My research involves a great deal of complex analysis and spectral theory, and I always feel a bit flustered when non mathematicians ask me what I study. It's hard to explain the math in layman's terms. I was thinking that, instead of explaining the math, I could explain some applications of my areas, and that would be much more entertaining and accessible. For complex analysis, I explain the applications of the poisson integral formula by saying "if you know the temperature of a metal disk at its edges, you can recover its temperature at any point." I'm at a loss for anything else because my education is in pure mathematics.