Low-dimensional topology and links of singularities

在自然界中，奇点与突然变化或灾难性事件有关。在复杂曲面的情况下，奇点的连杆是在曲面内环绕奇点或尖点的三维物体。这种物体通常具有反映奇点特性的复杂形状。在这种情况下，PI将研究一些与链路及其填充的代数和拓扑性质有关的问题，这些填充是具有给定边界的四维形状。该项目包括将这些形状应用于几何学中其他被广泛研究的问题，以及它们在某些新建筑中可能发挥的作用。这项研究在数学和科学的其他几个领域也有应用。PI与该项目相关的活动将对本科教育、课程开发、研究生和博士后培训做出重大贡献。PI将共同组织研究研讨会、会议、研讨会，并继续她在研究期刊《量子拓扑》（Quantum Topology）上的编辑工作。PI的合作研究将对三流形和四流形拓扑的重要领域做出重大贡献，并与代数几何和组合学相联系。该项目将进一步发展和扩展PI和她的合作者最近的成果，其中引入了新的视角和新的工具来研究奇异点链路的辛和接触拓扑。要解决的一个重要问题是在代数环境中出现的辛填充与那些具有更一般性质的辛填充之间的比较。除了填充物，辛共体也将被研究。PI计划将新开发的工具应用于奇异光滑四流形和奇异结表面的构造。另一个目标是进一步理解组合现象和辛现象之间的联系，特别是使用不变量，如Khovanov同调来检测“意外的”单因数分解和辛填充。该项目的目标包括研究一个长期存在的关于某些类型的填充物的有限性的猜想。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。

In nature, singularities are associated with sudden changes or catastrophic events. In case of a complex surface the link of a singularity is a three-dimensional object encircling the singularity, or the sharp point, within the surface. This object often has a complicated shape reflecting properties of the singularity. In this context, the PI will study a number of questions relating algebraic and topological properties of the link and its fillings that are four-dimensional shapes with given boundary. The project includes applications of these shapes to other much-studied questions in geometry and the role they may play in certain new constructions. The research has applications in several other areas of mathematics and science. The PI's activities related to the project will make significant contribution to undergraduate education, curriculum development, and graduate as well as postdoctoral training. The PI will co-organize research seminars, conferences, workshops, and continue her editorial work at Quantum Topology, a research journal. The PI’s collaborative research will make significant contributions to important areas in topology of three- and four-manifolds, with connections to algebraic geometry and combinatorics. This project will further develop and expand recent results of the PI and her collaborators, where a new perspective and novel tools were introduced to the study of symplectic and contact topology of links of singularities. One of the important questions to be addressed is the comparison between symplectic fillings that arise in the algebraic context and those that have more general nature. In addition to fillings, symplectic cobordisms will be studied. The PI plans to find applications of the newly-developed tools to constructions of exotic smooth four-manifolds and exotically knotted surfaces. Another goal is to understand further connections between combinatorial and symplectic phenomena, and in particular, to use invariants such as Khovanov homology to detect "unexpected" monodromy factorizations and symplectic fillings. The project goals include work on a long-standing conjecture about finiteness of certain types of fillings.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.