A number of undergraduate students at MSU have worked with me on a project understand "trapping" of random walkers in two dimensions. The ultimate goal of this research is to determine insect pest population densities from the number of individuals caught in traps --- the applied portion of this project is a collaboration with Dr. Jim Miller and Chris Adams in the MSU department of Entomology. To make contact with applications, the random walkers are thought of as insects. They are restricted to two dimensions because in the field the insects fly in a narrow vertical band inside of an orchard. The trap is a region of the plane in which the walkers remain stuck once they enter. In the field the trap is some sticky tape baited with pheromone that attracts the insects. It is probably more realistic to think of the trap as the cloud of pheromone, rather than just the sticky tape.

In keeping with the oft told joke, we may begin by considering a circular trap. Using such traps as models we can compute the trapping probability exactly and have had some success in fitting literature data. Of course real traps are not circular. In summer 2012, Trevor Steil studied on the question: can non-circular traps be thought of as "effectively" circular? You can read about his research and our "conformal radius conjecture" here. Trevor presented his results as a poster at the AMS/MAA meeting in San Diego in January 2013.

There is a limit to how much information one can extract from a single trap. To gather more data it may be useful to consider multiple traps that compete with each other. David Wegscheid began to explore what it means to quantify the "competition" between traps. You can read about his research here.