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

菲尔兹研究所是隶属于多伦多大学的国际数学研究中心。自1992年以来，该研究所的主要活动集中在为期一学期和一年的项目上，旨在将来自特定领域的世界领先的数学研究人员与学生、博士后研究员和年轻学者聚集在一起。2008年7月至12月，菲尔兹研究所将举办一项名为“算术几何、双曲几何及相关主题”的项目。这将是该研究所在此期间的主要活动。算术几何是代数几何方法在与数论相关的问题上的应用。具体地说，给定一个方程组，人们想要研究它的有理解(或给定数域上的解)，人们使用代数几何的方法来研究这个系统，并试图证明解的集合的有限或稀疏性。本程序的重点是使用丢番图近似中的Thue-Siegel-Roth定理的思想来实现这一点。双曲几何(如本程序所设想的)是研究从复平面到复代数流形的解析函数，再次通过代数几何的透镜进行研究。如果没有这样的函数(除了常量函数)，则流形称为双曲的。一个较弱的条件是显示这样的映射具有稀疏图像。本程序中使用的主要方法来自R.内万林纳在价值分配理论方面的工作。尽管以上两段描述了非常不同的数学领域，但他们的陈述和方法被发现非常相似，原因尚不完全清楚。此外，该计划还将包括阿拉克洛夫理论，这是一个在算术几何中特别有用的代数几何领域，它严重依赖双曲几何工具。这笔赠款将用于吸引美国数学家参加该计划及其两个研讨会，并在那里支持他们。这笔资金将主要用于支持没有其他支持来源的年轻数学家，但也可能帮助支付一些资深参与者的旅费。由于基本计划的资金主要来自加拿大来源，NSF资金的影响将得到高度杠杆化，允许初级和资金不足的美国研究人员以相对较小的旅行和生活费获得该计划。

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.