The objective of this project is to make progress on parts of the classification of noncommutative projective surfaces. This is an important problem, because many interesting new phenomena occur for surfaces that are absent for curves; thus understanding the surface case better will advance our understanding of noncommutative geometry in general. First, the PI plans to complete the classification of birationally commutative surfaces---which are noncommutative surfaces with a commutative function field of transcendence degree 2---in terms of the process of naive noncommutative blowing up. Next, one wants to study noncommutative surfaces containing a commutative curve of points---we will start by examining a class of subalgebras of elliptic regular algebras of dimension 3, which may be related to M. Van den Bergh's different notion of noncommutative blowing up. The PI plans to study fat point modules, about which there are many foundational questions. Finally, one should study the generalization of naive blowing up to higher dimensional noncommutative varieties.
Noncommutative projective geometry is a relatively new subject, which attempts to use some of the same intuition and techniques underlying commutative algebraic geometry, but in the noncommutative setting. Modern physics tells us the importance of considering operations that may not commute with each other, which means it matters which order you perform them. For example, one interpretation of the celebrated Heisenberg Uncertainty Principle is that position and momentum do not commute. There are many other connections between noncommutative algebra and physics. This project hopes to gain a better abstract understanding of certain kinds of structures in algebra (noncommutative graded rings of low dimension) which include among them some famous examples, such as the Sklyanin algebras, which are related to the study of the physical world.
