The geometric approach to the algebraic theory of quadratic
forms is the study of projective quadrics over arbitrary fields.
Function fields of quadrics have been central to the proofs of
fundamental results since the 1960's. Recently, more refined
geometric tools have been brought to bear on this topic, such as
Chow groups and motives, and have produced remarkable advances on a
number of outstanding problems. Several aspects of these new
methods are addressed in this volume, which includes an
introduction to motives of quadrics by A. Vishik, with various
applications, notably to the splitting patterns of quadratic forms,
papers by O. Izhboldin and N. Karpenko on Chow groups of quadrics
and their stable birational equivalence, with application to the
construction of fields with u-invariant 9, and a contribution in
French by B. Kahn which lays out a general framework for the
computation of the unramified cohomology groups of quadrics and
other cellular varieties.
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