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The Topology of Contact Type Hypersurfaces and Related Topics

接触型超曲面拓扑及相关主题

基本信息

批准号：
2105525
负责人：
Bulent Tosun
金额：
$ 15.64万
依托单位：
University of Alabama Tuscaloosa
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105525&HistoricalAwards=false
关键词：
Topology Contact Type Hypersurfaces Related

项目摘要

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.

该项目由拓扑学和已建立的激励竞争研究计划(EPSCoR)共同资助，围绕三维和四维空间的几何和拓扑、称为辛结构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中也自然而然地产生和应用。这些技巧起源于规范理论、Floer理论、全纯曲线技术，这些定理和猜想在光滑流形拓扑、双曲几何、动力学、多变量复变分析和复代数几何等领域都有应用和联系。在他广泛和协作研究的基础上，PI旨在研究许多位于低维辛拓扑/接触拓扑和光滑流形拓扑的交点上的独特问题和猜想，以及复杂的分析。本文的研究成果将极大地影响我们目前对低维几何拓扑的理解。作为该项目的一个组成部分，PI将帮助指导他研究领域的研究生和博士后研究员，通过组织研讨会、研讨会和会议来维持阿拉巴马大学活跃的拓扑学小组，并投入时间在塔斯卡卢萨建立一个数学圈。PI将调查低维光滑流形和定义在其上的某些几何/解析结构之间的潜在联系。这个项目的第一个长期研究目标是了解三维流形嵌入问题的辛和复几何方面，以及与之相关的辛/全纯刚性现象。具体地说，PI将致力于彻底解决Gompf的猜想，即这种嵌入对于非平凡的Brieskorn球面是不可能的，确定具有指定边界的接触型超曲面和有理凸Stein域的拓扑，并探索它们对光滑4-流形拓扑的影响。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而令人满意的联系。对于后一个项目，PI将专门研究闭合的、定向的3-流形上紧密和可填充接触结构的存在和分类这两个突出问题。后一个项目的许多特殊情况都是由于PI与他的合作者和该领域的其他研究人员的工作而被理解的，但它们之间的联系仍有待探索。该奖项反映了NSF的法定使命，并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，被认为值得支持。