I will survey some of the history of Dedekind sums, and show the current state of development of a theory of higher-dimensional differentiable Dedekind sums, which is joint work with Helaman Ferguson.

We use this theory to locate integer points in rational polytopes, one of the most fundamental problems in integer linear programming. I will give some examples in low dimensions.

To maximize accessibility, I will not assume any background at all in Dedekind sums.