Cutting and pasting in algebraic geometry

Given some class of “geometric spaces, we can make a ring as follows.
\begin{enumerate}
\item[(i)] {\em (additive structure)} When $U$ is an open subset of such a space $X$,
$[X] = [U] + [(X \setminus U)]$;
\item[(ii)] {\em (multiplicative structure)} $[X \times Y] = [X] [Y]$.
\end{enumerate}
In the algebraic setting, this ring (the “Grothendieck ring of varieties) contains
surprising structure, connecting geometry to arithmetic and topology. I will discuss some remarkable
statements about this ring (both known and conjectural), and present new statements
(again, both known and conjectural). A motivating example will be polynomials in one variable. (This talk is intended for a broad audience.) This is joint work with Melanie Matchett Wood.