Instanton homology in low-dimensional topology

这个项目将开发工具来研究关于相交曲面的几何问题，使用来自数学领域的新工具，这些领域以前没有在这一领域中应用过。自古以来，数学家们一直在研究三维欧氏空间中的球面、椭球面、抛物面和双曲面等曲面。希腊人熟悉的这些曲面都可以用二次方程用笛卡尔几何来描述。主要在20世纪和21世纪发展起来的现代代数几何为研究由高次方程定义的曲面提供了工具。虽然一个五次方程(例如)可以定义三个空间中的光滑曲面，但一对这样的方程将定义一对曲面，两个曲面的交点将是空间中的一条曲线。该项目将寻求回答长期存在的关于以这种方式产生的曲线可能出现奇点的问题，使用最初在描述原子和核尺度的自然基本力时出现的工具。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。项目活动将在以下具体领域开展。在与T.S.Mrowka的合作中，PI将开发空间三价图和一般三维流形中的纽结的瞬子同调的性质。特别是，将开发工具，使瞬子同源的计算比目前可能的更普遍。这个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)中纽结或图的补的基本群的表示有关。当使用局部系数系定义时，纽结和链环的瞬子同调对边界为给定纽结或链环的嵌入曲面的拓扑产生新的约束。具体地说，它提供了关于这类曲面的可能亏格及其奇点数量的信息。最终目标是开发这些工具，使其能够回答代数几何中有关代数曲线拓扑的长期存在的问题。例如，对于两个光滑的五次曲面是否可以在一条不可约的零亏格奇异曲线上相交的问题，PI将寻求否定的答案。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.