# Talk

Take a polynomial map $\mathbb{F}_q\to\mathbb{F}_q$ over a (large) finite field and compute the fraction of elements in its image. Most likely, you got $\approx 0.632$. Once we explain why that is, we arrive at a group theory question: Suppose a subgroup $G$ of the symmetric group $S_n$ has the same fraction of fixed-point-free elements as $S_n$ itself. Does it follow that $G=S_n$? The talk will be nontechnical. We will invoke a property of $e=2.71...$.