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Tuesday, February 10, 2015

**Abstract:** The Grothendieck ring of varieties is a fundamental object of study for algebraic geometers. As with all Grothendieck rings, one may hope that it arises as $\pi_0$ of a $K$-theory spectrum, $K(Var_k)$. Using her formalism of assemblers, Zahkarevich showed that this is in fact that case. I'll present an alternate construction of the spectrum that allows us to quickly see the $E_\infty$-structure on $K(Var_k)$ and produce various character maps out of $K(Var_k)$. I'll end with a conjecture about $K(Var_k)$ and iterated $K$-theory.