High Dimensional Expanders and Ramanujan Complexes

高维扩展器和拉马努金复合体

• 批准号: 1404257
• 负责人: Alexander Lubotzky
• 金额: $ 18万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1404257&HistoricalAwards=false
• 关键词: Dimensional; Expanders; Ramanujan; Complexes

Expander graphs have been a focus of study in mathematics and computer science during the last four decades. These sparse and yet highly connected graphs are of fundamental importance in building communication networks, in computer algorithms, and in the theory of error-correcting codes. The study of expander graphs has been drawing tools from deep mathematics, such as number theory, representation theory, and topology. In recent years the study of expanders also provided new problems for pure mathematicians. This interplay of mathematics and computer science has been very productive. The current proposal aims at taking the theory of expanders one step further by initiating a systematic study of high-dimensional expanders, which are simplical complexes of high dimensions having similar properties of expanders. The problems in the high-dimensional case are more difficult, but the theory we hope to develop can be expected to be powerful in applications. Specifically, the PIs plan to consider various ways to extend the definition of expanders, e.g., via spectrum, cohomology or topology. Of particular interest are the Ramanujan complexes, which generalize the Ramanujan graphs. Their extremal properties (proved by methods of representation theory and number theory) together with their remarkable symmetry are expected to be useful in solving several problems. Of special importance is Gromov's topological overlapping property. It is hoped that the extremal properties of Ramanujan complexes will help resolve some problems of basic importance in the theory of error-correcting codes, as well as in other areas of computer science.

在过去的四十年里，扩张图一直是数学和计算机科学研究的焦点。这些稀疏但高度连通的图在构建通信网络、计算机算法和纠错码理论中具有根本的重要性。扩展图的研究一直是从深层数学，如数论，表示论和拓扑学中提取工具。近年来展开式的研究也给纯数学工作者提出了新的问题。数学和计算机科学的这种相互作用是非常富有成效的。目前的建议旨在采取进一步的理论展开的系统研究的高维扩张，这是简单的复杂的高维具有类似的性质的扩张。在高维情况下的问题是更困难的，但我们希望发展的理论可以预期是强大的应用。具体来说，PI计划考虑各种方法来扩展扩展器的定义，例如，通过谱、上同调或拓扑。特别感兴趣的是拉马努金复形，它推广了拉马努金图。他们的极值性质（证明的方法表示论和数论）连同其显着的对称性，预计将是有用的，在解决一些问题。特别重要的是格罗莫夫的拓扑重叠性质。人们希望拉马努金复合物的极值性质将有助于解决纠错码理论以及计算机科学其他领域中的一些基本重要问题。