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

在2012年冬季和春季，菲尔兹研究所将开展一个关于伽罗瓦表示、丢番图方程和自同构形式的密集项目，这将是该研究所在此期间的主要活动之一。拟议的活动将为美国早期职业数学研究人员参与该计划提供支持。继Wiles和Taylor-Wiles在Shimura-Taniyama猜想上的开创性工作之后，伽罗瓦表示的模块化及其与Langlands程序的联系领域的思想和技术得到了非常迅速的发展；最近的例子包括Clozel、Harris、Shepherd-Barron和Taylor对Sato- Tate猜想的证明，以及Khare、Wintenberger和Kisin对Serre模块化猜想的证明。种种迹象表明，这些结果只是潜在收获的开始。自同构表示理论的新进展（例如由Ngo证明的基本引理），p进局部朗兰兹对应的新兴理论，p进模形式理论，变形理论和Shimura变体的局部结构的新思想，才刚刚开始被探索，未来几年可以期待重大的发展。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，早期职业的美国数学家将从了解和参与这些发展的机会中受益匪浅。数论是研究与整数性质有关的现象的数学分支。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。这些问题的答案通常可以用某些称为l函数的数学函数进行编码。数学家罗伯特·朗兰兹（Robert Langlands）提出了一系列关于l函数的猜想（或数学预言），预言任何l函数都应该由另一种称为自同构形式的数学函数产生。（数论学家将朗兰兹猜想的l函数与自同构形式之间的关系称为“互易律”。）朗兰兹发展了一系列强大的表征理论方法来研究他的猜想。这些方法利用自同构形式和l函数的许多对称性来分析它们的数学性质；这些方法已被纳入一个被称为“朗兰兹纲领”的数学体系，它是现代数论的中心领域之一，实际上也是现代纯数学的中心领域之一。在2012年冬季和春季，菲尔兹研究所将开展一个主题项目，名为伽罗瓦表示、丢番图方程和自同构形式，该项目将致力于研究朗兰兹程序。目前的活动将为早期职业的美国数学家参与菲尔兹项目提供支持，他们将因此有机会了解朗兰兹项目的最新发展，并发展必要的技能，帮助下一波发展做出贡献。

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.