Noncommutative Algebra

University Of Washington, Seattle WA

Investigators

Abstract

The principal investigator and his colleagues study noncommutative algebra and associated noncommutative geometry. The proposal consists of two topics: the classification of graded domains of Gelfand-Kirillov dimension three, or the classification of noncommutative projective surfaces; applications of dualizing complexes. The classification question is one of the most important questions in noncommutative algebra and noncommutative projective geometry. Derived categories has been used in many areas of mathematics, and it has been proven that the dualizing complex is an effective tool in studying noncommutative rings. More applications of dualizing complexes are expected to be discovered. Noncommutative algebra plays an important role in Mathematics and Physics. For example, quantum groups introduced in the 1980's are essentially a family of noncommutative and noncocommutative Hopf algebras. Some classes of subtle symmetries appeared in Mathematics and Physics can be expressed by using noncommutativity. The aim of this proposal is to develop effective methods, to understand the structure of noncommutative algebra, and to explore its interaction with other field such as statistical mechanics, quantum field theory, and noncommutative geometry.