Symplectic algebraic geometry and representation theory

提出的研究主题是辛代数几何、量子化和几何表示理论之间的相互作用。在此过程中，PI计划在看似无关的主题之间建立几个意想不到的联系。具体地说，PI将证明由于Bezrukavnikov的一个重要的范畴等价性的‘量子版本’。这个‘量子版本’将基仿射空间上的D-模范畴与仿射Grassman空间上的一个等变派生的层范畴联系起来。然后，PI期望通过基仿射空间上的微分算子代数找到动力Weyl群的自然解释。在Braverman、Maulik和Okounkov最近的工作的推动下，PI将在不同的方向上解决找到关于Springer解析的量子上同调的正则平坦连接作为Gauss-Manin连接的解释的问题。这里的思想是，Springer分辨率的正确镜像对偶是“非对易空间”，量子连接的镜像对偶是关于该非对易空间的周期循环同调的Gauss-Manin连接。拟议项目的主要特点之一是将一个数学分支开发的方法和结果应用于完全不同的数学领域。例如，这项研究涉及到各种代数技术在几何和表示论问题上的系统应用。因此，该项目的目标之一是加强“数学统一”的概念。拟议的工作不仅在数学的各个领域，而且在数学物理和理论物理，特别是在量子物理中，也应该具有深远的应用。数学科学作为一个整体，对社会做出了深远的贡献。目前的项目对促进不同学科之间的互动和增进我们的科学理解大有裨益。

The main theme of the proposed research is the interaction among symplectic algebraic geometry, quantization, and geometric representation theory. Along the way the PI plans to establish several unexpected links between seemingly unrelated topics. Specifically, the PI will prove a 'quantum version' of an important equivalence of categories due to Bezrukavnikov. This 'quantum version' relates a category of D-modules on the base affine space to an equivariant derived category of sheaves on the affine Grassmannian. The PI then expects to find a natural interpretation of dynamical Weyl groups in terms of the algebra of differential operators on the base affine space. In a different direction, motivated by recent work of Braverman, Maulik, and Okounkov, the PI will address the problem of finding an interpretation of the canonical flat connection on the quantum cohomology of the Springer resolution as a Gauss-Manin connection. The idea here is that the correct mirror dual of the Springer resolution is a "noncommutative space" and the mirror dual of the quantum connection is the Gauss-Manin connection on periodic cyclic homology of that noncommutative space.The proposed work opens up a wide variety of independent directions for research. One of the central features of the proposed projects is the application of methods and results developed in one branch of mathematics to a totally different area of mathematics. For example the research involves the systematic application of various algebraic techniques to problems in geometry and representation theory. Thus one of the goals of the project is to enhance the concept of the "unity of mathematics". The proposed work should also have far reaching applications not only in various areas of mathematics, but also in mathematical physics and theoretical physics, especially in quantum physics. The mathematical sciences, viewed as a whole, make a profound contribution to society. The current project goes a long way toward promoting the interactions between different disciplines and enhancing our scientific understanding.