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.

在过去的四十年中，扩展图一直是数学和计算机科学研究的焦点。这些稀疏但高度连接的图对于构建通信网络、计算机算法和纠错码理论至关重要。扩展图的研究一直在从深奥数学中汲取工具，例如数论、表示论和拓扑学。近年来对展开子的研究也为纯数学家提供了新的问题。数学和计算机科学的相互作用非常富有成效。目前的提案旨在通过启动对高维扩展器的系统研究，使扩展器理论更进一步，高维扩展器是具有与扩展器类似特性的高维单纯复形。高维情况下的问题更加困难，但我们希望发展的理论有望在应用中发挥强大作用。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。特别令人感兴趣的是拉马努金复合体，它概括了拉马努金图。它们的极值性质（通过表示论和数论的方法证明）及其显着的对称性预计将有助于解决一些问题。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体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。