Invariants of 3-Manifolds and Their Applications

DMS-0206464 Walter Neuman该项目的一部分涉及研究人员成功应用拓扑方法研究复杂曲面的奇异性和平面仿射曲线的拓扑结构的延续。一个需要解决的问题是在给定拓扑结构下奇点的显式解析描述。一般的问题是出了名的困难，但调查员和J. Wahl已经确定了显着广泛的一类拓扑结构，其中显着的进步可以取得。 另一个问题是二元多项式的全局拓扑的研究，以及分类问题，这是既有内在的兴趣，也有可能有助于著名的雅可比猜想的解决。 在这项工作中的一个重要工具是艾森巴德和调查员的拼接图和最近的推广。 在一个相当不同的方向，调查员将研究不变量的三维流形，涉及数论和代数，即所谓的字符和特征值品种和布洛欣变量。这些不变量之间的联系正在出现，这些不变量将形成拓扑学和代数/数论。代数几何，本质上是研究多项式族的零点集，一直是基础研究的重要领域，在控制论和通信等各种应用中也很重要。 这种零集的出现，例如，作为参数空间的过程和thecontrolspace的自动机。奇点（拓扑性质改变的地方）具有根本的意义。该项目有两个目标：为了更好地理解解析描述和奇点拓扑之间的联系，并研究低维流形的不变量，将拓扑与其他领域联系起来。

DMS-0206464Walter NeumanPart of this project concerns a continuation of the investigator'ssuccessful application of topological methods to the study ofsingularities of complex surfaces and the topology of plane affinecurves. One issue to be addressed is explicit analytic description ofsingularities with given topology. The general question is notoriouslydifficult, but the investigator and J. Wahl have identified aremarkably broad class of topologies where significant advance can bemade. Another issue is the study of the global topology oftwo-variable polynomials, as well as classification issues, which areboth of intrinsic interest and also have potential to contributeto the resolution of the famous Jacobian Conjecture. A significanttool in this work are the splice diagrams of Eisenbud and theinvestigator and a recent generalization of them. In a ratherdifferent direction, the investigator will study invariants of3-dimensional manifolds that relate to number theory and algebra,namely the so-called character and eigenvalue varieties and the Blochinvariant. Connections are emerging between these invariants that willinform both topology and algebra/number theory.Algebraic Geometry, which is essentially the study of the zero sets offamilies of polynomials, has always been an important area offundamental research, and is also important in such diverseapplications as control theory and communications. Such zero-setsarise, for instance, as the parameter spaces of processes and thecontrol spaces of automata. The singularities (places where the natureof the topology changes), are of fundamental significance. Thisproject has two thrusts: to understand better the link between theanalytic description and the topology of singularities, and toinvestigate invariants of low dimensional manifolds that link topologywith other fields.