Efficient Triangulations of Three-Manifolds

该项目涉及在研究和理解3-流形中开发和使用有效三角测量。3-流形的三角化决定了3-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，当一个3流形包含一个有趣的表面时，那么对于3流形的任何三角剖分，在三角剖分中法线表面都表现出相同的有趣特征。这为研究3流形和解决某些决策问题提供了强有力的方法。例如，法向曲面理论已经导致了确定两个包含注入表面的3-流形何时是同胚的理论解决方案，以及这些相同流形的分类，以及确定三球中的一个结是否为解结。然而，即使有有趣的法线面存在，给定的三角剖分也可能有许多无趣的法线面，这使得理论论证相当乏味，计算尝试几乎不可能。有效三角剖分的指导原则是减少不必要的法线面，提高基于三角剖分算法的计算效率。他们在这方面取得了显著的成功。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，这似乎与三流形的拓扑结构密切相关。本项目将探索有效的三角剖分，特别强调它们在三流形的同胚问题/分类问题和计算复杂性问题中的应用。低维拓扑汇集了许多数学研究领域，并为数学的相互作用和进步提供了一个共同的基础。它为大多数物理现象提供了一个自然的几何模型。这一领域的研究为计算几何、拓扑和复杂性理论做出了重要贡献。它对物理学、计算机可视化、医学和生物建模都有影响。

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.