The Topology of Contact Type Hypersurfaces and Related Topics

该项目由拓扑学和刺激竞争研究的既定计划（EPSCoR）共同资助，围绕三维和四维空间的几何和拓扑，称为辛和接触结构的数学对象，以及它们之间的相互作用。辛几何和接触几何不仅是经典物理某些方面的自然语言，而且在现代数学和数学物理的许多领域也自然出现并找到应用。这些技术源于规范理论、花理论、全纯曲线技术，这些定理和猜想在许多领域都有应用和联系，例如：光滑流形拓扑、双曲几何、动力学、多变量复分析和复杂代数几何。在他广泛的合作研究的基础上，PI旨在研究许多独特的问题和猜想，这些问题和猜想位于低维辛/接触拓扑和光滑流形拓扑的交叉点，以及复杂分析。提出的研究及其成果将极大地影响我们目前对低维几何拓扑的理解。作为这个项目的一个组成部分，PI将在他的研究领域帮助指导研究生和博士后，通过组织研讨会、讲习班和会议在阿拉巴马大学维持一个活跃的拓扑小组，并投入时间在塔斯卡卢萨发起一个数学圈。PI将研究低维光滑流形与在其上定义的某些几何/解析结构之间的潜在联系。本项目的第一个长期研究目标是了解4空间中3流形嵌入问题的辛和复几何方面，以及由此产生的相关辛/全纯刚性现象。具体来说，PI将致力于完全解决Gompf的猜想，即这种嵌入对于非平凡的Brieskorn球是不可能的，确定具有规定边界的接触型超曲面和合理凸Stein域的拓扑结构，并探索它们对光滑4流形拓扑结构的影响。第二个长期研究目标是在规范理论不变量、辛/接触几何和双曲几何之间建立具体和令人满意的联系。对于后一个项目，PI将专门研究两个突出的问题，即封闭定向3流形上紧密和可填充接触结构的存在和分类。由于PI与他的合作者和该领域的其他研究人员的工作，后一个项目的许多特殊情况得到了理解，但它们之间的联系仍有待探索。该奖项反映了美国国家科学基金会的法定使命，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, jointly funded by Topology and the Established Program to Stimulate Competitive Research (EPSCoR), centers around the geometry and topology of 3- and 4-dimensional spaces, mathematical objects known as symplectic and contact structures, and interactions between these. Symplectic and contact geometries are not just a natural language for some aspects of classical physics, but also naturally arise and find applications in many areas of modern mathematics and mathematical physics. The techniques spring from gauge theory, Floer theory, holomorphic curve techniques, and the theorems and conjectures find applications and connections in several fields, such as: smooth manifold topology, hyperbolic geometry, dynamics, complex analysis in several variables, and complex algebraic geometry. Building on his extensive and collaborative research, the PI aims to study many unique questions and conjectures that sit at the intersection of symplectic/contact topology and smooth manifold topology in low dimensions, and complex analysis. The proposed research and its outcomes will greatly impact our current understanding of geometric topology in low dimensions. As an integral part of this project, the PI will help mentor graduate students and postdoctoral fellows in his research area, maintain an active topology group at the University of Alabama by organizing seminars, workshops and conferences, and devote time to initiate a math circle in Tuscaloosa. The PI will investigate underlying connections between low dimensional smooth manifolds and certain geometric/analytic structures defined on them. The first long-term research objective of this project is to understand symplectic and complex geometric aspects of 3-manifold embedding problem in 4-space, and related symplectic/holomorphic rigidity phenomenon that develops. Specifically, the PI will work towards a complete resolution of Gompf’s conjecture that such embeddings are impossible for non-trivial Brieskorn spheres, determining the topology of contact type hypersurfaces and rationally convex Stein domains with prescribed boundary, and exploring their implications for smooth 4-manifold topology. The second long-term research objective concerns contributing concrete and satisfying connections between gauge theoretical invariants, symplectic/contact geometry and hyperbolic geometry. Towards this latter project, the PI will specifically work on two outstanding problems of existence and classification of tight and fillable contact structures on closed, oriented 3-manifolds. Many special cases of the latter project are understood due to work of the PI with his collaborators and other researchers in the area, but what links them remains to be explored.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.