Automorphic Forms: L-Functions and Related Geometry

自守形式：L 函数和相关几何

• 批准号: 1205036
• 负责人: Roger Howe
• 金额: $ 4.96万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-02-01 至 2013-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205036&HistoricalAwards=false
• 关键词: Automorphic; Forms; Functions; Related; Geometry

This conference is planned to take place 23 - 27 April, 2012, at Yale University in New Haven, CT. It will summarize, synthesize, and project into the future, the work of I.I. Piatetski-Shapiro in the area of automorphic forms, especially L-functions. The conference will be of five days duration, with five talks per day. The talks will investigate 5 main themes that reflect Piatetski-Shapiro's main areas of investigation: functoriality and converse theorems, explicit constructions and periods, p-adic L-functions, geometry, and analytic number theory. Each theme will continue throughout the conference. A volume of proceedings, intended to make these results more accessible and applicable, will be produced. Additional information can be found at the conference website: http://www.math.yale.edu/automorphicforms2012Automorphic forms are one of the most fascinating and mysterious of the many branches of number theory. Despite their complexity, they have had remarkable and widely varied applications. Results from the theory of automorphic forms were used in the recent proof by A. Wiles and R. Taylor, of "Fermat's Last Theorem" -- the statement that the sum of two perfect nth powers of whole numbers cannot be a perfect n-th power, if the power n is greater than 2 -- a problem that had been unsolved for over 300 years. Automorphic forms and L-functions are deeply related to prime numbers, which are whole numbers larger than one that have no factors except 1 and themselves. They are the building blocks for multiplication of whole numbers. Prime numbers have recently been used decisively in "public key cryptography", which is the basis of secure transactions on the internet. Prime numbers occur very irregularly among all whole numbers, and the subject of their distribution has attracted an immense amount of research. The Riemann zeta function, which offers a path to a refined understanding of the distribution of prime numbers among all positive integers, is the first example of an L-function. L-functions also provide a means of expressing subtle "reciprocity laws" that govern the solutions of polynomial equations. Automorphic forms are also the source of some of the most beautiful formulas in mathematics, for example, Jacobi's formula for the number of ways to express a whole number as a sum of four perfect squares. Finally, there are remarkable connections between the theory of automorphic forms and physics. The same mathematical structure that is foundational to quantum mechanics (the Heisenberg Canonical Commutation Relations), is also the setting for one of the most important methods for constructing automorphic forms. In addition, a variety of formulas from theoretical physics and related mathematics have been discovered in recent years to have strong connections to the theory or automorphic forms. I. I. Piatetski-Shapiro was a world leader in the theory of automorphic forms. The heart of his contributions involved establishing close connections between automorphic forms and L-functions, especially through his "Converse Theorem", which gave detailed conditions for a family of L-functions to be related to an automorphic form. This conference offers an opportunity to synthesize, disseminate, and build on Piatetski-Shapiro's work, to increase our understanding of these fascinating ideas.

本次会议计划于2012年4月23日至27日在康涅狄格州纽黑文的耶鲁大学举行。报告将总结、综合和预测国际劳工组织的工作。Piatetski-Shapiro在自守形式领域，特别是L-函数。会议为期五天，每天举行五场会谈。讲座将探讨反映Piatetski-Shapiro的主要研究领域的5个主题：泛函和匡威定理，明确的建设和时期，p-adic L-函数，几何和解析数论。每个主题将贯穿整个会议。为了使这些结果更容易获得和适用，将编制一卷会议记录。更多信息可以在会议网站上找到：http://www.math.yale.edu/automorphicforms2012Automorphic形式是数论众多分支中最迷人和神秘的分支之一。尽管它们很复杂，但它们具有显着且广泛的应用。自守形式理论的结果被A. Wiles和R.泰勒的“费马大定理”--如果n的幂大于2，则整数的两个完全n次幂之和不可能是完全n次幂--这个问题300多年来一直没有解决。自守形式和L-函数与素数有很深的关系，素数是大于1的整数，除了1和它们自己之外没有因子。它们是整数乘法的基本单元。最近，素数在“公钥密码学”中得到了决定性的应用，这是互联网上安全交易的基础。素数在所有整数中的出现非常不规则，它们的分布吸引了大量的研究。黎曼zeta函数是L函数的第一个例子，它提供了一条精确理解素数在所有正整数中分布的途径。L函数也提供了一种表达微妙的“互易定律”的方法，这些定律支配多项式方程的解。自守形式也是数学中一些最美丽的公式的来源，例如，雅可比的公式，用于将整数表示为四个完全平方和的方法的数量。最后，自守形式理论和物理学之间有着显著的联系。作为量子力学基础的数学结构（海森堡正则对易关系）也是构造自守形式的最重要方法之一。此外，近年来发现了许多来自理论物理和相关数学的公式与理论或自守形式有很强的联系。I. I.皮亚特茨基-夏皮罗是自守形式理论的世界领导者。他的贡献的核心是在自守形式和L-函数之间建立密切的联系，特别是通过他的“匡威定理”，该定理给出了一个L-函数族与自守形式相关的详细条件。这次会议提供了一个机会，综合，传播和建立在Piatetski-Shapiro的工作，以增加我们对这些迷人的想法的理解。