Efficient Triangulations of Three-Manifolds

DMS-0204707 William H.该项目涉及在3-流形的研究和理解中开发和使用有效的三角剖分。三维流形的三角剖分确定了三维流形中的一类曲面，称为法向曲面。一个基本的观察是，在大多数情况下，当一个3-流形包含一个有趣的表面，那么对于任何三角剖分的3-流形的相同的有趣的功能isshowed由一个正常的表面在三角剖分。这为研究三维流形和解决某些决策问题提供了一种强有力的方法。正常的表面理论，例如，导致theoreticalresolution确定当两个3-流形，其中包含injectivesurfaces，是同胚的，这些相同的流形的分类，并确定是否在三个球结是unknot。然而，即使存在感兴趣的法向曲面，给定的三角剖分也可能有许多不感兴趣的法向曲面，这使得理论论证相当繁琐，计算尝试几乎是不可能的。高效三角剖分的指导原则是减少不必要的法向曲面，提高三角剖分算法的计算效率。 他们在做这件事上表现出了巨大的成功。然而，更令人惊讶的是，有效三角剖分揭示了三流形的单顶点和理想三角剖分与新的组合结构之间的有趣关系，这个项目将探讨有效的三角剖分，特别是它们在三流形的同胚问题/分类问题中的应用，以及计算复杂性的问题。低-维拓扑将数学研究的许多领域汇集在一起，并为整个数学的互动和发展提供了共同基础。它为大多数物理现象提供了一个自然的几何模型。这一领域的研究对计算几何、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、医学和生物建模方面都有重要的影响。

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.