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.