Instanton homology in low-dimensional topology

这个项目将开发工具来研究相交表面的几何问题，使用以前没有看到在这一领域的应用数学领域产生的新工具。自古以来，数学家们就在三维欧氏空间中研究球面、椭球面、抛物面和双曲面等曲面。希腊人所熟悉的这些曲面，都可以用笛卡尔几何中的二次方程来描述。现代代数几何，主要是在20世纪和21世纪发展起来的，提供了研究由高次方程定义的曲面的工具。虽然一个五次方程（例如）可以定义三维空间中的光滑曲面，但一对这样的方程将定义一对曲面，并且两个曲面的交点将是空间中的曲线。该项目将寻求回答长期存在的问题，即以这种方式产生的曲线可能存在奇点，使用的工具首先出现在原子和核尺度上描述自然的基本力。这些工具也将用于解决有关网络流量的问题。同时，该项目将培训研究生，并向该领域的研究人员传播成果。与T. S. Mrowka，PI将开发空间三价图和一般三流形中的结的瞬子同调的性质。特别是，将开发的工具，使计算的瞬子同源性更一般比目前可能的。这种瞬子同调是在以前的工作中使用规范理论构造的，该规范理论与旋转群SO（3）中的纽结或图的补的基本群的表示有关。当使用局部系数系统定义时，结和链接的瞬子同调产生了对嵌入曲面的拓扑结构的新约束，嵌入曲面的边界是给定的结或链接。具体来说，它产生的信息可能属这种表面和他们的奇点的数量。最终的目标是发展这些工具，使它们能够回答代数几何中有关代数曲线拓扑的长期问题。例如，PI将寻求两个光滑五次曲面是否可以相交于亏格为零的不可约奇异曲线的否定答案。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will develop tools to study questions about the geometry of intersecting surfaces, using novel tools arising from areas of mathematics that have not previously seen application in this area. Since antiquity, mathematicians have studied surfaces such as spheres, ellipsoids, paraboloids and hyperboloids in three-dimensional Euclidean space. These surfaces, familiar to the Greeks, can all be described in cartesian geometry by equations of the second degree. Modern algebraic geometry, as developed primarily in the 20th and 21st centuries, provides tools to study surfaces defined by equations of higher degree. While a single equation of degree five (for example) may define a smooth surface in three-space, a pair of such equations will define a pair of surfaces, and the intersection of the two surfaces will be a curve in space. The project will seek to answer long-standing questions about the possible singularities of a curve arising in this way, using tools that first arose in the description of the fundamental forces of nature at the atomic and nuclear scale. These same tools will also be used in addressing questions about network flows. At the same time, the project will train graduate students and disseminate results to researchers in the area.The project activity will be in the following specific areas. In collaboration with T. S. Mrowka, the PI will develop properties of an instanton homology for spatial trivalent graphs and for knots in general three-manifolds. In particular, tools will be developed that will enable the calculation of instanton homology more generally than is currently possible. This instanton homology was constructed in previous work using a gauge theory related to representations of the fundamental group of complement of the knot or graph in the group of rotations, SO(3). When defined using a local coefficient system, instanton homology of knots and links yields new constraints on the topology of embedded surfaces whose boundary is a given knot or link. Specifically, it yields information about the possible genus of such surfaces and the number of their singularities. The final goal is to develop these tools to the point where they will answer long-standing questions in algebraic geometry concerning the topology of algebraic curves. For example, the PI will seek a negative answer to the question of whether two smooth quintic surfaces can intersect in an irreducible singular curve of genus zero.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.