Connections between Symplectic and Low Dimensional Topology

这个项目解决了三维和四维形状的几何和拓扑结构以及代表经典力学系统相空间的几何和拓扑结构的基本问题。它通过临时研究，培训和指导拓宽了基础数学技术的获取途径，并普遍促进了数学的多样性和支持性气氛。这些社区服务的研究工作，而且嵌入在各种教育，培训和辅导组件。该提案的另一个主要目标是促进妇女在数学、多样性和总体支持性环境方面的发展。除了增加关于这些问题的宣传和行政服务外，还通过每周社区建设和职业发展会议、年度辅导务虚会以及为教职员工开发公平培训模块来实现这一目标。 PI引入的辛范畴和伪全纯被子不变量为研究辛范畴与其他范畴的相关问题提供了一个一般框架，例如镜像对称中涉及的层范畴（因此编码辛几何和代数几何之间的关系），或拓扑场论中涉及的配边范畴（因此特别是构造3-和4-流形的不变量）。这项研究的一个核心目标是借助对奇异被子和更精细的代数结构的分析，将范畴结构扩展到一般的辛流形，这两种分析都是通过捕捉来自新颖的8字形气泡的障碍物来实现的。这个推广可以概括为一个A-无穷2-范畴的概念，它特别包含了辛流形的所有福谷A-无穷范畴。该提案的目的是对这一类别进行完全严格和可访问的构建。另一个目标是通过辛范畴和代数工具建立拓扑不变量的抽象构造原理，将它们相互联系起来，以及规范理论不变量。这为Ozsvath-Szabo的Heegard-Floer理论及其与Seiberg-Witten不变量的关系，以及唐纳森不变量和瞬子Floer同调的几何辛版本提供了统一的理解。此外，它可以帮助系统化未来的结构和证明这些不变量之间的关系。该提案还旨在将这一概念框架应用于某些具体情况。特别是，它解决了Atiyah-Floer类型的三维结构-巩固了该领域的指导愿景-并将其扩展到四维-为未来的发展提供了愿景。 更具体地说，这最后一部分的目的是建立辛类似物的唐纳森和Seiberg-Witten 4-流形不变量，预计将更适合计算。最后，该项目旨在巩固和提供可访问的微分拓扑基础的椭圆偏微分方程的模空间的正则化的论述。在最终确定了严格的Kuranishi型正规化蓝图后，重点将放在增加对多折叠技术的使用上。特别是，一组初级研究人员正在努力建立一个工具箱（如纤维产品和等变横截性）和样本应用程序从等变Gromov-Witten不变量通过阿诺德猜想到福谷类别。

This project addresses fundamental questions in the geometry and topology of three and four dimensional shapes as well as those that represent phase spaces of classical mechanics systems. It broadens access the foundational mathematical technologies through expository research, training, and mentoring, and generally promotes diversity and a supportive climate in mathematics. These community-serving research efforts are moreover embedded in a variety of educational, training, and mentoring components. Promoting women in mathematics, diversity and supportive climate in general, is another main goal of this proposal. Besides increasing advocacy and administrative service on such issues, this goal is pursued by means of weekly community building and career development meetings, an annual mentoring retreat, and the development of equity training modules for faculty.   The symplectic category and pseudoholomorphic quilt invariants introduced by the PI provide a general framework for studying questions relating the symplectic category to others, e.g. categories of sheaves involved in mirror symmetry (thus encoding relations between symplectic and algebraic geometry), or cobordism categories involved in topological field theories (thus in particular constructing invariants of 3- and 4-manifolds). One core goal of the research is to extend the categorical structures to general symplectic manifolds with the help of analysis for singular quilts and more refined algebraic structures - both motivated by capturing obstructions arising from novel figure eight bubbles. This extension can be summarized in a notion of A-infinity-2-category which in particular contains all Fukaya A-infinity-categories of symplectic manifolds. The proposal's aim is a fully rigorous and accessible construction of this category. Another goal is to establish an abstract construction principle for topological invariants via the symplectic category and algebraic tools for relating them to each other as well as gauge theoretic invariants. This provides a unified understanding for both Ozsvath-Szabo's Heegard-Floer theory and its relation to Seiberg-Witten invariants, as well as conjectural symplectic versions of Donaldson invariants and instanton Floer homology. In addition, it can help systematize future constructions and proofs of relationships between such invariants. The proposal also aims to apply this conceptual framework in some concrete situations. In particular, it addresses Atiyah-Floer type conjectures in dimension 3 - solidifying a guiding vision for the field - and extends them to dimension 4 - providing a vision for future developments.  More concretely, this last part aims at establishing symplectic analogues of both Donaldson and Seiberg-Witten 4-manifold invariants, which are expected to be more amenable to calculations. Finally, the project aims to solidify and provide accessible expositions of the differential-topological foundations for regularizations of moduli spaces of elliptic PDEs. After finalizing the blueprint for rigorous Kuranishi-type regularizations, the focus will lie on increasing access to the polyfold technology. In particular, a group of junior researchers is being involved in efforts to build a toolbox (such as fiber products and equivariant transversality) and sample applications from equivariant Gromov-Witten invariants via the Arnold conjecture to Fukaya categories.