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Tuesday, March 8, 2016

**Abstract:** In the early 1980’s Waldhausen proposed a program for studying the algebraic K-theory of the E(n) Bousfield localizations of the sphere using his localization theorem. The E(0) localization of the sphere, which is equivalent to the rational Eiilengberg-Maclane spectrum is the only case where the algebraic K-theory is known. Due to the work of McCarthy and Dundas, relative algebraic K-theory and relative topological cyclic homology are homotopy equivalent after p completion for certain maps of ring spectra. Topological cyclic homology is built out of topological Hochschild homology using the S^1-equivariant and cyclotomic structure. We therefore compute mod (p,v_1) homotopy of THH of the connective cover of the K(1)-local sphere, which is the p-completion of E(1)-local sphere. We construct a filtration of the connective cover of the K(1)-local sphere and then use a May-type spectral sequence to compute THH of the connective K(1)-local sphere. The connective K(1)-local sphere is a chromatic height 1 spectrum, so we expect height 2 phenomena in its algebraic K-theory according to the chromatic red-shift conjecture of Ausoni and Rognes. Though this is not visible in THH, it can already be seen by examining the homotopy fixed points with respect to the circle action. Specifically, I will show how the element v_2 appears in the homotopy fixed points with respect to the circle action.