This award supports the research of Michaela Vancliff to work in non-commutative algebra, with special emphasis on problems arising from the theory of AS-regular algebras and algebraic geometry. She is interested in the graded-module category viewed as a geometric space, with certain graded modules playing the role of geometric objects. The linear geometric modules (point modules, line modules, etc.) are parametrized by so-called linear schemes. Vancliff plans to study how the structure and role of higher-dimensional linear schemes generalize the structure and role of point schemes. Under prior NSF support, in work with T. Cassidy, Vancliff produced algebro-geometric techniques that allow the easy construction of certain AS-regular algebras (that generalize graded Clifford algebras) of any finite global dimension, naming such algebras graded skew Clifford algebras. Vancliff intends to study such AS-regular algebras of global dimension four that have finitely many points and a one-parameter family of line modules as a step towards classifying the line schemes that arise for "generic" quadratic AS-regular algebras of global dimension four. Her initial research with B. Shelton (under prior NSF support) suggests that such an algebra should have a line scheme that consists of exactly six elliptic curves, so if this is found to hold in general, then it would mimic the point scheme of generic quadratic AS-regular algebras of global dimension three (where it is one elliptic curve).
Mathematics in general is the study of patterns and frequently such patterns are described via systems of equations. For instance, systems of polynomial-style equations and their solutions play a critical role in almost every scientific field, such as statistical mechanics, elementary-particle physics, quantum mechanics, robotics, crystallography, networking, etc. Often, the solutions cannot be found by experimentation, and often they are not numbers but are functions (e.g., differential operators or matrices), and so, in general, they do not commute. The science of seeking methods that find all solutions to any system of polynomial-style equations in non-commuting variables is called non-commutative algebra. To find the solutions, the main idea is as follows. One associates to such a system of equations an entity, called an "algebra," that encodes all the properties of the original equations. Associated to this algebra are "modules," and these encode all the properties of the solutions to the equations. So, in order to find all the solutions, one should find all the modules for the associated algebra. In many of the applications, the algebras that arise in this way tend to share certain properties satisfied by commuting polynomials; such algebras are called AS-regular algebras and are the main focus of Vancliff's projects. One of the goals of the study of such algebras and their modules is to use geometric techniques to find certain modules (point modules, line modules, etc) of the AS-regular algebra, and then to use those modules to find the modules that give the solutions to the original system of equations. Vancliff's underlying goal is to improve on these geometric techniques and to understand better how they relate to the structure of the category of modules.
