This grant will support US-based speakers and other US-participants to attend the conference, Regulators III, to take place in July 2010 at the Centre de Recerca Matematica (CRM) in Barcelona, Spain. This is the culminating conference of a year-long program in arithmetic geometry at the CRM, and will attract many of the best researchers in the world. The funding will allow US-based researchers to learn of the latest developments in a field in which mathematicians from Europe and Asia have been playing an increasing role over the last 10 years. Recent developments in the subject include work on Beilinson?s conjecture on special values of L-functions of certain automorphic representations of a symplectic group in four variables, and the lack of surjectivity of certain regulators for generic surfaces of degree at least 5 in projective 3-space over p-adic fields.
Regulators help to measure the size of objects in arithmetic and geometry, and they are a unifying theme in the subject. The definition and computation of regulators is a fundamental part of several parts of mathematics. One recent theme in the subject has been to transport techniques from arithmetic to geometric situations, and vice-versa. These methods have allowed us to demonstrate the existence of interesting elements of important arithmetic and geometric objects.
Intellectual Merit: This award was used to support the travel and subsistence of US participants and speakers for the conference, Regulators III, which was held at the Universitat de Barcelona, Spain from July 12-22, 2010. The conference brought together a group of some 130 researchers from around the world. They reported on the latest results in the field in some 35 lectures during the conference. This gathering of experts from various aspects of the subject is helping to stimulate progress in the field and lead to new and fundamental results. A regulator is a way of capturing many of the properties of mathematical objects of various types and measuring their size and abundance. Many times, a particular mathematical object is very complicated and difficult to understand. A suitable regulator gives us information about the object that is easier to compute and understand, but may not be complete. However, in fortunate cases, we can glean enough information from suitable regulators to understand the object very well. The proceedings of the conference will be published in a book to appear in the series, Comtemporary Mathematics, published by the American Mathematical Society. There are some 15 high level articles that will assist this and the next generation to learn and advance the subject. Broader Impact: Most of the participants supported by the grant were students and young researchers in the field who had no other means of support. Several are women, who tend to be underrepresented in this particular subfield of mathematics.
