Title: Cutting and pasting in (algebraic) geometry
Speaker: Ravi Vakil (Stanford University)
Abstract: Given some class of "geometric space", we can make a ring as follows.
- (additive structure) When $U$ is an open subset of such a space $X$, $[X] = [U] + [(X \setminus U)]$;
- (multiplicative structure) $[X \times Y] = [X] [Y]$.
In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising "stabilization" 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.