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:

  1. the period-index problem on the relation between the dimension of a division algebra and its order in the Brauer group;
  2. 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, Ci-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:

A bound to kill the ramification over function fields
by  Alena Pirutka,  J. Algebra 377 (2013), 173-178  MR3008901
The period-index problem for twisted topological K-theory
by  Benjamin Antieau and Ben Williams,  Geom. Topol. 18 (2014), no. 2, 1115-1148  MR3190609
Division algebras and quadratic forms over fraction fields of two-dimensional Henselian domains
by  Yong Hu,  Algebra Number Theory 7 (2013), no. 8, 1919-1952  MR3134039