Low-dimensional topology and links of singularities

在自然界中，奇点与突然的变化或灾难性事件有关。在复杂曲面的情况下，奇点的链接是曲面内环绕奇点或尖点的三维物体。这个物体通常具有反映奇点性质的复杂形状。在这种情况下，PI将研究一些问题有关的代数和拓扑性质的链接及其填充物是四维形状与给定的边界。该项目包括将这些形状应用于其他备受研究的几何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的相关项目活动将作出重大贡献，本科教育，课程开发，研究生以及博士后培训。PI将共同组织研究研讨会，会议，研讨会，并继续她的编辑工作在量子拓扑学，研究杂志。PI的合作研究将为三流形和四流形拓扑学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及其合作者的最新成果，其中引入了新的视角和新的工具来研究奇点链接的辛和接触拓扑。要解决的重要问题之一是辛填充之间的比较，出现在代数背景下，那些具有更一般的性质。除了填充，辛配边将被研究。PI计划将新开发的工具应用于奇异光滑四流形和奇异打结曲面的构造。另一个目标是进一步了解组合和辛现象之间的联系，特别是使用不变量，如Khovanov同调，以检测“意想不到的”单值因子分解和辛填充。该项目的目标包括对一个长期存在的关于某些类型的填充物的有限性的猜想的研究。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。

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.