Computations of Chow-Witt groups for split quadrics and other smooth varieties

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Chow-Witt groups are refinements of the classical Chow groups. They contain important invariants of algebraic varieties. For example, they contain the "Euler class", which encodes the splitting behaviour of vector bundles over affine bases. The aim of this project is to compute the Chow-Witt groups of certain families of smooth varieties. Our initial focus will be on split quadrics, as results for these promise to have implications for the classical problem of the existence of sums-of-squares formulas. Next, we will investigate other classes of examples (other homogeneous spaces, smooth curves, ...). The Witt groups and the hermitian K-groups are already partially understood for these spaces, and the Chow-Witt groups are closely related to these via motivic spectral sequences.