I will report on joint work in progress with Sanath Devalapurkar andJeremy Hahn.  The singular cohomology of a closed, oriented manifoldsatisfies Poincaré duality, and the Galois cohomology of a localnumber field satisfies Tate-Poitou duality.  We prove duality theoremsfor syntomic cohomology and topological cyclic homology of a class ofring spectra, tentatively called higher local number rings, subject to anorientability hypothesis.  This class of ring spectra includes truncatedBrown-Peterson spectra, complex and real topological K-theory,topological modular forms, and their unramified extensions.  The dualitytheorems come in reduced, localized and filtered versions, analogous toknown refinements of Tate-Poitou duality in the case of classical rings.