This talk will provide a basic introduction to the rich, rather overwhelming, mathematical theory of waves. We will develop some models for real-world wave behavior, skim the history of research in wave theory over the last few centuries, and see particular results for waves in lattice structures. Some of these results are well-known and long-established; others are recent work with J. Douglas Wright.
The only prerequisite is a slight familiarity with the concept of the partial derivative.