## EVENT DETAILS AND ABSTRACT

**Midwest Topology Seminar**
**Title:** A topological proof of a theorem of Larsen and Lunts

**Speaker:** Inna Zakharevich

**Speaker Info:** University of Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

The Grothendieck ring of varieties $K_0(\mathrm{V})$ is defined to be
the free abelian group generated by varieties, modulo the relation
that for any variety $X$ and any subvariety $Y$, $[X] = [Y] + [X
\backslash Y]$. Multiplication is defined by $[X] [Y] = [X \times
Y]$. This is the universal additive invariant, in the sense that if
any function $\chi$ on varieties satisfies the formula $\chi(X) =
\chi(Y) + \chi(X \backslash Y)$ (such as point counting or Euler
characteristic) factors through this ring. When Kapranov introduced
his motivic zeta function he conjectured that it would always be
rational. However, in their 2002 paper Larsen and Lunts showed that
this is in fact generally not the case. The main step in their proof
was a computation of the quotient ring $K_0(V)/([\mathbb{A}^1])$. In
this talk we give an alternate proof of this computation by replacing
the ring $K_0(V)$ by an $E_\infty$ ring spectrum $K(V)$ and computing
$\pi_0$ of the cofiber of the map $K(V) \to K(V)$ induced by
multiplication by $\mathbb{A}^1$.

**Date:** Saturday, February 06, 2016

**Time:** 3:00 pm

**Where:** Swift 107

**Contact Person:** Lauren Bandklayder

**Contact email:** laurenb@math.northwestern.edu

**Contact Phone:**

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