Invariants in Low Dimensional Topology and Geometric Group Theory

这项建议的目的是促进对低维拓扑学、几何群论以及逻辑和计算机科学的基本理解。这个方案中的三个项目针对低维拓扑中一个非常有前途和活跃的领域，即经典纽结的Vassiliev不变量的研究。其目的一方面是为了揭示更多的内部结构，但另一方面也是为了使它们与拓扑问题更相关，例如经典纽结是否可以成为四维球中圆盘的边界。与这一领域有些关联的是四维流形中两维球面相交的Schneiderman-Teichner理论。与施奈德曼和泰克纳一起，私家侦探希望填补缺失的一步，并将这一理论完善为完整的分类方案。另一组项目围绕着被称为图形同调的代数对象，它与流形的不变量的研究、曲面的映射类群的研究、重要群OUT(F_N)的研究以及形变量化问题有关。P.I.建议一方面深入研究图同调的一般理论，另一方面分析与拓扑学相关的具体例子。更具体地说，P.I.将继续与Vogtmann合作，将Out(F_N)的已知有理同调类集合从2扩大到无穷大。最后，在探讨拓扑学和计算机科学之间的接口时，P.I.建议利用Mike Freedman的思想，采取一些步骤来区分P和NP。这一建议涉及到数学的几个核心领域的基础研究，包括拓扑学、群论和计算机科学。提案中的项目旨在在我们对一些发展最快的子学科的理解上取得重大突破。这些项目的一个共同主题是阐明看似不同的领域之间的联系，比如拓扑学和计算机科学。例如，P.I.将探索Mike Freedman的想法，将拓扑学应用于研究算法如何有效地解决问题。这项拟议的研究提供了充分的机会，让本科生和研究生有意义地参与进来。

The aim of this proposal is to advance fundamental understanding in low dimensional topology, geometric group theory, as well as logic and computer science. Three projects in this proposal target a very promising and active area of low dimensional topology, namely the study of Vassiliev invariants of classical knots. The aim is to, on the one hand, expose more internal structure, but on the other hand to make them more relevant to topological questions, such as whether a classical knot can be the boundary of a disk in the four dimensional ball. Somewhat related to this area is the Schneiderman-Teichner theory of intersections of two dimensional spheres in 4-manifolds. Together with Schneiderman and Teichner, the P.I. hopes to fill in a missing step and complete the theory to a complete classification scheme. Another cluster of projects revolves around the algebraic object called "graph homology," which has relevance to the study of invariants of manifolds, the study of the mapping class group of surfaces, the study of the important group Out(F_n), and to the deformation quantization problem. The P.I. proposes to make a deeper study of the general theory of graph homology, on the one hand, while on the other analyzing the specific examples of relevance to topology. More specifically the P.I. will continue a program, joint with Vogtmann, for enlarging the known set of rational homology classes for Out(F_n) from 2 to infinity. Lastly, probing the interface between topology and computer science, the P.I. proposes to take some steps toward distinguishing P from NP, utilizing ideas of Mike Freedman.This proposal concerns fundamental research in several core areas of mathematics, including topology, group theory and computer science. The projects in the proposal aim at making significant breakthroughs in our understanding of some of the most rapidly developing subdisciplines. A common theme to the projects is the elucidation of connections between seemingly disparate fields, such as topology and computer science. For example, the P.I. will explore ideas of Mike Freedman to apply topology to the study of how efficiently alogrithms can solve problems. The proposed research offers ample oppurtunity to meaningfully involve undegraduates and graduate students.