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.
Santa Barbara, March 1994