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 give an introduction to the Grothendieck ring of varieties, giving a survey of a small portion of interesting results in the field due to various people (Kapranov, Cheah, Larsen-Lunts, Liu-Sebag, Bittner, Borisov, Litt, ...). Time permitting, I will discuss joint work with Melanie Matchett Wood. (This lecture is intended for a broad audience.)