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Tuesday, May 5, 2015

**Abstract:** T. Pirashvili defined the higher Hochschild homology groups of a commutative ring (with coefficients in the ring itself or more generally a bimodule) as the homotopy groups of the Loday construction of that ring (and module) evaluated on spheres. Having strictly associative and commutative ring spectra allows us to do the same for spectra, obtaining higher topological Hochschild homology. I would like to discuss some basic calculations (joint with I. Bobkova, K. Poirier, B. Richter, and I. Zakharevich) of higher Hochschild homology, namely of Z/p[x] and Z/p[x]/x^a when p divides a. I would then like to explain how these lead to a calculation (joint with B. Dundas and B. Richter) of higher topological Hochschild homology of number rings with coefficients in their residue fields, and to a re-calculation of higher topological Hochschild homology of finite fields (done originally by other methods by M. Basterra and M. Mandell).