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In recent joint work with Calmès, Dotto, Harpaz, Land, Moi, Nardin and Nikolaus we developed a formalism for Grothendieck-Witt spectra of stable categories, that is very amenable to computation, in particular enjoying a tight relation to Witt-(or L-)spectra. After briefly recalling the set-up, I will explain how this theory recovers the classical Grothendieck-Witt animae of ordinary rings, which are defined as group completions of moduli spaces of unimodular forms over R. In combination these statements allow us to solve a number of open problems, and allow access to the stable homology of orthogonal and symplectic groups over the integers for example.

The comparison itself is a hermitian analogue of Quillen's `+=Q´ theorem and the Gillet-Waldhausen theorem, though our proof proceeds very differently: It is based on ideas from the theory of cobordism categories in manifold topology, of which we provide an algebraic analog based on Ranicki's algebraic surgery theory.

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