Low-Dimensional and Contact Topology of Links of Surface Singularities

本专题着重于三维和四维空间中曲面几何的几个相互关联的主题。在一个由代数方程定义的复杂曲面上，在一个光滑区域中的尖点称为孤立奇点。在自然界中，奇点是在突变或灾难事件发生时产生的，因此从不同的角度研究奇点对所有科学分支都很重要。在几何学中，一个被称为奇点链的物体环绕着表面上的尖点;这个物体在没有尖点的情况下类似于一个球体，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支持的其他结构。一个特别的重点将是在物理学中出现的称为接触结构的平面集合。Plamenevskaya的研究将有助于数学的几个领域：低维拓扑，辛和接触拓扑，代数几何和组合学。 她与该项目有关的其他活动将为本科教育和研究生及博士后培训做出重大贡献。Plamenevskaya正在努力提高本科生对当前拓扑学研究的认识，并促进妇女参与数学。她共同组织研究研讨会，会议和讲习班，为不同的观众，从研究生到杰出的研究人员。 具体地说，PI将与奇异连杆上的复切线引起的正则接触结构一起工作。该项目的一个特定目标是了解Stein填充/cobordisms等接触结构和平滑/变形的表面奇异性之间的关系，并解决各种问题的奇异性理论和辛和接触拓扑的相互作用。PI以前的工作（与Ghiggini和Golla一起）表明，具有约化基金属环的有理表面奇点，其变形理论很好地理解，精确地对应于平面接触结构类，其填充可以通过表面超同态来研究。这打开了大门，延长某些结果从代数几何辛拓扑以及应用辛拓扑奇异理论。 Plamenevskaya将使用一系列工具来研究这些问题，包括低维结构，如开卷分解和辛帽，复杂和辛线排列，以及某些Floer同调和组合不变量。她打算发现上述结构和思想之间的进一步联系，这可能会导致新的结果映射类组和组合线安排。Plamenevskaya还将研究一些相关的问题，如某些结构格上同调（组合理论介绍了由Nemethi链接的奇点）和他们的关系Heegaard弗洛尔同源性。 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。

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.