Instanton homology in low-dimensional topology

低维拓扑中的瞬子同调

• 批准号: 2304877
• 负责人: Peter Kronheimer
• 金额: $ 40万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304877&HistoricalAwards=false
• 关键词: Instanton; homology; low; dimensional; topology

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.

该项目将开发工具来研究相交曲面几何问题，使用以前从未在该领域得到应用的数学领域产生的新颖工具。自古以来，数学家就研究了三维欧几里得空间中的球体、椭球体、抛物面和双曲面等表面。这些希腊人所熟悉的曲面都可以用笛卡尔几何中的二次方程来描述。主要发展于 20 世纪和 21 世纪的现代代数几何提供了研究由高阶方程定义的曲面的工具。虽然单个五次方程（例如）可以定义三空间中的光滑表面，但一对这样的方程将定义一对表面，并且两个表面的交集将是空间中的曲线。该项目将寻求回答有关以这种方式出现的曲线可能出现奇点的长期存在的问题，使用最初在原子和核尺度上描述自然基本力时出现的工具。这些相同的工具也将用于解决有关网络流的问题。同时，该项目将培训研究生并向该领域的研究人员传播成果。该项目活动将在以下具体领域进行。 PI 将与 T. S. Mrowka 合作，开发空间三价图和一般三流形中的结的瞬子同调特性。特别是，将开发能够比目前更普遍地计算瞬子同源性的工具。这种瞬子同源性是在之前的工作中使用规范理论构建的，规范理论与旋转群中结或图的基本补群的表示相关，SO(3)。当使用局部系数系统定义时，结和链接的瞬子同源性对边界为给定结或链接的嵌入表面的拓扑产生新的约束。具体来说，它产生有关此类表面的可能属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，对于两个光滑五次曲面是否可以在不可约奇异曲线零格中相交的问题，PI 将寻求否定答案。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值进行评估，被认为值得支持以及更广泛的影响审查标准。