The proposal concerns two topics in the area of noncommutative algebra, namely, noncommutative projective geometry and the theory of noncommutative Iwasawa algebras. The principal investigator proposes to work on some central projects in the field such as Artin's program on noncommutative projective surfaces and the classification of quantum projective three-spaces (or the classification of connected graded Artin-Schelter regular algebras of global dimension four). The principal investigator proposes to study the module theory of noncommutative Iwasawa algebras and to understand the connection between the algebraic side and the number-theoretic side of the noncommutative Iwasawa theory. Another goal of the proposal is to search for various ring-theoretic and homological invariants that reflect and predict the structure of noncommutative algebras.
Noncommutativity is a very common and increasingly important phenomenon in Mathematics and other disciplines. The subject of noncommutative algebra is the mathematical foundation for characterizing and understanding noncommutativity. As more details of noncommutativity are to be revealed, it is essential to examine the structure of noncommutative algebras. Recent developments in noncommutative algebraic geometry furnish new ideas and methods for the study of noncommutative algebras. Part of the proposal emphasizes some classification problems of noncommutative algebras using techniques from algebraic geometry. The project has applications to other subjects such as algebraic geometry, number theory and mathematical physics, and will enhance broader understanding of these subjects.
