English

The Atiyah-Segal completion theorem states that the completion of equivariant complex K-theory may be computed using the Borel construction, which is a generalization of the construction of the classifying space. This thesis investigates whether there is an analogue of the Atiyah-Segal completion theorem for Hermitian K-theory, also known as Grothendieck-Witt theory. We compute the higher Grothendieck-Witt groups of the motivic classifying space of a split torus over an arbitrary field of characteristic not two. We also compute the higher Grothendieck-Witt groups of projective bundles over a divisorial base scheme for which two is an invertible global section, as well as the higher Grothendieck-Witt groups of even Grassmannians over a field of zero characteristic. These computations rely on semi-orthogonal decompositions that behave well with respect to duality. Using these results, we prove the Atiyah-Segal completion theorem for Grothendieck-Witt theory in the special case of the split torus. This completion theorem is an important first step in finding a more general Atiyah-Segal completion theorem, moreover it is likely to be a key ingredient in the proof of such a generalization.