Connections between Symplectic and Low Dimensional Topology

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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.