My PhD research focused on Groebner Basis techniques for noncommutative rings and applications to Hochschild cohomology of finite dimensional algebras. I continued to be active in this area for several years after completing my doctoral work. Early in my career at Salisbury University, I started a collaboration with departmental colleague Kathleen Shannon involving visualization of concepts from abstract algebra. This quickly took some interesting twists and turns and ultimately became a much deeper and richer project than we expected early on in the work. This led me into a new area of interest, namely cellular automata generated over algebraic structures. I have been playing with these discrete dynamical systems for years now. Over the last three decades I have also dabbled in other areas, including working with a student on a calculus-based physics project, developing a C++ lab manual with one of my colleagues, and writing about mathematics education with other faculty. These are all described in more detail below. For a list of publications please click here .
In this research I used techniques from noncommutative Groebner Basis theory to construct projective resolutions of an algebra over its enveloping algebra. This led me to a construction which is now known in the research literature as the Bardzell resolution . It is an explicit minimal projective resolution for monomial algebras presented as quotients of path algebras. This resolution provides a combinatorial way to resolve the algebra as a bimodule over itself. I constructed it based on chains of overlapping paths in the underlying quiver, with differentials defined by concatenation and truncation of these paths. The construction exposes how relations overlap, which directly controls the behavior of syzygies, periodicity, and growth in resolutions. Avoiding the size and complexity of the bar resolution, this construction is an efficient tool for explicit calculations of Hochschild homology and cohomology. This has led me and others to precise results about alternating syzygy patterns and homological dimensions that would be hard to see using general-purpose resolutions. Many later works use it to define comparison morphisms, describe cup products and Gerstenhaber brackets, and extend techniques to broader classes such as string, gentle, and truncated quiver algebras. In this light, it serves not just as a computational device but as a conceptual framework linking combinatorics of quivers with homological and deformation-theoretic properties of algebras. The Bardzell resolution is still used by current researchers because it turns abstract homological questions about monomial algebras into explicit, tractable, and structurally revealing computations.
Soon after I joined the faculty at Salisbury University, Kathleen Shannon showed me a program she wrote to render colored images of Pascal's Triangle modulo and integer n. Group theoretic connections soon became clear and Kathleen and I genreralized the construction to incorporate other finite groups into what we called PascGalois triangles. This led to the PascGalois Project, funded by two grants from the National Science Foundation, to create course enhancement materials for visualizing concepts from abstract algebra and other topics from the undergraduate mathematics curriculum. Don Spickler joined the project and wrote an expanded software package, PascGaloisJE, which has served as the project software for over 20 years. As a result, a plethora of ideas for undergraduate research have flowed ever since. Other faculty colleages, both at SU and other institutions, joined on to create academic enhancement materials and project ideas for undergraduate research. Two one-week PascGalois summer research retreats, modeled as "mini-REU's", were offered at New College of Florida for undergraduates around the country. The most important contributors in the end, of course, have been the large number of students, both at SU and other institutions, who have participated in the project. Many of the ideas and mathematical results generated by participating students and faculty can be found in my book Beyond Pascal's Triangle which will soon be published by the American Mathematical Society in the Dolciani Mathematical Expositions series. For more information, see the PascGalois Project Website.
The triangular structues initially examined during the early years of the PascGalois Project turned out to be special cases of infinite 1-dimensional cellular automata. This opened the doors to significantly expand and generalize our first constructions. A cellular automaton is a type of discrete dynamical system which sits in a lattice space and evolves in discrete time steps according to a "local rule". My work focuses on such systems where the underlying alphabet possess algebraic structure, such as a finite group. Particular areas of interest include 1) fractal patterns and dimensions of infnite systems and 2) periodic and transient behaviors in finite linear systems. Although cellular automata can generate high levels of complexity on the global scale (some are capable of universal computation), locally they are relatively simple and easy to simulate. This makes them particulary suitable for undergraduate student exploration and research. By utilizing techniques from discrete mathematics, abstract algebra, number theory, dynamical systems, and even fractal geometry, a range of interesting questions can be pursued by both professional mathematicians as well as students at different stages of their undergraduate career.
Much of my work over the years has focused on expository, instructional, and educational themed projects and writing. My colleague Dean DeFino and I published Lab Manual to Accompany Starting Out with C++. This 280 page book was witten as a lab companion resource for the well-known introductory C++ text by Tony Gaddis. Resource papers related to undergraduate mathematics, e.g., A Ring Construction Using Finite Directed Graphs and Can We Learn Calculus From a Jerk (with student Paula Kenyon) can be found in journals such as PRIMUS. Work with K-12 teachers and students has led to a range of projects over the years, two of which are summarized in articles co-authored appearing in PDS Partners: Bridging Research to Practice and the Doceamus Column of the Notices of the American Mathematical Society. Much of my K-12 outreach work has involved grant writing to the Maryland Higher Education Commission. For a list of funded grants, please click here.