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Tuesday, January 19, 2016

**Abstract:** In Balmer's framework of tensor triangular geometry, the prime thick tensor ideals in a tensor triangulated category $\mathcal{C}$ form a space which admits a continuous map to the Zariski spectrum $\mathrm{Spec}^h(\mathrm{End}^\bullet_u(1))$ of homogeneous prime ideals in the graded endomorphism ring of the unit object. (Here the grading is induced by an element $u$ of the Picard group of $\mathcal{C}$.) If $\mathcal{C}$ is the stable motivic homotopy category and $u$ is the punctured affine line, then this endomorphism ring is the Milnor-Witt K-theory ring $K^{MW}_*(F)$ of the base field $F$. I will describe work by my student, Riley Thornton, which completely determines $\mathrm{Spec}^h(K^{MW}_*(F))$ in terms of the orderings of $F$. I will then comment on work in progress which uses the structure of this spectrum to study the thick subcategories of the stable motivic homotopy category.