Grant: Discovery Grants Program - Individual

Algebraic geometry is the study of solution spaces of polynomials in several variables. This includes the geometry of familiar shapes like parabolas, spheres, and curves in the plane defined by a polynomial. One tries to study the solutions by relating them to other spaces via mappings or parametrizations. This subject is highly controlled by the algebra of polynomials. Geometric statements about the space of solutions correspond directly to algebraic statements about the polynomial equations one is solving. In noncommutative algebra, applying operations in different orders can yield different results. We have that x times y is not necessarily equal to y times x. Noncommutative algebra, just as in the study of the algebra of polynomials, is often motivated by geometric problems and intuition. In this setting, the field is called noncommutative algebraic geometry and it uses the techniques and ideas of (commutative) algebraic geometry applied to noncommutative algebra. For example, we may try to solve noncommutative equations with matrices and try to understand that space of solutions. More concretely, the equation yx-xy=1, which describes the quantum mechanical relation between position and momentum, has no matrix solutions, but the equation yx+xy=0 has many. In general, an important goal of noncommutative algebraic geometry is to understand the space of solutions to polynomial equations where the variables need not commute. In addition to having immediate applications in noncommutative algebra, there are applications to algebraic geometry and physics. We will generalize deep results in algebraic geometry to the geometry of orders. Since orders are finite over their centres, they are largely controlled by commutative algebraic geometry. However, statements that do not directly use the large centre tend to generalize to the general case. In noncommutative algebraic geometry, we take the category of quasi-coherent sheaves as the basic object. This includes the module category of a noncommutative ring as an analogue of an affine variety. For projective varieties, we take the category of graded modules modulo finite dimensional modules over a graded ring. There are other constructions that do not fit into this framework. The proposed research involves classification of noncommutative varieties, particularly in low dimension. Classification is a large part of commutative algebraic geometry. There are several results that say that noncommutative curves are finite over their centres, since one variable commutes with itself. Classification in dimension two is understood for orders, and we propose to establish a conjectural classification for general noncommutative surfaces. In addition, we propose to classify Artin-Schelter regular algebras of low dimensions, analogs of projective spaces. The classification of these algebras in dimension three was the birth of this subject as studied by Artin, Stafford, Van den Bergh and others.