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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.

本项目研究所有更高维度的Legendrian和Contact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，热力学系统的演化，以及一般实值函数及其一阶变化的研究。除了在控制理论和哈密顿动力学中的经典应用，接触拓扑学最近还在流体力学和化学中发现了令人惊讶的应用，包括胆甾相液晶的研究。该项目旨在通过操纵传奇波阵面来提供对这些接触结构的了解。这些波前本质上是几何的，该项目的一部分旨在系统地描述操纵这些波前的有效算法。这将极大地提高我们根据前图的组合学定性研究许多动力系统的能力。前图中的语法问题使我们能够将奇点理论的力量带入接触结构的研究中。因此，经典的问题，如给定的哈密顿系统的稳定性，周期轨道的数目和截面曲面的存在，将有可能通过前图组合提供的前端发展视角来回答。从技术上讲，这个项目将研究余维2接触子流形的分类，并在所有更高的维度上发展Legendrian Kirby演算和Weinstein结构的研究。该项目解决了高维接触结理论中存在和不唯一的问题，温斯坦手柄及其高维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，通过对Stein结构、数学物理的研究，包括对同调镜面对称性和奇点理论的应用。本项目中提出的方法也将直接应用于高维辛拓扑中Floer理论不变量的计算，以及拉格朗日填充和包裹Fukaya范畴的研究。该项目的中心工具结合了高维几何拓扑学、h-原理理论和勒让德不变量。该项目包括关于代数Legendrian环的拉格朗日填充分类的一个中心猜想和关于高维接触子流形的存在唯一性h-原理。研究这些猜想所采用的技术包括通过操纵Legendrian阵面和褶皱奇点得到的显式结构，以及使用伪全纯曲线来计算它们的接触不变量。此外，该项目还将包括高维接触结不变量的开发和微局部鞘理论的使用，作为与簇代数理论的指导联系。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。