Low-Dimensional and Contact Topology of Links of Surface Singularities

这个项目集中于三维和四维曲面几何中的几个相互关联的主题。在由代数方程定义的复杂曲面上，在其他光滑区域中的尖点状锐点称为孤立奇点。在自然界中，当突然变化或灾难性事件发生时，奇点就会出现，因此从不同角度研究奇点对科学的所有分支都是重要的。在几何学中，称为奇点链接的对象围绕曲面上的尖点；该对象在没有尖点的情况下类似于球体，但通常具有编码奇点属性的复杂形状。PI将研究这些形状的属性以及它们所支持的附加结构。一个特别的焦点将是物理学中出现的称为接触结构的平面的集合。Plamenevskaya的研究将有助于数学的几个领域：低维拓扑学、辛拓扑、接触拓扑学、代数几何和组合学。她与该项目相关的其他活动将对本科教育以及研究生和博士后培训做出重大贡献。Plamenevskaya正在努力提高本科生对拓扑学研究现状的认识，并为女性参与数学做出贡献。她为不同的受众共同组织研究研讨会、会议和研讨会，从研究生到杰出的研究人员。具体地说，PI将与奇点链接上的复杂切线引起的正则接触结构一起工作。该项目的一个特别目标是了解这种接触结构的Stein填充/密线与曲面奇点的平滑/变形之间的关系，并解决关于奇点理论与辛拓扑和接触拓扑学相互作用的各种问题。PI以前的工作(与Ghiggii和Golla)表明，具有简化基本周期的有理曲面奇点，其变形理论是众所周知的，精确地对应于一类平面接触结构，其填充可以通过曲面微分同胚来研究。这为将某些结果从代数几何推广到辛拓扑以及将辛拓扑应用于奇点理论打开了大门。Plamenevskaya将使用一系列工具来探索这些问题，包括低维结构，如开卷分解和辛帽、复线排列和辛线排列，以及某些Floer同调和组合不变量。她打算发现上述结构和想法之间的进一步联系，这可能会进一步导致在绘制类群和线条排列的组合方面的新结果。Plamenevskaya还将研究一些相关的问题，例如格上同调中的某些结构(Nemethi为奇点链接引入的组合理论)以及它们与Heegaard Floer同调的关系。几位合作者将参与国际和平基金会的工作。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 focuses on several interrelated topics in geometry of surfaces in dimensions three and four. On a complex surface defined by algebraic equations, a cusp-like sharp point in an otherwise smooth region is called an isolated singularity. In nature, singularities occur when sudden changes or catastrophic events happen, thus the study of singularities from various viewpoints is important to all branches of science. In geometry, an object known as the link of singularity encircles the sharp point on the surface; this object resembles a sphere in the absence of a sharp point but typically has a complicated shape encoding properties of the singularity. The PI will study the properties of such shapes and the additional structures they support. A particular focus will be on collections of planes called contact structures that arise in physics. Plamenevskaya's research will contribute to several areas of mathematics: low-dimensional topology, symplectic and contact topology, algebraic geometry, and combinatorics. Her other activities related to the project will make significant contributions to undergraduate education and graduate and postdoctoral training. Plamenevskaya is working to increase awareness of current research in topology among undergraduate students, as well as contribute to participation of women in mathematics. She co-organizes research seminars, conferences, and workshops for a diverse audience, from graduate students to distinguished researchers. Specifically, the PI will work with the canonical contact structure induced by the complex tangencies on the link of singularity. A particular goal of the project is to understand the relation between Stein fillings/cobordisms of such contact structures and smoothings/deformations of the surface singularity, and to address a variety of questions concerning the interplay of the singularity theory and symplectic and contact topology. The PI's previous work (with Ghiggini and Golla) shows that rational surface singularities with reduced fundametal cycle, whose deformation theory is well-understood, correspond precisely to the class of planar contact structures, whose fillings can be studied via surface diffeomorphisms. This opens the door to extending certain results from algebraic geometry to symplectic topology as well as to applications of symplectic topology to singularity theory. Plamenevskaya will pursue these questions using a range of tools, including low-dimensional constructions such as open book decompositions and symplectic caps, complex and symplectic line arrangements, and certain Floer-homological and combinatorial invariants. She intends to discover further connections between the aforementioned constructions and ideas, which may additionally lead to new results on mapping class groups and combinatorics of line arrangements. Plamenevskaya will also study a number of related questions, such as certain constructions in lattice cohomology (a combinatorial theory introduced by Nemethi for links of singularities) and their relation to Heegaard Floer homology. Several collaborators will be involved in the PI's work.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.