PECASE: Galois Representations and Modular Forms

研究者之前在椭圆曲线和伽罗瓦表示法上的工作导致了几个问题的方向，这些问题最终涉及理解伽罗瓦表示法的本质，无论是从几何的角度还是从变形理论的角度。其中一个问题是找到Coleman-Mazur特征曲线的概念模理论解释。研究者还建议研究在伽罗瓦表示的变形理论中纳入半稳定性（在方丹的意义上）等条件的问题，继续从怀尔斯的工作中发展出来的发展路线。在一个稍微不同的方向上，Buzzard最近在许多例子中观察到，特征型的斜率似乎比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的框架，研究者建议确定这种现象的一般性质。除了研究这些问题外，研究者还继续努力支持高中学生对数学的积极兴趣。通过在当地一所学校的个人联系，他定期安排会议，与学生们就各种有趣的数学思想（来自各种学科：数论、几何、概率论等）进行非正式的小组讨论。这样做的目的是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，但可以在基本的上下文中呈现。研究者还为这些学生提供有关夏季数学课程和研究机会的信息，以便他们能够体验数学作为科学探究的一个活生生的领域。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商务就不可能实现。RSA密码系统和椭圆曲线分解算法就是这方面的两个突出应用。我们对椭圆曲线的理论认识的改进有望导致沿着这些路线的进一步应用。研究者的科学工作涉及椭圆曲线理论中自然产生的几个问题，部分目的是继续发展用于最近解决椭圆曲线理论中最重要问题之一的Shimura-Taniyama猜想的技术。研究者还定期拜访当地的高中生，向他们展示在学校里不常遇到的重要数学概念，比如RSA密码系统的内部工作原理，以及概率在罕见疾病医学测试设计中的作用。研究者还为这些学生提供有关夏季数学教育、研究和工作机会的信息，并为希望参加几个著名的高中科学研究竞赛的学生提供指导。教师职业发展计划使研究者能够继续他的科学工作，同时使他能够让一些高中生更深入地了解数学及其在现代社会中的重要作用。

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.