Vojta's Conjectures (14w5129)

Arriving in Banff, Alberta Sunday, September 28 and departing Friday October 3, 2014

Organizers

(Michigan State University)

(University of Waterloo)

(University of California at Berkeley)

Umberto Zannier (Scuola Normale Superiore)

Objectives

As this workshop would be the first mathematical gathering explicitly centered on Vojta's conjectures, it affords a number of unique opportunities. A primary objective of the workshop is to bring together experts in diverse fields and to facilitate collaboration and the exchange of ideas amongst participants. The invited participants will include experts in Diophantine approximation, Nevanlinna theory, algebraic geometry, and arithmetic geometry, among others. As the participants will have varied backgrounds, in addition to typical research talks, some lectures will be requested to provide introductions/surveys of various topics (e.g., the Nevanlinna-Diophantine dictionary). This should be especially useful for the younger researchers and participants we expect to invite. Indeed, another aim of the workshop is to help build a strong foundation for younger participants, and in particular an increased awareness of the current approaches to all aspects of Vojta's conjectures. To this end, colleagues from the invitation list will be solicited for information on strong upcoming graduate students and postdoctoral researchers to be included in the workshop.As mentioned in the overview, the workshop will feature talks and research that attack Vojta's conjectures and its related problems from a multitude of directions: the Thue-Siegel-Roth method, function field techniques, Kim's approach to Siegel's theorem, Mochizuki's work, analogues from Nevanlinna theory, Arakelov theory, etc. While no workshop can be completely comprehensive, we nonetheless hope to have representatives of the current techniques and advances in the field.Vojta's conjectures concern fundamentally deep and important problems. The one-dimensional case of the conjectures implies the ABC conjecture, which by itself has taken on an increasingly central role in number theory. The increasing number of papers in recent years proving results conditional on Vojta's conjectures or the ABC conjecture underscores their importance. Special known cases of the conjectures (e.g., Roth's theorem, Faltings' theorem) are applied throughout mathematics and form part of the core of modern number theory.Given the recent flurry of activity around Vojta's conjectures and the lack of a previous comparable workshop, the workshop is very timely and highly relevant to many mathematical communities. It is an exciting time to be working in the area, with problems that require an ever-expanding toolbox of techniques drawn from the most recent advances in diverse areas of mathematics. The workshop will provide a unique opportunity for a wide range of mathematicians sharing a common interest in Vojta's conjectures to discuss, collaborate, and make advances on this important topic.