High Dimensional Expanders and Ramanujan Complexes

在过去的四十年里，扩展图一直是数学和计算机科学研究的焦点。这些稀疏但高度连通的图在构建通信网络、计算机算法和纠错码理论中具有重要的基础意义。展开图的研究一直在从深度数学中提取工具，如数论、表示理论和拓扑学。近年来，展开器的研究也为纯数学提供了新的问题。数学和计算机科学的这种相互作用非常有成效。目前的建议旨在通过启动高维展开子的系统研究，使展开子理论更进一步，高维展开子是具有类似展开子性质的高维简单复合体。高维情况下的问题更加困难，但我们希望发展的理论可以在应用中发挥强大的作用。具体来说，pi计划考虑各种方法来扩展扩展器的定义，例如，通过频谱、上同调或拓扑。特别有趣的是拉马努金复形，它推广了拉马努金图。它们的极值性质（由表示论和数论的方法证明）和它们显著的对称性被期望在解决一些问题上有用。特别重要的是Gromov的拓扑重叠性质。希望拉马努金配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的一些基本问题。

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.