A workshop on "Arithmetic of abelian varieties in families" will be held at the Centre Interfacultaire Bernoulli (CIB) in Lausanne, Switzerland during November 12 - 16, 2012, taking advantage of the presence of researchers at the semester-long program on "Rational points and algebraic cycles" there during Fall 2012: see http://rational.epfl.ch/workshop2.php The main topics of the workshop will be the distribution of arithmetic quantities (e.g., Mordell-Weil groups, Selmer groups, and Shafarevich-Tate groups) associated to elliptic curves and higher-dimensional abelian varieties, and the related questions of existence of rational points on varieties fibered in genus 1 curves.
The theme of the supported workshop is about the rational number solutions to structured polynomial equations, a topic that was of interest to the ancient Greeks but is still largely a mystery today. Recently several groups of researchers working independently have made progress on these questions using a variety of methods, studying the statistical behavior of families of equations all at once instead of trying to understand individual equations in isolation. The grant will enable the PIs to bring the key researchers and their ideas together, and to involve young researchers so that the new ideas are assimilated into the next generation.
". The topic of the conference lies in arithmetic geometry which studies the rational number solutions to polynomial equations, continuing a line of fundamental research that began with the ancient Greeks and that starting in the 20th century found unforeseen applications to cryptography and error-correcting codes. Even simple-looking equations, such as polynomial equations of degree 3 in two variables, are not yet fully understood, but recently it was realized that significant progress could be made by studying equations not in isolation, but by studying the statistical behavior in families of equations. The topic turned out to be a timely one because of recent advances from different directions. The conference consists of 18 one-hour lectures covering the wide variety of approaches to the subject. The organizers of the conference were successful in adding the top U.S. researchers in the field to the list of participants along with the top researchers from other countries, in part due to the NSF funding. The funding also provided support to several young mathematicians and mathematicians from underrepresented groups who otherwise would have been unable to attend. For some of these, the opportunity to attend this conference has been eye-opening, and has led to advances in their own research. The breaks between the talks led to some fortuitous interactions; for example, when one participant posed three open questions, another participant succeeded in answering two of them before the end of the conference.
