PECASE: Galois Representations and Modular Forms

PECASE：伽罗瓦表示和模形式

• 批准号: 0093542
• 负责人: Brian Conrad
• 金额: $ 25万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0093542&HistoricalAwards=false
• 关键词: PECASE; Galois; Representations; Modular; Forms

The investigator's previous work on elliptic curves and Galois representations leads in the direction of several questions which are ultimately concerned with understanding the nature of Galois representations, either from the point of view of geometry or deformation theory. One such problem is to find a conceptual moduli-theoretic interpretation of the Coleman-Mazur eigencurve. The investigator also proposes to study the problem of incorporating conditions such as semi-stability (in the sense of Fontaine) in the deformation theory of Galois representations, continuing a line of development growing out of the work of Wiles. In a somewhat different direction, Buzzard has recently observed in numerous examples that the slopes of eigenforms seem to possess much more structure than conjectured by Gouvea-Mazur. These surprising observations do not fit into any general framework, and the investigator proposes to determine the general nature of such phenomena. In addition to studying these problems, the investigator continues his efforts in the direction of supporting active student interest in mathematics at the high school level. Through personal contacts at a local school, he arranges regular meetings in which he leads informal group discussions with students on an assortment of interesting mathematical ideas (taken from a wide variety of disciplines: number theory, geometry, probability, etc.). The idea is to expose students to important and interesting concepts which are not usually encountered in the classroom but which can be presented in an elementary context.The investigator also provides these students with information about summer math programs and research opportunities, in order that they can experience mathematics as a living field of scientific inquiry. Number theory is the branch of mathematics which is concerned with the properties of whole numbers. It abounds in deep and unsolved problems, particularly concerning properties of prime numbers and geometric objects called elliptic curves. Prime numbers and the theory of elliptic curves also lie at the heart of modern cryptographic systems, without which secure diplomatic transmissions and Internet commerce would be impossible. The RSA cryptosystem and the elliptic curve factorization algorithm are two such prominent applications in this context. Improvements in our theoretical understanding of elliptic curves is expected lead to further applications along these lines. The investigator's scientific work is concerned with several questions naturally arising from the theory of elliptic curves, and partly aims to continue the development the techniques that were used to recently settle the Shimura-Taniyama Conjecture, one of the most important problems in the theory of elliptic curves. The investigator also regularly visits with local high school students, showing them important mathematical ideas that are not usually encountered in school, such as the inner workings of the RSA cryptosystem and the role of probability in the design of medical tests for rare diseases. The investigator also provides these students with nformation about summer opportunities for education, research, and work in mathematics, and offers guidance for students who wish to take part in several prestigious high school science research competitions. The Faculty Career Development Program makes it possible for the investigator to continue his scientific work while at the same time enabling him to give some high school students a deeper appreciation for mathematics and its important role in modern society.

研究者先前关于椭圆曲线和伽罗瓦表示的工作引导了几个问题的方向，这些问题最终涉及从几何或形变理论的角度来理解伽罗瓦表示的性质。一个这样的问题是找到科尔曼-马祖尔特征曲线的概念性模理论解释。研究者还建议研究在伽罗瓦表示的形变理论中加入半稳定条件(方丹意义下)的问题，延续了Wiles工作的发展脉络。在一个略有不同的方向上，Buzzard最近在大量的例子中观察到，特征形式的斜率似乎比Gouvea-Mazur猜想的具有更多的结构。这些令人惊讶的观察结果不符合任何一般框架，研究人员建议确定此类现象的一般性质。除了研究这些问题外，调查者继续努力支持学生在高中阶段对数学产生积极的兴趣。通过在当地一所学校的私人联系，他安排定期会议，在会上他与学生就各种有趣的数学思想(来自各种学科：数论、几何、概率等)进行非正式的小组讨论。这个想法是为了让学生接触到重要而有趣的概念，这些概念通常不会在课堂上遇到，但可以在初级环境中呈现。调查者还为这些学生提供了关于暑期数学项目和研究机会的信息，以便他们能够将数学作为科学探究的活领域来体验。数论是研究整数性质的数学分支。它充满了深刻的和未解决的问题，特别是关于素数和称为椭圆曲线的几何对象的性质。素数和椭圆曲线理论也是现代密码系统的核心，没有它们，安全的外交传输和互联网贸易就不可能实现。RSA密码体制和椭圆曲线分解算法就是这方面的两个突出应用。在我们对椭圆曲线的理论理解方面的改进有望导致沿着这些线的进一步应用。研究人员的科学工作涉及椭圆曲线理论自然产生的几个问题，部分目的是继续发展最近用来解决下村-谷山猜想的技术，下村-谷山猜想是椭圆曲线理论中最重要的问题之一。研究人员还定期走访当地的高中生，向他们展示在学校里通常不会遇到的重要数学思想，如RSA密码系统的内部工作原理，以及概率在设计罕见疾病医学测试中的作用。调查员还为这些学生提供有关暑期教育、研究和数学工作机会的信息，并为希望参加几个著名的高中科学研究比赛的学生提供指导。学院职业发展计划使研究人员有可能继续他的科学工作，同时使他能够让一些高中生更深入地欣赏数学及其在现代社会中的重要作用。