#
Deformation theory, patching, quadratic forms, and the Brauer group

January 17 to January 21, 2011
at the

American Institute of Mathematics,
San Jose, California

organized by

Daniel Krashen and Max Lieblich

## Original Announcement

This workshop will focus on the interaction between algebraic geometry and the structure theory
of fields, particularly the use of deformation theory and patching.
Formal techniques in algebraic geometry provide a strong link
between moduli theory, various kinds of local-to-global principles,
and several classical problems in algebra. The goal of this workshop
is to bring together researchers in algebra,
number theory, and algebraic geometry to study two main problems in
the arithmetic of fields:

- the period-index problem on the relation between the dimension of a
division algebra and its order in the Brauer group;
- the
*u*-invariant problem on the maximal dimension of an anisotropic
quadratic form.

A central question related to these problems is what dependence the
period-index relation and the *u*-invariant have on various measures
of the
dimension of a field (e.g. cohomological dimension, *C*_{i}-property).
In particular, how do the period-index relation and the *u*-invariant
change upon taking field extensions of positive transcendence degree?
In addition, how do these problems interact with standard
local-to-global conjectures? To what extent to they reflect a general
theory of 0-cycles on homogeneous varieties?
These problems have seen a flurry of activity in recent years rooted
in moduli theory, infinitesimal deformation theory, and patching.
Significant further progress on these problems seems within reach if a
critical mass of workers with diverse backgrounds can be brought to
bear on them.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop: