Topics in noncommutative algebra 2022: homological regularities

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Many phenomena in sciences, specially in mathematics and physics, can be described by noncommuting variables, that is, the product XY of two variables X and Y may not equal to the product YX in the opposite order. A noncommutative algebra is a mathematical concept that encodes such a phenomenon, and the subject of noncommutative algebra naturally has many applications in quantum mechanics, quantum field theory, string theory, and so on. The study of noncommutativity has become more and more common in modern science and technology. For example, a quantum group is equivalent to a Hopf algebra whose underlying algebraic structure is noncommutative and/or whose underlying coalgebraic structure is noncocommutative. The study of noncommutative algebras is both important and challenging. In many cases, noncommutative algebras arise as noncommutative analogues and generalizations of classical objects coming from other fields such as commutative algebra, algebraic geometry, and Lie theory. To understand the structure of noncommutative algebras, the PI will investigate invariants that are defined by homological means. These invariants capture hidden structures and symmetries of noncommutative algebras. This research project combines ideas and methodology from several areas of mathematics such as algebraic geometry, commutative algebra, linear algebra, homological algebra, and combinatorics. The proposed research activities will make contributions to teaching, undergraduate and graduate student training, and outreach. A major part of the project concerns homological invariants such as Castelnuovo-Mumford regularity, Tor-regularity, and Artin-Schelter regularity for connected graded algebras. A weighted version of homological regularities with an extra parameter will also be introduced that incorporates different classical homological concepts. This is a new research direction in noncommutative algebra with connections to representation theory, noncommutative algebraic geometry, and noncommutative invariant theory. The PI will investigate and prove homological identities involving these invariants for connected graded algebras and their associated categories, and thus expand the foundations for this theory. One particular sub-project is the classification of non-Artin-Schelter regular algebras that are close to being Artin-Schelter regular. In addition this project concerns other active research topics in noncommutative algebra: the automorphism problem, the cancellation problem, operad theory, and noncommutative discriminants.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.