Given some class of "geometric spaces", we can make a ring as follows.

1. (additive structure)  When U is an open subset of such a space X, [X] = [U] + [(X \ U)];
2. (multiplicative structure)  [X x Y] = [X] [Y].

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 the case of "points on a line" --- polynomials in one variable.  (This talk is intended for a broad audience.)  This is joint work with Melanie Matchett
Wood.