Special Meeting: Galois Representations, Diophantine Equations, and Automorphic Forms

在2012年冬季和春季，菲尔德研究所将开展一项关于伽罗瓦表示、丢番图方程和自同构形式的密集课程，这将是该研究所在此期间的主要活动之一。拟议的活动将为职业生涯早期的美国数学研究人员参与该计划提供支持。在Wiles和Taylor-Wiles关于Shimura-Taniyama猜想的开创性工作之后，Galois表示的模性及其与朗兰兹程序的联系方面的思想和技术已经有了非常迅速的发展；最近的例子包括Clozel、Harris、Shepherd-Barron和Taylor证明Sato-Tate猜想，以及Khare、Wintenberger和Kisin证明Serre的模性猜想。所有迹象都表明，这些结果只是潜在收获的开始。自同构表示理论的新进展(例如，Ngo证明的基本引理)、新兴的p-进局部朗兰兹对应理论、p-进模形理论以及形变理论和Shimura簇的局部结构的新思想的结果才刚刚开始探索，并有望在未来几年内取得重大进展。所有这些发展，无论是现有的还是潜在的，都将作为菲尔兹计划的一部分进行探索，早期的美国数学家将从了解并参与这些发展的机会中受益匪浅。数论是数学的一个分支，研究与整数性质有关的现象。一个典型的数论问题是确定某个感兴趣的方程的整数解的个数。这类问题的答案通常可以编码成称为L函数的某些数学函数。数学家罗伯特·朗兰兹发展了一系列关于L函数的猜想(或数学预测)，这些猜想(或数学预测)预言，任何L函数都应该由另一种称为自同构形的数学函数产生。(数学家将朗兰兹猜想的L函数与自同构形之间的关系称为“互易定律”。)朗兰兹发展了一系列强有力的表征理论方法来研究他的猜想。这些方法利用自同构形和L函数的许多对称性来分析它们的数学性质；这些方法已被并入一套被称为“朗兰兹程序”的数学中，该程序是现代数论的中心领域之一，实际上也是现代纯数学的核心领域之一。在2012年冬季和春季，菲尔兹研究所将开展一个主题计划，题为伽罗瓦表示、丢番图方程和自同构形式，致力于研究朗兰兹计划。目前的活动将支持职业生涯早期的美国数学家参加菲尔兹计划，从而使他们有机会了解朗兰兹计划的最新发展，并发展必要的技能，以帮助促进下一波发展。

During the Winter and Spring of 2012, the Fields Institute will mount an intensive program on the subject of Galois Representations, Diophantine Equations, and Automorphic Forms, and this will be one of the principal activities of the Institute during that period. The proposed activity will provide support for the participation of early career U.S. mathematical researchers in the program. There has been a very rapid development of ideas and techniques in the area of modularity of Galois representations and its connections with the Langlands program, following the pioneering work of Wiles and Taylor-Wiles on the Shimura-Taniyama conjecture; recent examples include the proof of the Sato--Tate conjecture by Clozel, Harris, Shepherd-Barron, and Taylor, and the proof of Serre's modularity conjecture by Khare, Wintenberger, and Kisin. There is every indication that these results are only the beginning of the potential harvest. The consequences of new progress in the theory of automorphic representations (e.g. the fundamental lemma, proved by Ngo), the emerging theory of a p-adic local Langlands correspondence, the theory of p-adic modular forms, and new ideas in deformation theory and the local structure of Shimura varieties, have only begun to be explored, and significant developments can be expected in the next few years. All of these developments, both existing and potential, will be explored as part of the Fields program, and early career U.S. mathematicians will benefit greatly from the chance to learn about, and indeed to participate in, these developments.Number theory is the branch of mathematics that studies phenomena related to properties of whole numbers. A typical number theoretic question is to determine the number of whole number solutions of some equation of interest. The answers to such questions can often be encoded in certain mathematical functions known as L-functions. The mathematician Robert Langlands has developed a series of conjectures (or mathematical predictions) regarding L-functions, which predict that any L-function should arise from another kind of mathematical function called an automorphic form. (Number theorists refer to Langlands conjectured relationship between L-functions and automorphic forms as a ``reciprocity law''.) Langlands developed an array of powerful representation theoretic methods to study his conjectures. These are methods that exploit the many symmetries of automorphic forms and L-functions to analyze their mathematical properties; these methods have been incorporated into a body of mathematics known as ``the Langlands program'', which is one of the central areas of modern number theory, and, indeed, of modern pure mathematics. In Winter and Spring of 2012, the Fields Institute will run a thematic program, titled Galois Representations, Diophantine Equations, and Automorphic Forms, which will be dedicated to studying the Langlands program. The present activity will provide support for early career U.S. mathematician to participate in the Fields program, who will thereby be provided with the opportunity both to learn about recent developments in the Langlands program, and to develop the skills necessary to help contribute to the next wave of developments.