I will explain how the study of group schemes in derived algebraic geometry leads naturally to the construction of equivariant cohomology theories. In particular, I will sketch a purely algebraic construction of equivariant K-theory and provide an interpretation of Snaith's theorem, which relates complex K-theory to the suspension spectrum of CP^{infty}.

Let $X$ be a Euclidean building of dimension $m$. The Tits-Solomon theorem implies that metric spheres of small diameter are $(m-1)$-spherical. We show that this generalizes to metric spheres of arbitrary size. If $X$ is irreducible, we want to show that horospheres are also $(m-2)$-connected, and we may already have a proof. This is joint with Kevin Wortman (Yale).