Mathematical Sciences: Index Reduction Formulas for Central Simple Alge>ras.

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This work is concerned with the study of index reduction formulas for central simple algebras on passage to the function field of a projective variety. The type of variety under consideration is a twisted form of G/P where G is a split reductive algebraic group and P is a parabolic subgroup of G. The index reduction formula will follow from a knowledge of the Grothendieck group of the variety of interest, which has been computed in general only when G is connected. The principal investigator will investigate the Grothendieck group of the twisted forms of G/P when G is not connected, where it is clear from examples that some new phenomena not seen in the connected case will arise. The principal investigator will also study central simple algebras with involution, and the associated generalized even Clifford algebras. This research is in the general area of ring theory. A ring is an algebraic object having both an addition and a multiplication defined on it. Although the additive operation satisfies the commutative law, the multiplicative operation is not required to do so. An example of a ring for which multiplication is not commutative is the collection of nxn matrices over the integers. The study of noncommutative rings has become an important part of algebra because of its increasing significance to other branches of mathematics and physics.