在数学和物理中，形变量子化是一个重要的思想和工具。本项目拟利用Poisson代数及其形变量子化理论研究一些非交换代数的结构及同调性质。拟通过光滑代数上的Poisson结构来寻求新的Artin-Schelter正则代数或Calabi-Yau代数的例子，为高维Artin-Schelter正则代数的分类提供新的思路。试图揭示Poisson代数的Poisson（上）同调与其形变量子化代数的Hochschild（上）同调之间的关系，特别地，研究两个上同调环作为Gerstenhaber代数或Batalin-Vilkovisky代数的同构问题。研究Poisson代数的模导子的同调trace，讨论Poisson代数的模导子与形变量子化代数的Nakayama自同构的关系。 本项目还将探讨形变量子化与Koszul对偶、非交换不变量理论的关系，揭示Poisson代数与其形变量子化代数之间的深刻联系。

Deformation quantization is an important tool in mathematics and physics. The main purpose in this project is to investigate homological properties and structures of some noncommutative algebras by using Poisson algebras and their deformation quantization theory. We will try to use Poisson structures over smooth algebras to construct new examples of Artin-Schelter regular algebras or Calabi-Yau algebras, which may lead a new strategy for the classification of Artin-Schelter regular algebras. We will explore the relation between Poisson (co)homology of Poisson algebras and Hochschild (co)homology of the deformation quantization algebras. In particular, we will study when the two cohomology rings are isomorphic as Gerstenhaber algebras or Batalin-Vilkovisky algebras. We also try to define homological trace for (modular) derivations, and explore the relation between the modular derivations of Poisson algebras and the Nakayama automorphisms of the deformation quantization algebras. Furthermore, the connections between deformation quantization, Koszul duality and noncommutative invariant theory will also be discussed.