PECASE: Galois Representations and Modular Forms

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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.