Fields Program on Arithmetic Geometry, Hyperbolic Geometry, and Related Topics: International U.S. Participation

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The Fields Institute is an international center for mathematical research affiliated with the University of Toronto. Since 1992, a major part of the Institute's activity has focused on semester- and year-long programs designed to bring the world's leading mathematical researchers from a particular area together with students, postdoctoral fellows, and young academics.In July--December, 2008, the Fields Institute will hold a program entitled "Arithmetic Geometry, Hyperbolic Geometry, and Related Topics." This will be the Institute's principal activity during that period.Arithmetic geometry is the application of methods of algebraic geometry to questions of relevance to number theory. Specifically, given a system of equations for which one wants to study its rational solutions (or solutions over a given number field), one studies this system using methods of algebraic geometry and attempts to show finiteness or sparseness of the set of solutions. The focus of this program is to do so using ideas stemming from the Thue-Siegel-Roth theorem in diophantine approximation.Hyperbolic geometry (as envisioned by this program) is the study of analytic functions from the complex plane to complex algebraic manifolds, again studied through the lens of algebraic geometry. If there are no such functions (other than constant functions), then the manifold is said to be hyperbolic. A weaker condition would be to show that such maps have sparse image. The main methods used in this program stem from the work of R. Nevanlinna in value distribution theory.Although the above two paragraphs describe very different areas of mathematics, their statements and methods have been found to be very similar, for reasons that are not fully understood. In addition, the program will also encompass Arakelov theory, an area of algebraic geometry that is of particular use in arithmetic geometry, and which relies heavily on tools of hyperbolic geometry.This grant will be used to bring US mathematicians to this program and its two workshops, and support them while there. The funds will be used principally to support young mathematicians who do not have other sources of support, but may also help cover travel expenses of some senior participants. Since the basic program is primarily funded through Canadian sources, the impact of NSF funding will be highly leveraged, allowing junior and underfunded US researchers to access the program at the relatively small cost of their own travel and subsistence.