Combinatorics and Number Theory III

题目：组合与数论iii [i]：李文清，宾州州立大学该提案包含两个项目，一个是数论，另一个是数论在组合学中的应用。第一个项目是关于非同余群的模形式及其与同余群形式的联系。对于非同余群形式的算术理解很少。基于他们的数值数据，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形基础来满足。PI，Long和Yang最近的一项联合工作给出了第一个二维例子，建立了非同余群和同余群形式之间的asdd同余关系。为了向前推进，PI计划系统地研究非同余群的尖形结构，探索它们与同余群的形式的关系，并解决真正存在于非同余群上的代数尖形的猜想，即它们的傅里叶系数具有无界分母。第二部分是关于自同构形式与拉马努金超图的相互作用，以及超图与LDPC码之间的联系。拉马努金超图是拉马努金图的高维类似物，具有广泛的应用。PI提出研究拉马努金图中存在的组合学和数论之间的丰富相互作用是否扩展到拉马努金超图中。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关于从附图到附超图刻画LDPC码伪码字的合作工作。做数论的基础研究，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，一直是PI的长期研究目标。对这些领域之间相互作用的研究已经证明是相当富有成效的。PI应用数论的深层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。这一建议是PI努力追求同一总体主题的延续。本文的第一个课题是基础研究，了解非同余子群的模形式算法。第二个项目是探索自同构形式和拉马努金超图之间的算术和联系的相互作用，扩展PI之前在拉马努金图上的成功。一个潜在的应用是LDPC代码。LDPC码高效的编解码算法及其广泛的应用使其成为编码理论研究的热点。第二个项目的主要目标是使用拉马努金超图构建良好的LDPC码，并理解由快速解码算法产生的“错误”解码单词。计划在2007年召开一次会议，传播这项建议的结果。

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.