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.

纯数学研究项目的重点是更高维度的抽象几何结构，称为接触和象征性拓扑。该田间交织了严格的几何问题，例如研究周期性轨道或计数通过给定点的曲线，并具有柔性的剪切结构。该提案中进行的研究的一部分是根据图解的微积分对这些结构的组合描述的发展，以及其在这些僵化几何问题上的应用。这提供了新的示例和计算的可访问来源，并为严格的高维结构建立了基础。由于其组合性质，这种研究线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了将代数几何学和几何拓扑结合的已知元素与接触和符号拓扑的新思想相结合的技术，而拟议的研究的结果为数学研究的每个中央领域做出了贡献。该项目的第一部分是在前投影中开发了一个传统的演算，包括更高维度的雷迪德动作，legendrian的手柄滑动和存在宽松图表的标准。通过进一步建立与Bi-Lefschetz纤维的关系，该工具将为Weinstein结构的（子）灵活性提供有效的组合标准，并计算代数不变性物质，例如Legendrian接触式差异分级代数和包装的Fukaya类别。这结合了仿射代数几何形状和高维接触手术理论的思想。该研究项目的第二部分重点是检测高维模型的紧密度和过度扭转，包括边界的Legendrian Open Books和过度扭转模型的小社区。该项目将使用适合每个问题的假酚类技术与研究Legendrian Submanifold的新想法相结合，包括研究Legendrian Cobordism及其与Weinstein Cobordism的关系。在这两个项目中，对特殊位置的兼容开放书籍的研究都起作用，并且将使用渐近霍明型技术研究一般存在定理。此外，该项目将包括对正式同型类别和相关H-principles完全确定的灵活类恩格尔结构的研究。