弦拓扑是近十几年来兴起的一门学科，主要研究光滑流形的路径空间上的代数结构。本研究项目主要利用弦拓扑的背景，研究辛几何和非交换几何中出现的类似的代数结构，比如Batalin-Vilkovisky代数，非交换Poisson代数，等等。具体问题包括：1、研究辛几何中Fukaya范畴的Hochschild同调和循环同调上的代数结构，以及它们与辛同调、切触同调之间的关系；在余切丛情形，研究它们跟弦拓扑之间的关系。2、研究在非交换几何领域中，李代数和结合代数对应的非交换空间上的辛结构和Poisson结构，以及它们在弦拓扑情形（做为非交换空间的特例）的拓扑学意义。在这个研究过程中，我们并将研究A-无穷范畴的导出范畴意义下的Morita理论，并应用到Fukaya范畴的计算中；同时我们还会研究Calabi-Yau流形中田刚-Todorov引理在非交换几何下的表达形式。

String topology is mathematical field that arises in the past decade, which studies the algebraic structures on the path space of smooth manifolds. This program mainly studies some algebraic structures in symplectic geometry and noncommutative geometry, such as Batalin-Vilkovisky algebras, noncommutative poisson algebras, which already appear in and/or are inspired by string topology. Main topics of the program include: 1) studying the algebraic structures on the Hochschild homology and cyclic homology of Fukaya categories that appear in symplectic geometry, and their relationships with symplectic homology and contact homology, as well as their relationships with string topology in the case of cotangent bundles; 2) studing the symplectic and Poisson structures in noncommutative associative and Lie spaces, and their topological signifcance in string topology. In this process, we will study the derived Morita theory of A-infinity categories and apply it to the computation of Fukaya categories, and will also study the noncommutative version of the Tian-Todorov lemma for Calabi-Yau manifolds.