P r e f a c e

During the decade of the 1980s profound connections were discovered

relating modern algebraic geometry and algebraic AT-theory to arithmetic

problems. Indeed, the phrase "arithmetic algebraic geometry" was popu-

larized during that time and is now used by many to denote an entire branch

of 20th century number theory. In addition, these same developments in

algebraic geometry and AT-theory greatly influenced research into the arith-

metic of fields in general, in particular the algebraic theory of quadratic forms

and the theory of finite-dimensional division algebras. The purpose of the

1992 AMS Summer Research Institute was to provide the research commu-

nity with both a broad overview of the tools from algebraic geometry and

AT-theory that proved to be the most powerful in solving problems in the the-

ory of quadratic forms and division algebras, as well as provide a forum for

exposition of recent research.

The three week institute had six week-long speakers: three in Af-theory,

R. Swan, A. A. Suslin, and A. S. Merkurjev; and three in algebraic geometry,

J.-L. CoUiot-Thelene, W. Raskind, and D. Saltman. A substantial portion

of their lectures are reproduced in their expository articles in this volume.

The editors hope these articles will help introduce young researchers to these

important tools. In addition the institute had a series of individual research

lectures, many of which can be found here. The organizers would like to

thank the NSF for financial support, as well as the AMS and conference

coordinator Wayne Drady for their efforts in administering the institute.

Bill Jacob

Alex Rosenberg

Santa Barbara, March 1994

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