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

January 17 to January 21, 2011

at the

American Institute of Mathematics, Palo Alto, California

organized by

Daniel Krashen and Max Lieblich

This workshop, sponsored by AIM and the NSF, 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.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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