Algebraic and hermitian K-theory in motivic homotopy theory

Algebraic K-theory space, as a motivic space, has a known
geometric model given by (Z copies of) the infinite Grassmannian. In the new
geometric model that we offer, the infinite Grassmannian is replaced by the
Hilbert scheme of the infinite affine space, stabilized with respect to
degree. Although the geometry of the Hilbert scheme is much more
complicated, this model leads to new insights on K-theory. For example, we
obtain an analogous model for the hermitian K-theory space, given by the
Hilbert scheme of finite Gorenstein subschemes that are equipped with an
orientation. Turns out, this model works also in charactersitic 2, while the
previously known model, given by the infinite orthogonal Grassmannian,
represents hermitian K-theory only over a base where 2 is invertible. This
is joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Burt
Totaro.