Noncommutative Algebra

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Homological methods are used more and more in many areas. Recentdevelopments connected with triangulated (derived) categorieshave enormous impacts to commutative algebra, algebraic geometryand noncommutative algebra. These lead us to search for newhomological-combinatorial structures of algebraic and geometricobjects. Some new theories discovered in the noncommutative settinghave unexpected applications to commutative algebra and algebraicgeometry, while many classical theories in commutative algebra andalgebraic geometry have been generalized to the noncommutativesetting via the homological vehicle. The proposal emphasizes thehomological aspects of the subject. The Principal Investigatorstudies invariants such as the homological integrals of Hopfalgebras and the Nakayama order of rigid Gorenstein rings, furtherdevelops the theory of dualizing complexes over noncommutativealgebras and spaces, and will try to complete the classificationof quantum projective three-spaces.Noncommutative algebra in general is a powerful tool in mathematicsand physics. Current research in algebraic geometry, number theory,string theory and topology require further and deeper understandingof several classes of noncommutative algebras, such as, quantumgroups, noncommutative Iwasawa algebras and Artin-Schelter regularalgebras. The proposal consists of some outstanding questions innoncommutative algebra, with an eye on applications to other areas.The Principal Investigator will communicate and collaborate withexperts in different fields. The project has broader impacts onresearch, education, and training graduate and undergraduate students.