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以及代表经典力学系统相空间的基本问题。它通过说明性研究、培训和指导扩大了基础数学技术的获取范围，并总体上促进了数学中的多样性和支持性气氛。这些服务于社区的研究努力还嵌入了各种教育、培训和辅导部分。促进妇女在数学、多样性和支持性气氛中的普遍存在，是这项建议的另一个主要目标。除了在这类问题上增加宣传和行政服务外，这一目标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师开发公平培训模块来实现。例如，PI引入的辛范畴和伪全纯被子不变量为研究与其他问题相关的辛范畴提供了一般框架，例如涉及镜像对称的轮子范畴(从而编码辛几何和代数几何之间的关系)，或涉及拓扑场论的余边范畴(从而特别地构造了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-这两种结构的动机都是捕获来自新的八字泡泡的障碍。这一推广可以概括为A-无穷-2-范畴的概念，它特别包含了辛流形的所有Fukaya-A-无穷-范畴。该提案的目的是为这一类别建立一个完全严谨和容易接近的建筑。另一个目标是通过辛范畴和用于将它们相互联系的代数工具以及规范理论不变量来建立拓扑不变量的抽象构造原理。这为Ozsvath-Szabo的Heegard-Floer理论及其与Seiberg-Witten不变量的关系，以及Donaldson不变量和瞬子Floer同调的猜想辛版本提供了统一的理解。此外，它还有助于将这些不变量之间的关系的未来构造和证明系统化。该提案还旨在将这一概念框架应用于一些具体情况。特别是，它解决了3维的Atiyah-Floer型猜想--巩固了该领域的指导愿景--并将它们扩展到4维--为未来的发展提供了一个愿景。更具体地说，这最后一部分的目的是建立Donaldson和Seiberg-Witten的4-流形不变量的辛类似，预计它们更易于计算。最后，该项目的目的是巩固和提供椭圆型偏微分方程模空间正则化的微分拓扑基的易于理解的论述。在敲定严格的仓西型正规化蓝图后，重点将放在增加获得多折叠技术上。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)，并从等变Gromov-Witten不变量到Fukaya范畴的样本应用。

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.