Instanton homology in low-dimensional topology

该项目将开发工具来研究相交曲面的几何问题，使用以前未在该领域应用的数学领域产生的新工具。自古以来，数学家就在三维欧几里得空间中研究球体、椭球、抛物面和双曲面等曲面。这些为希腊人所熟悉的曲面，在笛卡尔几何中都可以用二次方程来描述。现代代数几何主要是在20世纪和21世纪发展起来的，它提供了研究由高次方程定义的曲面的工具。一个单一的五次方程（例如）可以定义三维空间中的光滑曲面，一对这样的方程将定义一对曲面，两个曲面的交点将在空间中形成一条曲线。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的可能奇点，使用首先在原子和核尺度上描述自然基本力的工具。这些相同的工具也将用于解决有关网络流的问题。同时，该项目将培养研究生，并将成果传播给该地区的研究人员。项目活动将在下列具体领域进行。在与t.s. Mrowka的合作中，PI将开发空间三价图和一般三流形中的结的瞬时同调性质。特别是，将开发工具，使计算的瞬子同源性比目前可能的更普遍。这种瞬子同调是在以前的工作中使用与旋转群中结或图的补的基本群表示相关的规范理论构造的，SO(3)。当使用局部系数系统定义时，结点和连杆的瞬时同调产生了对其边界为给定结点或连杆的嵌入曲面拓扑结构的新约束。具体地说，它产生了关于这些曲面的可能属和它们的奇点数目的信息。最终目标是开发这些工具，使其能够回答代数几何中关于代数曲线拓扑的长期问题。例如，对于两个光滑的五次曲面能否相交于一条不可约的零属奇异曲线，PI将寻求一个否定的答案。该奖项反映了美国国家科学基金会的法定使命，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.