I will describe a conjecture of Carlsson's that provides an explicit constructive model of the homotopy type of the completed algebraic K-theory spectrum (KF)_p of an arbitrary field F. This is achieved using exclusively the semilinear representation theory of the absolute Galois group G_F. Unlike other approaches, such as the Quillen-Lichtenbaum conjecure, which focus on the hotomopy groups of (KF)_p, Carlsson's approach addresses the entire homotopy type. The conjecture is known to hold in two special cases; in this talk I will outline the proof of the most recently established case. |