CAREER: Representation theory of symplectic singularities

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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.