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# Francis Brown

### Mixed Tate motives over the integers

Abstract. In this talk, I will outline a proof of the following theorem: the category of mixed Tate motives over $\mathbb{Z}$ is generated by the fundamental group of the projective line minus 3 points. This implies a conjecture due to Deligne and Ihara on the action of the absolute Galois group on the pro-l fundamental group. The method of proof also implies a conjecture due to M. Hoffman, which states that  every multiple zeta value $\zeta(n_1,...,n_r)$  is a $\mathbb{Q}$-linear combination of $\zeta(a_1,..,a_s)$ where $a_i=2$ or $3$.

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