Constructions in Higher-Dimensional Contact Topology

这个纯数学研究项目的重点是高维的抽象几何结构，即接触拓扑和辛拓扑。该领域将刚性几何问题，如研究周期轨道或通过给定点的计数曲线，与灵活的剪切粘贴结构交织在一起。在这个提议中进行的部分研究是在图解微积分方面发展这些结构的组合描述，并将其应用于这些刚性几何问题。这为新的例子和计算提供了一个可访问的来源，并为严格的高维结构奠定了基础。由于其组合性质，这条研究路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和几何拓扑中的已知元素与接触和辛拓扑中的新思想相结合的技术，并且所提出的研究结果有助于数学研究的每个中心领域。该项目的第一部分是在前投影中发展Legendrian微积分，包括高维Reidemeister移动，Legendrian手柄滑动和松散图存在的准则。通过进一步建立与bi-Lefschetz纤维的关系，该工具将为Weinstein结构的（亚）柔性和代数不变量（如Legendrian接触微分梯度代数和包裹的Fukaya范畴）的计算提供有效的组合准则。它结合了仿射代数几何和高维接触外科理论的思想。本课题的第二部分重点研究高维模型的紧度和过扭度检测，包括有边界的Legendrian开书和过扭模型的小邻域。该项目将使用适用于每个问题的伪全纯技术，并结合来自Legendrian子流形研究的新思想，包括Legendrian协阵及其与Weinstein协阵的关系的研究。在这两个项目中，研究特殊位置上的相容开本，并利用渐近全纯技术研究一般存在定理。此外，本项目还将研究完全由形式同伦类决定的一类灵活的Engel结构，以及相关的h-原理。

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.