Fukaya范畴的非交换代数几何研究

Fukaya category is a symplectic topological invariant which is closely related to algebraic geometry, representation theory ,topology and mathematical physics. Kontsevich’s homological mirror symmetry conjecture says that the Fukaya category is equivalent to the derived category of coherent sheaves of its mirror as Calabi-Yau categories which put Fukaya category in a more fundamental footing. We will study the algebraic,geometric and dynamical side of Fukaya category in the perspective of non-commutative algebraic geometry. More precisely, we will investigate: 1) representation theory of A-infinity structures and 2-A-infinity structure; 2) the Moyal quantization of Fukaya categories and their stability conditions in interesting examples; 3) non-commutative integrable systems and their applications to Fukaya categories.

Fukaya范畴是重要的辛拓扑不变量，它和代数几何、表示论、拓扑、数学物理关系密切。特别是在Kontsevich的同调镜像对称猜想中辛结构的Fukaya范畴和镜像复流形的凝聚层范畴作为Calabi－Yau范畴导出等价。这置Fukaya范畴于更基本的地位。本项目拟在非交换代数几何的框架下研究Fukaya范畴的代数、几何以及动力学方面。具体来说：1） A-无穷结构的表示论和2-A-无穷结构；2）Fukaya范畴的Moyal量子化和稳定性条件；3）非交换可积系统及其在Fukaya范畴上的应用。

本项目是在非交换代数几何的框架下研究Fukaya范畴的代数、几何与分析。具体地来说，我们研究了（1）有限维和无穷维Morse理论的高阶结构，这对于理解量子场论中的拓扑缺陷会很有帮助；（2）Fukaya范畴的Moyal量子化和Moyal量子化的resurgence性质，这是我们在形式量子化方案的收敛性方面尝试；（3）Chern-Simons不变量的模性质与resurgence理论Stokes现象之间的关系，这或许会对研究模形式提供新的方法；（4）复化开普勒问题的单值性群问题亦会从半经典量子力学的角度对氢原子光谱提供新的理解。