangulations of three-manifolds and new combinatorial structures,which seem to be intimately related to the topology of the three-manifold.This project will explore efficient triangulations with particularemphasis of their application to the Homeomorphism Problem/ClassificationProblem for three-manifolds and issues of computational complexity.Low-dimensional topology brings together many areas of mathematicalresearch and provides a common ground for interaction and advance acrossall of mathematics. It provides a natural geometric model of most physicalphenomena. Research in this area is making significant contribution tocomputational geometry and topology and complexity theory. It hasconsequences in physics, computer visualization and medical and biologicalmodeling.