Noncommutative Algebra

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The interaction between noncommutative algebra and other subjects is one of the main themes in modern noncommutative algebra. The categorical and homological approaches to noncommutative algebras have been very successful. The Principal Investigator is studying the structure of noncommutative algebras and associated noncommutative schemes, in particular, the homological aspects of those objects. One of the main projects of this proposal is the classification of Artin-Schelter regular algebras of dimension four. A new approach to this question is to use A-infinity algebras.Many physical phenomena have been explained by noncommutative equations. New topological invariants have been discovered by using noncommutative Hopf algebras. Some de-singularization problems in algebraic geometry have been interpreted as derived equivalences induced by a Morita context in noncommutative algebra. Many questions in string theory have been formulated by using noncommutative algebras. One may also use noncommutative algebra to encode models for networks, quantum computing and chemical molecules. Further and deeper understanding of noncommutative algebras will be greatly beneficial to many areas in mathematics, physics and other disciplines.