CAREER: Legendrian and Contact Topology in Higher Dimensions

本课题研究所有高维的Legendrian拓扑和接触拓扑。接触结构是控制半经典射线光学中光波前行为的基本几何结构，是热力学系统的演化，通常是实值函数及其一阶变化的研究。结合其在控制理论和哈密顿动力学中的经典应用，接触拓扑最近也在流体力学和化学中发现了令人惊讶的应用，包括胆甾型液晶的研究。该项目旨在通过对勒让德波前的操作来理解这些接触结构。这些波前本质上是几何的，本项目的一部分旨在系统地描述有效的算法来操纵这些波前。这将极大地提高我们在前图组合方面定性研究许多动力系统的能力。用前图来表述问题使我们能够将奇点理论的力量引入到接触结构的研究中。因此，经典问题，如给定哈密顿系统的稳定行为，周期轨道的数量和截面表面的存在，将有可能通过前面图的组合学提供的前端发展观点得到回答。从技术上讲，本项目将研究协维-2接触子流形的分类，并发展所有高维的Legendrian Kirby微积分和Weinstein结构的研究。该项目解决了高维接触结理论中的存在性和非唯一性问题，Weinstein柄体及其高维Legendrian波前的构造和操作。该项目包括这些目标在复杂几何中的新应用，通过研究斯坦因结构、数学物理，包括在同调镜像对称和奇点理论中的应用。本项目中提出的技术也将立即应用于高维辛拓扑中的花理论不变量的计算以及拉格朗日填充和包裹的Fukaya范畴的研究。该项目的核心工具结合了高维几何拓扑、h原理理论和Legendrian不变量。该项目包括一个关于代数Legendrian连杆的拉格朗日填充分类的中心猜想和高维接触子流形的存在唯一性h原理。用于研究这些猜想的技术包括通过操纵勒让德锋面和褶皱奇点获得的显式构造，以及使用伪全纯曲线来计算它们的接触不变量。此外，该项目还将纳入高维接触结的不变量的发展，并使用微局部轴理论作为簇代数理论的指导连接。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。

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.