CAREER: Representation theory of symplectic singularities

PI计划研究辛奇异变形量子化的表示理论。这个计划的长期研究目标是证明PI及其合作者的猜想，即辛奇点上存在对偶运算。目前，这个对偶性最重要的性质是它对相伴代数表示理论的影响：与奇点相连的“范畴O”应该是Koszul对偶的（即，诱导出范畴的一个非常特殊的等价）。这种二重性的许多特殊情况是很好理解的，但它们之间的联系仍有待研究。这项研究也被捆绑在探索这些奇点的个别例子;在这些我们的建议将是一个“几何化”和“分类”的著名对偶在数学上，如舒尔-外尔对偶，秩级对偶和盖尔对偶。我们的观点也提供了一个富有成效的方法来不同的主题，如辛反射代数的表示理论和Rouquier-Khovanov-Lauda分类的量子群，并有应用，因为远的低维拓扑。世纪最令人震惊的发现之一是量子力学的发现，以及将非对易观测量引入物理学。 数学家们在这些观察的基础上建立了一个抽象的“变形量子化”和“非对易几何”理论。“PI的工作是关于一方面非交换空间的经典极限几何与另一方面可以描述相关物理系统的状态空间之间的关系。 PI还提出了一个教育组件，该组件基于Wordpress创建了一个数学展示的网站平台。这将使博客，维基，和一系列的网站，不？不太适合这些类别，更接近于一个动态版本的主页免费和容易获得的数学社区在专业设置。这个奖项是共同资助的代数和数论计划和拓扑计划。

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.