Orthogonal Grassmannians and hermitian K-theory in A¹-homotopy theory of schemes

In this work we prove that the hermitian K-theory is geometrically representable in the A^1 -homotopy category of smooth schemes over a field. We also study in detail a realization functor from the A^1 -homotopy category of smooth schemes over the field R of real numbers to the category of topological spaces. This functor is determined by taking the real points of a smooth R-scheme. There is another realization functor induced by taking the complex points with a similar description although we have not discussed this other functor in this dissertation. Using these realization functors we have concluded in brief the relation of hermitian K-theory of a smooth scheme over the real numbers with the topological K-theory of the associated topological space of the real and the complex points of that scheme: The realization of hermitian K-theory induced taking the complex points is the topological K-theory of real vector bundles of the topological space of complex points, whereas the realization induced by taking the real points is a product of two copies of the topological K-theory of real vector bundles of the topological space of real points