Workshop on Graphs and Arithmetic

数论和组合学之间的相互作用有着悠久的历史。在过去的二十年里，自同构形式和数论的深层结果被用来构造（最优）扩展器，这些扩展器在计算机科学和通信网络中有着广泛的应用。这些技术被推广到构造高维的类似物。最近，图的zeta函数已经扩展到复合体。它们包含组合对象的拓扑信息和谱信息，因此当且仅当组合对象谱极值时，黎曼假设被满足。此外，近年来算术组合学的令人兴奋的发展为构造好的展开式族提供了新的工具，这些展开式族反过来又用于获得深入的数论结果。同时，可拓的概念在群论和计算机科学中被扩展到一个不同的新背景。2010年3月8-12日，一个关于图和算术的研讨会将在加拿大蒙特利尔的CRM举行，回顾最近在扩展器和数论方面令人兴奋的发展。重点将放在组合学、群论和数论之间的相互联系上。理论和应用都将被强调。17位受邀演讲者来自美国。为了支付受邀演讲者的费用和部分支持研究生和最近的博士学位，寻求NSF的支持来补充CRM承诺的资金。

There is a long history of interaction between number theory and combinatorics. In the past two decades, deep results in automorphic forms and number theory were used to construct (optimal) expanders, which are known to have wide applications in computer science and communication networks. These techniques were generalized to construct higher dimensional analogues. Very recently, zeta functions for graphs have been extended to complexes. They contain topological and spectral information of the combinatorial objects so that the Riemann hypothesis is satisfied if and only if the object is spectrally extremal. Furthermore, exciting developments in arithmetic combinatorics in recent years provide new tools to construct families of good expanders, which in turn are used to obtain deep number theoretic results. At the same time, the concept of expansion is extended in group theory and computer science to a different new context.A workshop on graphs and arithmetic will take place March 8-12, 2010, at CRM,Montreal, Canada, to review recent exciting developments in expanders and number theory. The focus will be on the interconnections between combinatorics, group theory and number theory. Both theories and applications will be emphasized. Seventeen invited speakers are from the US. To cover the expenses of the invited speakers and partially support graduate students and recent doctorates, support from the NSF is sought to supplement the fundscommitted by the CRM.