Representation Theory of Groups and Applications

PI计划继续研究群表示论中的各种主题，特别强调与Kazhdan性质T或pro-finite群相关的问题。该项目的主要目的是研究凯莱图的扩展性质，并了解pro-finite和离散群的各种表示论性质。该项目还涉及研究几何群论的核心对象，如自由群的自同构群。PI将使用代数，组合，几何和概率工具，以减少提出的问题，在组合学和随机游走理论的问题，并将结果应用于重要的开放问题的图论。作为该项目的副产品，这些领域的问题可能会得到解答。扩展器是广泛用于计算机科学的高度连接稀疏图，从并行计算到密码学。马古利斯给出了第一个明确的建设扩张，以下平斯克的观察，随机稀疏图的扩张。这种构造在80年代由马古利斯、卢博茨基、菲利普斯和萨尔纳克改进，他们利用深层数论结果构造了拉马努金图（最优扩展器）。在随后的几年里，膨胀器的应用范围和深度都有所增加-在过去的十年里，出现了几种全新的和意想不到的发展路线，该领域经历了爆炸性的增长。首席研究员计划从事以扩展器为中心的项目，并涉及算术，群论和组合学之间的互利互动。

The PI plans to continue working on various topics in the representation theory of groups, with special emphasis on problems related to the Kazhdan property T or pro-finite groups. The main aims of the project are to study the expansion properties of Cayley graphs and to understand various representation theoretic properties of pro-finite and discrete groups. The project also involves studying objects central to the geometric group theory, like automorphism groups of free groups. The PI will use algebraic, combinatorial, geometric and probabilistic tools to reduce the proposed problems to questions in combinatorics and theory of random walks and apply the results to important open problems in graph theory. As a byproduct of the project, questions in these areas are likely be answered.Expanders are highly-connected sparse graphs widely used in Computer Science, in areas ranging from parallel computation to cryptography. Margulis gave the first explicit construction of expanders, following Pinsker's observation that random sparse graphs are expanders. This construction was improved in the eighties by Margulis, Lubotzky, Phillips and Sarnak who constructed Ramanujan graphs (optimal expanders) using deep number-theoretic results. In the ensuing years the scope and depth of applications of expanders has increased -- over the past decade several completely new and unexpected lines of development have emerged and the field has undergone explosive growth. The principal investigator plans to pursue projects centered on expanders and involving mutually beneficial interactions between arithmetic, group theory and combinatorics.