Combinatorics and Number Theory III

组合数学与数论 III

• 批准号: 0457574
• 负责人: Wen-Ching Li
• 金额: $ 10.5万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-15 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0457574&HistoricalAwards=false
• 关键词: Combinatorics; Number; Theory; III

Title: Combinatorics and Number Theory IIIPI: Wen-Ching Winnie Li, Penn State UniversityAbstract.This proposal contains two projects, in number theory and itsapplications to combinatorics. The first project concernsmodular forms for noncongruence groups and their connections to formsfor congruence groups. The arithmetic of the forms for noncongruence groups is little understood. Based on their numerical data, Atkin and Swinnerton-Dyer (ASD) suggest very interesting congruence relations to be satisfied by a basis of cusp forms for a noncongruence group. A recent joint work by the PI,Long and Yang gives the first two-dimensional example establishing the ASDcongruence relations between forms for noncongruence and congruence groups. To move forward, the PI plans to systematically study the structure of the cusp forms for noncongruence groups, to explore their relationshipwith forms for congruence groups, and to tackle the conjecture that algebraic cusp forms genuinely living on noncongruence groups are distinguishedby their Fourier coefficients having unbounded denominators. The second is on the interplay between automorphic forms and Ramanujan hypergraphs, and connections between hypergraphs and LDPC codes. Ramanujan hypergraphs are higher dimensional analogue of Ramanujan graphs, which are known to have broad applications. The PI proposes to study whether the rich interplay between combinatorics and number theory, which exists for Ramanujan graphs, extends to Ramanujan hypergraphs. On the applied side, the PI plans to construct good LDPC codes using Ramanujan hypergraphs, and to generalize her joint work with Koetter, Vontobel, and Walker on characterizing pseudo-codewords for LDPC codes from attached to graphs to attached to hypergraphs.It has been the PI's long term research goal to do fundamental research in number theory and to seek applications of number theory to graph theory and coding theory, especially to solve real world problems. The study of interplay between these areas has turned out to be quite fruitful. The PI has applied deep results in number theory to construct efficient communication networks; and conversely, investigations in graph theory inspired very interesting and unexpected results in number theory. This proposal is a continuation of the PI's effort to pursue the same general theme. The firstproject of the proposal lies in basic research, to understand the arithmeticsof modular forms for noncongruence subgroups. The second project is to explore the interplay in arithmetic and connections between automorphic forms and Ramanujan hypergraphs, extending PI's previous success on Ramanujan graphs. A potential application is to LDPC codes. The very efficient encoding and decoding algorithms for LDPC codes together with their extensive applications make them very hot research topic in coding theory. A primary goal of the second project is to construct good LDPC codes using Ramanujan hypergraphs, and to understand the "wrongly" decoded words arising from the rapid decoding algorithm. A conference is planned in 2007 to disseminate the results of this proposal.

标题：组合学和数论 IIIPI：Wen-Ching Winnie Li，宾夕法尼亚州立大学摘要。本提案包含两个项目，涉及数论及其在组合学中的应用。第一个项目涉及非同余群的模块化形式及其与同余群形式的联系。非同余群的形式的算术知之甚少。根据他们的数值数据，Atkin 和 Swinnerton-Dyer (ASD) 提出了非常有趣的同余关系，可以通过非同余群的尖点形式的基础来满足。 PI、Long 和 Yang 最近的一项联合工作给出了第一个二维示例，在非同余群和同余群的形式之间建立了 ASD 同余关系。为了进一步推进，PI计划系统地研究非同余群尖点形式的结构，探索它们与同余群形式的关系，并解决真正存在于非同余群上的代数尖点形式的特点是其傅立叶系数具有无界分母。第二个是自同构形式和拉马努扬超图之间的相互作用，以及超图和 LDPC 码之间的联系。 拉马努金超图是拉马努金图的高维模拟，众所周知，拉马努金图具有广泛的应用。该 PI 提议研究拉马努金图中存在的组合学和数论之间丰富的相互作用是否扩展到拉马努金超图。在应用方面，PI 计划使用 Ramanujan 超图构造良好的 LDPC 码，并推广她与 Koetter、Vontobel 和 Walker 的联合工作，将 LDPC 码的伪码字特征从附加到图到附加到超图进行推广。 PI的长期研究目标是进行数论基础研究，并寻求数论在图论和编码理论中的应用，特别是解决现实世界的问题。对这些领域之间相互作用的研究已证明是非常富有成果的。 PI应用了数论的深入成果来构建高效的通信网络；相反，图论的研究激发了数论中非常有趣和意想不到的结果。该提案是 PI 努力追求同一主题的延续。该提案的第一个项目在于基础研究，以了解非同余子群的模形式的算术。第二个项目是探索算术中的相互作用以及自同构形式和拉马努扬超图之间的联系，扩展 PI 之前在拉马努扬图上的成功。 LDPC 码是一个潜在的应用。 LDPC码非常高效的编解码算法及其广泛的应用使其成为编码理论中的热门研究课题。第二个项目的主要目标是使用拉马努金超图构造良好的 LDPC 码，并理解快速解码算法产生的“错误”解码字。计划于 2007 年召开一次会议来传播该提案的结果。