Workshop: Towards a Local Proof of the Local Langlands Correspondence

研讨会：本地朗兰通讯的本地证明

• 批准号: 1207440
• 负责人: Liang Xiao
• 金额: $ 2.57万
• 依托单位: University of Chicago
• 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: 2012-04-01 至 2013-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207440&HistoricalAwards=false
• 关键词: Workshop; Towards; Local; Proof; Langlands

The organizers intend to hold a weekend workshop at the University of Illinois at Chicago, aimed at graduate students and young researchers, on the subject of the Local Langlands Correspondence. Recent work on the special fiber of the stable reduction of the Lubin-Tate tower by J. Weinstein using p-adic Hodge theory and p-divisible groups appears finally to make possible a purely local proof of the correspondence. Five experts will be invited to give talks on different pertinent subjects, which in total present the complete story behind the local correspondence: introductory material; C. Bushnell and P. Kutzko's type theory; M. Strauch's Jacquet-Langlands correspondence via the Lefschetz trace formula; Weinstein's aforementioned results on the stable reduction of the Lubin-Tate tower, and his joint work with M. Boyarchenko on the geometry of the special fiber. The workshop will be held on May 12-13, 2012. More information can be found at the conference website: http://math.uchicago.edu/~lxiao/workshop_site/Number theory is the branch of pure mathematics devoted to the study of whole numbers and their relations; a particular goal is often to understand whether a given algebraic equation has a whole number solution. The apparent simplicity of such problems belies their complexity, and a box of intricate tools has been collected over the past 400 years to attack them. Chief among them is a 'correspondence' proposed by R. Langlands in the 1960s which, once established, would allow number theorists to exploit methods from a wide range of other, seemingly unrelated mathematical fields. For example, the 300 year old problem 'Fermat's Last Theorem' was successfully tackled by A. Wiles in the 1990s by establishing part of Langlands' correspondence. Now, although the correspondence remains mysterious in general, it can be split into pieces which can be studied 'one prime at a time', called 'local correspondences': since prime numbers are the atoms of the whole numbers, a common approach in number theory is to attack a problem one prime at a time. These local correspondences can be investigated using geometry and are much better understood; the main intent of the organizers is to host a weekend workshop where graduate students and young researchers can learn about the latest developments in the field.

组织者打算在伊利诺伊大学芝加哥分校举办一个周末研讨会，针对研究生和年轻研究人员，主题是当地朗兰兹通信。最近由J.Weinstein利用p-进Hodge理论和p-可除群对Lubin-Tate塔的稳定约化的特殊纤维所做的工作似乎终于使这种对应的纯局部证明成为可能。将邀请五位专家就不同的相关主题进行演讲，这些主题总共介绍了当地通信背后的完整故事：介绍性材料；C.Bushnell和P.Kutzko的类型理论；M.Strauch通过Lefschetz迹公式的Jacquet-朗兰兹通信；Weinstein关于鲁宾-塔特塔稳定缩减的前述结果，以及他与M.Boyarchenko在特殊纤维几何方面的联合工作。研讨会将于2012年5月12-13日举行。更多信息可以在会议网站上找到：http://math.uchicago.edu/~lxiao/workshop_site/Number理论是纯数学的一个分支，致力于研究整数及其关系；一个特殊的目标通常是了解给定的代数方程是否有整数解。这些问题表面上的简单掩盖了它们的复杂性，在过去的400年里，人们收集了一盒复杂的工具来攻击它们。其中最主要的是R·朗兰兹在20世纪60年代提出的一项“通信”理论，该理论一旦建立，将允许数字理论家利用其他广泛的、看似无关的数学领域的方法。例如，A·威尔斯在20世纪90年代通过建立朗兰兹的部分通信，成功地解决了300年前的费马最后定理问题。现在，尽管这种对应关系在总体上仍然是个谜，但它可以被分成几个片断，可以一次研究一个素数，称为“局部对应”：因为素数是整个数的原子，所以数论中的一种常见方法是一次解决一个素数问题。这些当地通信可以用几何学进行调查，而且更容易被理解；组织者的主要目的是举办一个周末研讨会，让研究生和年轻的研究人员可以了解该领域的最新发展。