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Collaborative Research: FRG: Class numbers, Hyperbolic Manifolds and Dynamics

合作研究：FRG：类数、双曲流形和动力学

基本信息

批准号：
0139816
负责人：
Ted Chinburg
金额：
$ 15.69万
依托单位：
University of Pennsylvania
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2002
资助国家：
美国
起止时间：
2002-07-01 至 2006-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139816&HistoricalAwards=false
关键词：
Collaborative Research FRG Class numbers

项目摘要

DMS-0139816Ted ChinburgThe problem of constructing infinitely many number fields of classnumber one descends from Gauss's work on quadratic forms, and remainsone of the major unsolved problems in algebraic number theory. Thishas been the inspiration for many new ideas and techniques inalgebraic number theory and arithmetic geometry. The proposers plan asustained attack on this problem using geometric methods based uponrecent advances in the theory of hyperbolic manifolds coupled with anatural broadening of the classical Bianchi- Hurwitz theorem.Possible important mathematical by-products include new techniquesfor estimating class numbers and an improvement of theunderstanding of properties of the trace fields of hyperbolicmanifolds.Further consequences of this work of a somewhat broader impact alsoseem possible. The most obviously relevant applications seem tocome from the direction of cryptography. The difficulty of factoringnumbers into primes is the basis for the RSA cryptosystem. The fastestknown algorithm for factoring large whole numbers into primes is theso-called number field sieve, and the techniques from algebraic numbertheory involved in this sieve were developed in part from work on classnumbers. This project could impact such algorithms and progress onfactoring problems would have significant implications for cryptography.This award is jointly funded by the programs in Topology and Algebra,Number Theory, & Combinatorics.

DMS-0139816Ted Chinburg 构造第一类无穷多个数域的问题源自高斯关于二次形式的工作，并且仍然是代数数论中未解决的主要问题之一。这是代数数论和算术几何中许多新思想和技术的灵感来源。提议者计划使用基于双曲流形理论的最新进展以及经典 Bianchi-Hurwitz 定理的自然扩展的几何方法来持续解决这个问题。可能的重要数学副产品包括用于估计类数的新技术以及对双曲流形迹域属性的理解的改进。这项工作的进一步后果似乎也有可能产生更广泛的影响。最明显相关的应用似乎来自密码学方向。将数字分解为素数的困难是 RSA 密码系统的基础。将大整数分解为素数的已知最快算法是所谓的数域筛，该筛涉及的代数数论技术部分是从类数的工作中开发出来的。该项目可能会影响此类算法，并且因式分解问题的进展将对密码学产生重大影响。该奖项由拓扑与代数、数论和组合学项目共同资助。