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，奇点的链接是包围曲面内奇点或尖点的三维对象。该对象通常具有反映奇点属性的复杂形状。在这种情况下，PI将研究一些与具有给定边界的四维形状的链接及其填充的代数和拓扑性质相关的问题。该项目包括将这些形状应用于几何中其他备受研究的问题，以及它们在某些新结构中可能扮演的角色。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、课程开发以及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、会议和研讨会，并继续她在研究期刊Quantum Topology的编辑工作。PI的合作研究将在与代数几何和组合数学有关的三维和四维流形拓扑的重要领域做出重要贡献。这个项目将进一步发展和扩展PI和她的合作者的最新成果，其中引入了一个新的视角和新的工具来研究奇异链环的辛拓扑和接触拓扑。要解决的一个重要问题是比较在代数背景下出现的辛填充和具有更一般性质的辛填充之间的比较。除了填充外，还将学习辛边线。PI计划寻找新开发的工具在构造奇异光滑四流形和奇异纽结曲面方面的应用。另一个目标是了解组合现象和辛现象之间的进一步联系，特别是使用不变量，如Khovanov同调来检测“意想不到的”单调因式分解和辛填充。该项目的目标包括关于某些类型填充物的有限性的长期猜想的工作。该奖项反映了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。

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.