Equivariant Chow groups: applications to quadratic forms and
algebraic groups -- Burt Totaro

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For an algebraic group G, we want
to classify G-bundles over arbitrary fields. For example,
PGL(n)-bundles over a field are equivalent to projective bundles
(or, in algebraic terms, to central simple algebras), and O(n)-bundles
over a field are equivalent to quadratic forms.

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Grothendieck was able to relate these algebraic problems
to the topology of the group G. I will describe an improved proof
of Grothendieck's theorem, using the simple geometric
definition of equivariant Chow groups.

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Using Grothendieck's theorem plus some new calculations,
we get information about G-bundles over arbitrary fields,
for all types of Lie groups (including E_8).
In the case of orthogonal groups and spin groups,
we get new results about splitting fields for quadratic forms.