Mathematical Sciences: Efficient Triangulations and Normal Surface Theory

小行星9704833雅科 这个项目涉及有效的三角剖分和正常的表面理论的研究和理解的三维流形的发展和使用。 主要目标是开发一个算法来决定两个3-流形是否同胚，从而提供一个完整的3-流形分类。 高效的三角剖分导致组合方法，产生类似于当已知3流形具有丰富的几何结构（双曲结构）时所获得的结果。 所有的3-流形都承认三角剖分，那些最不被理解的正是承认有效三角剖分的类，因此，该项目提供了新的方法来研究3-流形的最不被理解的情况。 低维流形提供了大多数物理现象的几何模型。 要求对所有可能的这类模型进行描述是很自然的。 近世纪以来，2-流形的分类（所有可能模型的精确列表）已经被理解。 在这项工作进行了50年之后，人们发现不可能对所有的四维模型进行分类。 这个项目解决了第三维度的问题，这个问题仍然是开放的。 ***

9704833 Jaco This project involves the development and use of efficient triangulations and normal surface theory in the study and understanding of 3-manifolds. The primary objective is to develop an algorithm to decide if two 3-manifolds are homeomorphic, thereby providing a complete classification of 3-manifolds. Efficient triangulations lead to combinatorial methods that yield results analogous to those obtained when a 3-manifold is known to have a rich geometry (hyperbolic structure). All 3-manifolds admit triangulations, those which are least understood are precisely the class which admit efficient triangulations; hence, this project provides new methods to study the least understood cases of 3-manifolds. Low-dimensional manifolds provide the geometric models of most physical phenomena. It is natural to ask for a description of all possible such models. For nearly a century, the classification (precise listing of all possible models) of 2-manifolds has been understood. Following this work by fifty years, it was learned that it is not possible to classify all 4-dimensional models. This project attacks the problem in dimension 3, which remains open. ***