Shifted Symplectic & Poisson Structures and their Quantisations in the context of Derived Algebraic Geometry

拟议的博士项目是关于研究移位辛和泊松结构及其量子化的例子在派生代数几何的背景下。这是一个及时的研究项目连接到代数几何，量子代数和数学物理领域的现代发展。这个项目中使用的技术是从纯数学中获得的。在这个项目的第一阶段，学生将学习交换微分分次代数的几何和这些结构上的n-形式和导子的定义。然后，这些工具可以用来定义n-移位辛和泊松结构。一个主要的例子是探讨如何在李代数和拟李双代数结构的非退化配对可以从这个框架中恢复。在这些主要目标实现之后，下一步将是应用上述技术来研究涉及更高代数结构的更新颖和更丰富的例子，例如微分分次李代数。

The proposed PhD project is about studying examples of shifted symplectic and Poisson structures and their quantisations in the context of derived algebraic geometry. This is a timely research project connecting to modern developments in the areas of algebraic geometry, quantum algebra and mathematical physics. The techniques used in this project are obtained from pure mathematics. In the first stage of this project, the student will learn about the geometry of commutative differential graded algebras and the definitions of n-forms and derivations on these structures. These tools can then be used to define n-shifted symplectic and Poisson structures. A main example to be explored is how non-degenerate pairings on a Lie algebra and quasi-Lie bialgebra structures can be recovered from this framework. After these primary goals have been achieved, the next step will be to apply the above techniques to investigate more novel and richer examples involving higher algebraic structures, such as differential graded Lie algebras.