Proposed Research

I was a graduate student at the Mas-sachusetts Institute of Technology. The rst three years of these studies were supported by an NSF Graduate Student Fellowship. My research there led to a Ph.D. thesis entitled \Noncommutative ruled surfaces." My thesis research describes certain classes of graded rings which arise as homogeneous coordinate rings of noncommutative quantizations of algebraically ruled surfaces. Ruled surfaces, including rational surfaces, play a central role in the classical theory of commutative algebraic surfaces. I use Van den Bergh's concept of a bimodule sheaf to construct noncommutative analogs of ruled surfaces. Bergh, and others attempting to classify graded rings of Gelfand-Kirillov-dimension 3 which are \nearly commutative" in some sense. For example, 1] describes those graded rings of GK-dimension 3 which are of nite global dimension and Gorenstein: these rings can be described as homogeneous coordinate rings of quantum projective planes or of quantum quadric surfaces (i.e. quantum P 1 P 1). My goal was to develop a method for constructing GK-dimension 3 graded rings as homogeneous coordinate rings of an arbitrary quantum ruled surface, which is a noncommutative analog of a ruled surface. Since dimension 3 com-mutative rings arise naturally as homogeneous coordinate rings of algebraic surfaces, it is reasonable to expect that noncommutative GK-dimension 3 graded rings may be constructed as homogeneous coordinate rings of quan-tized versions of algebraic surfaces. Many examples of GK-dimension 3 rings can be formed in this fashion, including homogeneous coordinate rings of quantum quadric surfaces, quotients of 4-dimensional Sklyanin algebras, and quotients of the homogenized enveloping algebra of the Lie algebra sl 2 .