It is a fair assumption that many of us in the math department enjoyed math puzzles in our youth and this helped to bring us to where we are. I know I did (and do!). I recently had to solve a classic style of problem: find the nth term of a sequence of integers. I tried everything I knew but only had a pile of scratched out notes to show for it. And then I was told about generating functions. Although not a total panacea for all things sequential, generating functions provide a staightforward blueprint for deriving nth-term formulas and more. I will present a few basic examples and some notes on the excellent book I used as a reference, but the majority of the talk will discuss my particular problem and its solution via generating functions. The main goal will be to impress upon younger grad students the power of this method where other more familiar methods fail.

I'll discuss rigorous nonlinear stability results for traveling waves in a class of reaction-diffusion systems that arise in chemical reaction models. The class includes systems in which there is no diffusion in some equations. The results are detailed enough to show, for example, that the results of adding some heat or adding some reactant to a combustion front are different.