Mathematical Sciences: Efficient Triangulations and Normal Surface Theory

9704833 JACO这个项目涉及在研究和理解三维流形中开发和使用有效的三角剖分和法面理论。其主要目的是开发一种算法来判定两个3-流形是否同胚，从而提供3-流形的完整分类。有效的三角剖分导致了组合方法，其结果类似于当已知三维流形具有丰富的几何(双曲结构)时所获得的结果。所有的三维流形都允许三角剖分，而那些最不被理解的恰恰是允许有效三角剖分的类，因此，这个项目为研究最不被理解的三维流形情况提供了新的方法。低维流形提供了大多数物理现象的几何模型。要求描述所有可能的此类模式是很自然的。近一个世纪以来，人们已经了解了2-流形的分类(所有可能的模型的精确列表)。在这项工作进行了50年后，人们了解到，不可能对所有的4维模型进行分类。这个项目解决了维度3中的问题，而维度3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. ***