CAREER: Representation theory of symplectic singularities

计划研究辛奇点变形量子化的表示理论。这个项目的长期研究目标是证明PI和他的合作者关于辛奇点上存在对偶运算的猜想。目前，这个提出的对偶最重要的性质是它对关联代数的表示理论的影响：附加在奇点上的“范畴O”应该是Koszul对偶（即，引出一个非常特殊的范畴等价）。这种对偶性的许多特殊情况已被很好地理解，但它们之间的联系仍有待研究。这项研究也与这些奇点的个别例子的探索有关；在这些方面，我们的建议是对数学中众所周知的对偶进行“几何化”和“分类”，例如Schur-Weyl对偶、秩-水平对偶和Gale对偶。我们的观点也为辛反射代数的表示理论和量子群的Rouquier-Khovanov-Lauda分类等各种主题提供了富有成效的方法，并且在低维拓扑领域也有应用。20世纪最令人震惊的发现之一是量子力学的发现，以及将不可交换的可观测性引入物理学。数学家们在这些观察的基础上创建了“变形量子化”和“非交换几何”的抽象理论。PI的工作一方面是关于非交换空间的经典极限几何之间的关系，另一方面是关于可以描述相关物理系统的状态空间。PI进一步提出了一个教育组件，创建一个基于Wordpress的数学展示网站平台。这将使博客、维基和其他一系列的网站都不具备这种功能。T非常适合这些类别，并且更类似于一个动态版本的主页，可以在专业设置中免费方便地提供给数学社区。该奖项由代数与数论项目和拓扑学项目共同资助。

The PI plans to research the representation theory of deformation quantizations of symplectic singularities. The long term research objective of this program is proving the conjecture of the PI and his collaborators that there is a duality operation on symplectic singularities. At moment, the most important property of this proposed duality is its effect on the representation theory of associated algebras: "categories O" attached to the singularities should be Koszul dual (that is, induce a very special equivalence of categories). Many special cases of this duality are well-understood, but what links them remains to be investigated. This research is also tied up in the exploration of individual examples of these singularities; in these our proposal would be a "geometrification" and "categorification" of well-known dualities in mathematics, such as Schur-Weyl duality, rank-level duality and Gale duality. Our perspective also provides a fruitful approach to topics as diverse as the representation theory of symplectic reflection algebras and the Rouquier-Khovanov-Lauda categorification of quantum groups, and has applications as far afield as low-dimensional topology. One of the most shocking discoveries of the 20th century was the discovery of quantum mechanics, and the introduction of non-commuting observables into physics. Mathematicians have built on these observations to create an abstract theory of "deformation quantization" and "noncommutative geometry." The PI's work is about the relationships between the geometry of classical limits of noncommutative spaces on one hand, and state spaces that could describe related physical systems on the other. The PI further proposes an educational component which creates a website platform for mathematical exposition, based on Wordpress. This will make blogs, wikis, and a whole range of of websites that don?t quite fit in those categories, and more closely resemble a dynamic version of a homepage freely and easily available to the mathematical community in a professional setting.This award is cofunded by the Algebra and Number Theory program and the Topology program.