Efficient Triangulations, Decision Problems & Algorithms

这一研究项目的主要目的是进一步发展有效的三角剖分及其在三维流形研究和理解中的应用。0-和1-有效的三角剖分在3-流形的三角剖分、决策问题、算法、计算复杂性、Heegaard分裂和Dehn填充等方面产生了许多新的方法和结果。我们建议进一步发展我们对0-和1-有效三角剖分的理解，并形成g-有效三角剖分的概念(G1)。我们的目标是在有效三角剖分和三维流形上的几何结构之间建立明确的联系，使用有效三角剖分来更好地理解三维流形的Heegaard分解，并使用我们的方法将给定的三角剖分简化为有效的三角剖分来实现三维流形的同胚问题的解的简化(3-流形的分类)。特别地，我们提出了判定三维流形是否为Haken流形、三维流形的JSJ分解和识别Haken流形的新算法。三维流形研究的主要问题是分类，即对所有的三维流形进行一张完整的、不重复的列表。对三维流形的研究和理解是本课题的主要目的。特别是，我们知道三个流形可以被认为是以一种非常有组织的方式组装在一起的构建块的联合。例如，四面体可以用作构建块；在这种情况下，四面体的集合及其如何组合在一起的信息称为三维流形的三角剖分。所有三个流形都可以三角剖分。因此，解决分类问题的主要策略之一是找到识别特定三维流形的方法，该流形在其众多可能的三角剖分中被赋予了三个流形。三维流形拓扑学的应用范围从DNA中的蛋白质打结和解结问题到空间(宇宙)的形状问题。特别是，后者很可能会涉及到认识我们宇宙的三重流形的问题。因此，对三维流形的分类和理解具有深远的应用和意义。

The principal thrust of this research project is further development of efficient triangulations and their applications to the study and understanding of 3-manifolds. 0- and 1-efficient triangulations have lead to a number of new methods and results on triangulations of 3-manifolds, decision problems, algorithms, computational complexity, Heegaard splittings, and Dehn fillings. We propose to develop further our understanding of 0- and 1-efficient triangulations and to form a notion of g-efficient triangulations (g 1). Our goals are to make an explicit connection between efficient triangulations and geometric structures on 3-manifolds, to use efficient triangulations for a better understanding of Heegaard splittings of 3-manifolds, and to use our methods that reduce a given triangulation to an efficient triangulation to achieve a simplification to the solution of the Homeomorphism Problem for 3-manifolds (Classification of 3-manifolds). In particular, we propose new algorithms for deciding if a 3-manifold is a Haken manifold, for the JSJ decomposition of a 3-manifold and for the recognition of Haken manifolds.Three-dimensional manifolds are mathematical objects which are locally modeled on familiar three-dimensional space. The major problem in the study of three-manifolds is their classification, which is to make a complete list of all three-manifolds without duplications. The study and understanding of three-manifolds toward such a classification is the main objective of this project. In particular, we know that three-manifolds can be considered as a union of building blocks fitting together in a very organized way. For example, tetrahedra may be used as the building blocks; in this case, the collection of tetrahedra and the information about how they fit together is called a triangulation of the three-manifold. All three-manifolds can be triangulated. Thus one of the major strategies toward solving the classification problem is to find methods to recognize a particular three-manifold, having been given the three-manifold in one of its many possible triangulations. Applications of topology of three-manifolds range from questions of protein knotting and unknotting in DNA to the issue of the shape of space (the universe). In particular, the latter may very well come to an issue of recognizing the three-manifold that is our universe. Thus the classification and understanding of three-manifolds has far reaching applications and implications.