Constructions in Higher-Dimensional Contact Topology

高维接触拓扑结构

• 批准号: 1841913
• 负责人: Roger Casals Gutierrez
• 金额: $ 7.57万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-21 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1841913&HistoricalAwards=false
• 关键词: Constructions; Higher; Dimensional; Contact; Topology

This research project in pure mathematics focuses on abstract geometric structures in higher dimensions, known as contact and symplectic topology. The field intertwines rigid geometric problems, such as studying periodic orbits or counting curves passing through a given point, with flexible cut-and-paste constructions. Part of the research conducted in this proposal is the development of a combinatorial description of these structures in terms of a diagrammatic calculus, and its applications to these rigid geometric problems. This provides an accessible source of new examples and computations, and establishes the ground for strictly higher-dimensional constructions. Due to its combinatorial nature, this line of research is naturally conducive to participation from undergraduate students and young researchers. Another part of this project uses techniques that combine known elements from algebraic geometry and geometric topology with new ideas from contact and symplectic topology, and the outcome of the proposed research contributes to each of these central fields of mathematical research. The first part of this project is the development of a Legendrian calculus in the front projection, including higher-dimensional Reidemeister moves, Legendrian handle-slides and criteria for the existence of loose charts. By further establishing a relation with bi-Lefschetz fibrations, this tool will provide effective combinatorial criteria for (sub)flexibility of Weinstein structures and the computation of algebraic invariants such as the Legendrian contact differential graded algebra and the wrapped Fukaya category. This combines ideas from affine algebraic geometry and higher-dimensional contact surgery theory. The second part of this research project focuses on the detection of tightness and over-twistedness of higher-dimensional models, including bordered Legendrian open books and small neighborhoods of over-twisted models. The project will use pseudoholomorphic techniques adapted to each problem in combination with new ideas coming from the study of Legendrian submanifolds, including the study of Legendrian cobordisms and their relation to Weinstein cobordisms. In both these projects the study of compatible open books in special position have a role, and a general existence theorem will be studied with asymptotically holomorphic techniques. In addition, the project will include the study of a flexible class of Engel structures, completely determined by formal homotopy class, and related h-principles.

这个纯数学研究项目侧重于更高维度的抽象几何结构，称为接触和辛拓扑。该领域将严格的几何问题与灵活的剪切和粘贴结构交织在一起，例如研究周期轨道或计算通过给定点的曲线。该提案中进行的部分研究是根据图解演算对这些结构进行组合描述，及其在这些严格的几何问题中的应用。这提供了新示例和计算的可访问来源，并为严格的高维结构奠定了基础。由于其组合性质，这一研究方向自然有利于本科生和年轻研究人员的参与。该项目的另一部分使用的技术将代数几何和几何拓扑的已知元素与接触和辛拓扑的新思想相结合，并且所提出的研究成果对数学研究的每个核心领域都有贡献。该项目的第一部分是在前投影中开发传奇微积分，包括高维雷德迈斯特移动、传奇手柄幻灯片和松散图表存在的标准。通过进一步建立与双 Lefschetz 纤维的关系，该工具将为 Weinstein 结构的（子）灵活性和代数不变量（例如 Legendrian 接触微分分级代数和包裹 Fukaya 范畴）的计算提供有效的组合标准。这结合了仿射代数几何和高维接触手术理论的思想。该研究项目的第二部分侧重于检测高维模型的紧密性和过度扭曲度，包括有边框的传奇开放书和过度扭曲模型的小邻域。该项目将使用适合每个问题的伪全纯技术，并结合来自传奇子流形研究的新思想，包括传奇配边主义及其与韦恩斯坦配边主义的关系的研究。在这两个项目中，对特殊位置的兼容开放书的研究都发挥着作用，并且将使用渐近全纯技术来研究一般存在定理。此外，该项目还将包括研究灵活的恩格尔结构类，完全由形式同伦类和相关的 h 原理决定。