# COW: Cutting and pasting in algebraic geometry

• Ravi Vakil (Stanford)
• Thursday 21 April 2016, 14:15-15:15
• CMS MR13.

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)] (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 is joint work with Melanie Matchett Wood.

This talk is part of the Algebraic Geometry Seminar series.