Mathematical Sciences: Efficient Triangulations and Normal Surface Theory

Jaco这个项目涉及到在研究和理解3流形中发展和使用有效三角剖分和法线曲面理论。主要目标是开发一种算法来确定两个3-流形是否同胚，从而提供一个完整的3-流形分类。有效的三角剖分导致组合方法产生的结果类似于已知具有丰富几何形状（双曲结构）的3流形所获得的结果。所有的3-流形都允许三角剖分，那些最不容易理解的恰恰是那些允许有效三角剖分的类别；因此，这个项目提供了新的方法来研究最不了解的3流形的情况。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。近一个世纪以来，人们已经理解了2-流形的分类（所有可能模型的精确列表）。在这项工作之后的五十年里，人们认识到不可能对所有的四维模型进行分类。这个项目在维度3中解决问题，它仍然是开放的。＊＊＊

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. ***