As a Loyola student, you have the opportunity to work alongside our talented professors to partner in collaborative research. Learn more about some recent research and projects currently underway.
Dr. Jeremy Thibodeaux and senior mathematics student Michael Hennessey have derived a system of differential equations that model certain blood cell and particle populations in the body when it is infected with Dengue virus. The model aims to capture the relevant physiological processes to provide researchers a tool to develop more effective antiviral drugs and treatments in the fight against Dengue Fever.
My research involves using graph theory to study rings. A ring is a set within which we can add, subtract, and multiply. Strangely enough, sometimes in a ring the product of two non-zero elements is zero! For example, this can happen when we multiply matrices.
We can represent this situation using a graph. A graph is just a bunch of point with lines connecting them. Graphs can be used to describe many situations, such as the patterns of city streets, computer networks, and the structure of molecules, among other things. In my research, we write down all the zero divisors in a ring and connect two of them if their product is zero. We can then study the graph to find patterns which can give information about the ring. Here are two fairly simple graphs.
Over the last two years two of my students have received $4,000 grants to study these graphs. The study of these graphs provides a good introduction to undergraduate research.
Dr. Michael Kelly is currently working on research in the area of topology known as Fixed Point Theory. A collaboration with Professor D. Goncalves of the University of Sao Paulo, Brasil has lead to a research article which will appear in the journal Bulletin of the Mexican Math. Society. This collaboration is ongoing and also involves a research project which is related to the graphic on the left.
Second year mathematics major Linda Hexter is currently working under the supervision of Dr. Kelly. She is working on a project exploring idempotent matrices and implications to a problem in fixed point theory regarding homotopy idempotents.
Professor Thibodeaux and senior student Savannah Logan '14, worked on a project to investigate relationships between derivatives and algebraic structures called zero divisors. They first derived a formula for the number of zero divisors in the set of upper triangular matrices whose entries are from some subset of the whole numbers. They then determined the rate at which this number grows as a function of the size of the matrices and the size of the subset of whole numbers. Those rates were then compared with the derivatives of those functions, as is done with marginal cost and revenue in economics. Finally, using Gronwall’s inequality, a result was presented concerning when it is appropriate to use derivatives to approximate these discrete quantities.
Zero divisors are objects that arise in one of the most abstract areas of mathematics. Surprisingly, investigators are able to study zero divisors using computational and geometric techniques. One of the geometric techniques involves diagrams called zero divisor graphs. Since 1988 there has been a plethora of articles on this topic. Dr. Thibodeaux and Dr. Tucci have published a joint paper in Communications in Algebra (Volume 42, Issue 9, 2014), “Zero Divisor Graphs of Finite Direct Products of Finite Rings,” which adds the continuing discussion on Zero divisors, and they are completing a second. Both of them are submitting papers which are co-authored with students.