Essential Surfaces in Knot Exteriors: The Lopez Conjecture

提案ID：DMS-9978071提案标题：结外表面：洛佩兹猜想。主要研究人员：威廉·H·雅各布博士摘要：该项目涉及斯坦福大学和美国数学研究所的一个特殊学期，由四名高级研究人员、两名博士后研究员和两名高级研究生组成的团队，对结外表面的本质表面进行紧张和合作的研究。在过去的二十年里，两名高级调查人员一直在致力于一个程序，通过结外部关键表面的属性来检测特定的结。他们的方法在识别这种性质方面非常成功，但并没有导致能够实现这种结的方法。另外两名高级研究人员在过去三年中开发了实现具有有趣特性的结的新方法。这个项目将把这两种方法结合起来，以解决这一数学研究领域中最突出的问题之一。低维拓扑学将许多数学研究领域聚集在一起，并为所有数学领域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、拓扑学和复杂性理论做出了重要贡献。它在物理学、计算机可视化和医学建模方面产生了影响。

Proposal ID: DMS-9978071Proposal Title: Surfaces in knot exteriors: The Lopez Conjecture.Principal Investigator: Dr. William H. JacoAbstract: This project involves a Special Semester at Stanford University and The American Institute of Mathematics bringing together a team of four senior investigators, two postdoctoral fellows and two advanced graduate students for an intense and collaborative study of essential surfaces in knot exteriors. Two of the senior investigators have been working on a program over the past two decades to detect certain knots by properties of essential surfaces in the knot exterior. Their methods have been very successful in identifying such properties but do not lead to methods that will enable one to realize such knots. The other two senior investigators have in the last three years developed new methods for realizing knots having interesting properties. This project will bring these two methods together to address one of the most outstanding problems in this area of mathematical research.Low-dimensional topology brings together many areas of mathematical research and provides a common ground for interaction and advance across all of mathematics. It provides a natural geometric model of most physical phenomena. Research in this area is making significant contribution to computational geometry and topology and complexity theory. It has consequences in physics, computer visualization and medical modeling.