CAREER: Legendrian and Contact Topology in Higher Dimensions

职业：高维中的勒让德和接触拓扑

基本信息

批准号：
1942363
负责人：
Roger Casals Gutierrez
金额：
$ 45万
依托单位：
University of California-Davis
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1942363&HistoricalAwards=false
关键词：
CAREER Legendrian Contact Topology Higher

项目摘要

The present project studies Legendrian and contact topology in all higher dimensions. Contact structures are the underlying geometric structures reigning the behaviour of light wavefronts in semiclassical ray optics, the evolution of thermodynamical systems and, in general, the study of real-valued functions and their first-order variations. In conjunction with its classical applications to control theory and Hamiltonian dynamics, contact topology has also recently found surprising applications to fluid mechanics and chemistry, including the study of Cholesteric Liquid Crystals. This project aims at providing an understanding of these contact structures through the manipulation of Legendrian wavefronts. These wavefronts are geometric in nature and part of this project aims at systematically describing efficient algorithms to manipulate these fronts. This will significantly enhance our ability to qualitatively study many dynamical systems in terms of the combinatorics of front diagrams. Phrasing problems in terms of front diagrams allows us to bring the strength of the theory of singularities into the study of contact structures. Thus, classical questions such as the stable behaviour of a given Hamiltonian system, the number of periodic orbits and the presence of surfaces of section will potentially be answered through the front-end development perspective offered by the combinatorics of front diagrams.In technical terms, this project shall be studying the classification of codimension-2 contact submanifolds and developing the study of Legendrian Kirby Calculus and Weinstein structures in all higher dimensions. The project addresses existence and non-uniqueness questions in higher-dimensional contact knot theory, the construction and manipulation of Weinstein handlebodies and their higher-dimensional Legendrian wavefronts. The project includes new applications of these goals to complex geometry, through the study of Stein structures, mathematical physics, including applications to Homological Mirror Symmetry, and singularity theory. The techniques proposed in this project will also have immediate applications to the computation of Floer-theoretic invariants in higher-dimensional symplectic topology and the study of Lagrangian fillings and the wrapped Fukaya category. The central tools in the project combine higher-dimensional geometric topology, the theory of h-principles and Legendrian invariants. The project includes a central conjecture on the classification of Lagrangian fillings for algebraic Legendrian links and an existence and uniqueness h-principle for higher-dimensional contact submanifolds. The techniques to be employed in the study of these conjectures include explicit constructions obtained via the manipulation of Legendrian fronts and wrinkled singularities, and the use of pseudo-holomorphic curves in order to compute their contact invariants. In addition, the project will also incorporate the development of invariants for higher-dimensional contact knots and the use of microlocal sheaf theory as a guiding connection to the theory of cluster algebras.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.

本项目研究Legendrian和所有更高维度的接触拓扑。接触结构是在半经典射线光学器件，热力学系统的演化以及对实价函数及其一阶变化的研究中，在半经典光学光学中的行为统治着的基本几何结构。结合其在控制理论和哈密顿动力学上的经典应用，接触拓扑最近还发现了对流体力学和化学的惊人应用，包括研究胆固醇液晶。该项目的目的是通过操纵Legendrian Wavefronts来理解这些接触结构。这些波前本质上是几何形状，该项目的一部分旨在系统地描述有效的算法来操纵这些方面。这将显着增强我们从前图的组合方法来定性研究许多动态系统的能力。在前图方面，措辞问题使我们能够将奇异性理论的强度带入接触结构的研究中。 Thus, classical questions such as the stable behaviour of a given Hamiltonian system, the number of periodic orbits and the presence of surfaces of section will potentially be answered through the front-end development perspective offered by the combinatorics of front diagrams.In technical terms, this project shall be studying the classification of codimension-2 contact submanifolds and developing the study of Legendrian Kirby Calculus and Weinstein structures in all higher dimensions.该项目解决了高维接触结理论，温斯坦车把及其高维传统波兰的构建和操纵中的存在和非唯一性问题。该项目包括通过研究Stein结构，数学物理学，包括对同源镜对称性的应用和奇异理论，包括这些目标在复杂几何形状上的新应用。该项目中提出的技术还将立即应用于高维象征性拓扑中浮动理论不变性的计算以及Lagrangian填充物的研究和包裹的福卡亚类别。该项目的中心工具结合了高维几何拓扑，H-Principles和Legendrian不变性的理论。该项目包括有关代数Legendrian链接拉格朗日填充物分类的中央猜想，以及对高维触点子曼属的存在和独特性H原则。这些猜想的研究中要采用的技术包括通过操纵Legendrian阵线和皱纹的奇异性获得的明确结构，以及使用伪旋晶曲线以计算其接触不变性。此外，该项目还将结合不变的较高接触结的发展，以及将微局部捆绑理论用作与集群代数理论的指导联系。该奖项反映了NSF的法定任务，并被认为是通过使用该基金会的知识分子和更广泛影响的评估来评估的支持，并被认为是值得的。