Representation Theory of Groups and Applications

群是对称概念的数学抽象。自从150年前开始对群体进行研究以来，它们已经被证明在数学和其他学科（如数学物理、晶体学和密码学）的大量背景中具有根本性的重要性。由于群体结构的多样性，我们对群体还有很多需要学习的地方。该研究项目涉及研究群体的新方法，包括构建以前未知的群体和调查一些已知但高度复杂的群体的详细结构。预计该项目将导致扩展图的新例子，高度连接的稀疏图广泛应用于计算机科学领域，从并行计算到纠错码到密码学。本课题主要研究群的表示理论，重点研究自由群的自同构群的上同调问题。另一个重点是与Kazhdan性质t相关的问题。研究者旨在使用代数、组合、几何和概率工具将研究中的问题简化为组合学和随机游走理论中的问题，并将结果应用于图论中的重要开放问题。本课题的主要目的是研究Cayley图的展开性质，了解前有限群和离散群的表示理论性质。该项目涉及研究几何群论的中心对象，包括自由群的自同构群和映射类群。这位研究者计划开展以扩张器为中心的项目，这是一个在过去十年中经历了爆炸式增长的研究领域。这项工作将涉及算术、群论和组合学之间的互利互动。

Groups are the mathematical abstraction of the notion of symmetry. Since the beginning of the study of groups 150 years ago, they have proved to be of fundamental importance in an extraordinarily large number of contexts in mathematics and other disciplines such as mathematical physics, crystallography, and cryptography. Because of the extraordinary diversity of possible group structures, we still have much to learn about groups. This research project concerns new approaches to studying groups, including the construction of previously unknown groups and investigation of the detailed structure of some well-known but highly complex groups. It is anticipated that project will lead to new examples of expander graphs, highly-connected sparse graphs widely used in computer science in areas ranging from parallel computation to error-correcting codes to cryptography.This research project concerns topics in the representation theory of groups, with special emphasis on problems related to the cohomology of the automorphism group of the free group. Another emphasis is on questions related to the Kazhdan property T. The investigator aims to use algebraic, combinatorial, geometric, and probabilistic tools to reduce the problems under study to questions in combinatorics and theory of random walks and to apply the results to important open problems in graph theory. The main aims of the project are to study the expansion properties of Cayley graphs and to understand representation theoretic properties of pro-finite and discrete groups. The project involves studying objects central to geometric group theory, including automorphism groups of free groups and mapping class groups. The investigator plans to pursue projects centered on expanders, a field of study that has undergone explosive growth in the past decade. The work will involve mutually beneficial interactions among arithmetic, group theory, and combinatorics.