Noncommutative Algebra

University Of Washington, Seattle WA

Investigators

Abstract

Homological methods are used more and more in many areas. Recent developments connected with triangulated (derived) categories have enormous impacts to commutative algebra, algebraic geometry and noncommutative algebra. These lead us to search for new homological-combinatorial structures of algebraic and geometric objects. Some new theories discovered in the noncommutative setting have unexpected applications to commutative algebra and algebraic geometry, while many classical theories in commutative algebra and algebraic geometry have been generalized to the noncommutative setting via the homological vehicle. The proposal emphasizes the homological aspects of the subject. The Principal Investigator studies invariants such as the homological integrals of Hopf algebras and the Nakayama order of rigid Gorenstein rings, further develops the theory of dualizing complexes over noncommutative algebras and spaces, and will try to complete the classification of quantum projective three-spaces. Noncommutative algebra in general is a powerful tool in mathematics and physics. Current research in algebraic geometry, number theory, string theory and topology require further and deeper understanding of several classes of noncommutative algebras, such as, quantum groups, noncommutative Iwasawa algebras and Artin-Schelter regular algebras. The proposal consists of some outstanding questions in noncommutative algebra, with an eye on applications to other areas. The Principal Investigator will communicate and collaborate with experts in different fields. The project has broader impacts on research, education, and training graduate and undergraduate students.