Mathematics Colloquium

Suppose a function $\{1,\dots,N\} \to \mathbb R$ has the property that when we take discrete derivatives $k$ times, the result is identically zero. It is fairly well-known that this is equivalent to being a polynomial of degree $k-1$. It's not too unnatural to ask: what does the function look like if, instead, the iterated derivative is required to be zero just a positive proportion of the time? Such \emph{approximate polynomials} have a richer structure, related to nilpotent Lie groups.

On an unrelated note: given an $n \times n$ chessboard, how many ways are there to arrange $n$ queens on it, so that no two attack each other?

I'll outline how both these questions are connected to what's known as \emph{higher order Fourier analysis}, and explain more generally what higher order Fourier analysis is and what it can be used for (other than potentially placing queens on chessboards).