近年来随着超对称理论物理的发展而诞生的微分分次几何不仅拓展了几何学的研究范围，而且给出了数学其他方向新的研究方法。本项目旨在申请者关于微分分次几何Atiyah和Todd同调类的工作基础上，继续几项新的研究：一方面，我们在微分分次几何的框架下研究可积分布的Hochschild上同调以及它的形变量子化；另一方面，我们研究Lie代数胚对的1维Chern-Simons模型的Batalin-Vilkovisky量子化，并证明Lie代数胚对版本的Caldararu猜想。

Inspired by the development of supersymmetry in theoretical physics, the born of differential graded geometry does not only extend research of geometry but also implies new methods in other fields of Mathematics. Based on my work on the Atiyah and Todd classes in differential graded geometry, we plan several new projects: On the one hand, we study Hochschild cohomology of integrable distributions and their deformation quantization in the field of differential graded geometry; On the other hand, we study Batalin-Vilkovisky quantization of 1-dimensional Chern-Simons models associated to Lie algebroid pairs, and prove the Caldararu conjecture for Lie algebroid pairs.