Below is a brief description of my current research; hopefully, it should be comprehensible, even to non-mathematicians. This PowerPoint presentation has a few more details, for a general math audience.

The theory of expander graphs is a hot topic in mathematics and computer science, and with good reason: these objects enjoy a wide range of applications. Think of a communications network as a graph, by which we mean a collection of dots (vertices) and line segments (edges) connecting them. Each vertex represents a person (or a telephone, or a computer) in the network; two vertices are connected by an edge if they can communicate directly with one another. The graph to the right, for example, has 10 vertices, each of which can communicate directly with 3 of the other vertices. We therefore say that this graph is regular and has degree 3.

When designing a communications network, one naturally wishes that any message originating somewhere within it will propagate as rapidly as possible. There are various measurements which describe how well a graph spreads messages. One such quantity is called the spectral gap of the graph. Roughly speaking, the larger the spectral gap is, the better the graph is as a communications network. If one attempts to create regular graphs with more and more vertices but without increasing the degree, the spectral gap may very well become infinitesimally small as the communications networks become poorer and poorer. A family of expander graphs is an infinite sequence of regular graphs, each with the same degree, such that the spectral gap does not vanish altogether in the limit. Roughly speaking, graphs with the best possible spectral gap are called Ramanujan graphs. Families of Ramanujan graphs are the most desirable families of expander graphs.