权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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和接触拓扑在所有更高的维度。接触结构是控制半经典射线光学中光波前行为、光学系统演化以及一般实值函数及其一阶变分研究的基本几何结构。结合其经典的应用控制理论和哈密顿动力学，接触拓扑最近也发现了令人惊讶的应用流体力学和化学，包括胆甾相液晶的研究。该项目旨在通过操纵勒让德波前来理解这些接触结构。这些波前在本质上是几何的，这个项目的一部分旨在系统地描述操纵这些波前的有效算法。这将大大提高我们的能力，定性研究许多动力系统的组合方面的前图。用前图来表述问题，使我们能够把奇点理论的力量运用到接触结构的研究中。因此，经典的问题，如一个给定的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，将有可能通过前端开发的角度提供的组合前图回答。本项目将研究余维的分类-2切触子流形，并发展了所有高维的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的存在性和非唯一性问题，温斯坦天体及其高维勒让德波前的构造和操纵。该项目包括这些目标的复杂几何的新应用，通过斯坦因结构，数学物理，包括应用同调镜像对称，和奇点理论的研究。在这个项目中提出的技术也将有直接的应用程序在高维辛拓扑的Floer理论不变量的计算和拉格朗日填充和包裹福谷范畴的研究。该项目的核心工具结合了联合收割机高维几何拓扑学、h-原理理论和勒让德不变量。该项目包括一个关于代数勒让德链接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性和唯一性h-原理。在研究这些结构时所采用的技术包括通过勒让德阵面和褶皱奇点的操纵获得的显式构造，以及为了计算它们的接触不变量而使用的伪全纯曲线。此外，该项目还将纳入更高维度接触结不变量的开发以及微局部层理论的使用，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价值和更广泛的影响审查标准进行评估，被认为值得支持。