Low-dimensional topology and links of singularities

低维拓扑和奇点链接

• 批准号: 2304080
• 负责人: Olga Plamenevskaya
• 金额: $ 37.27万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304080&HistoricalAwards=false
• 关键词: Low; dimensional; topology; links; singularities

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.

在自然界中，奇点与突然变化或灾难性事件相关，在复杂表面的情况下，奇点的链接是围绕奇点的三维对象，或表面内的尖点。该对象通常具有复杂的形状。在这种情况下，PI 将研究一些与链接及其填充物（具有给定边界的四维形状）相关的代数和拓扑属性的问题。该研究在数学和科学的其他几个领域都有应用，该项目将为本科生教育、课程开发和研究生做出重大贡献。以及博士后培训。 PI 将共同组织研究研讨会、会议、讲习班，并继续她在研究期刊 Quantum Topology 的编辑工作。 PI 的合作研究将为三元和三元拓扑的重要领域做出重大贡献。四流形，与代数几何和组合学相关。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颖的工具来研究奇点的辛和接触拓扑。要解决的重要问题之一是代数环境中出现的辛填充与具有更一般性质的填充之间的比较除了填充之外，辛配边也将是。研究人员计划将新开发的工具应用于奇异的光滑四流形和奇异的结曲面的构造，特别是使用诸如 Khovanov 之类的不变量。同源性来检测“意外的”单向分解和辛填充。该项目的目标包括对某些类型填充的有限性的长期猜想进行研究。该奖项反映了这一点。通过使用基金会的智力价值和更广泛的影响审查标准进行评估，NSF 的法定使命被认为值得支持。