Representation Theory of Groups and Applications

群表示论及其应用

• 批准号: 1601406
• 负责人: Martin Kassabov
• 金额: $ 22.93万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601406&HistoricalAwards=false
• 关键词: Representation; Theory; Groups; Applications

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.

组是对称概念的数学抽象。自从150年前的小组研究开始以来，在数学和其他学科（例如数学物理学，晶体学和密码学）中，它们在数量非常大的情况下被证明至关重要。由于可能的群体结构的非凡多样性，我们仍然有很多了解小组的知识。该研究项目涉及研究小组的新方法，包括构建以前未知的群体以及对一些知名但高度复杂的群体的详细结构进行调查。可以预料，项目将导致扩展器图的新示例，这是在计算机科学中广泛使用的稀疏图，范围从平行计算到错误纠正码到密码学，这一研究项目涉及组的主题，该群体的主题，群体的主题，群体的主题，特别强调了与自由组的自动形态群体的共同体相关的问题。另一个重点是与Kazhdan财产有关的问题。研究人员的目的是使用代数，组合，几何和概率工具，以将所研究的问题和随机步行理论的问题减少到随机步行理论中，并将结果应用于重要的开放问题在图理论中。该项目的主要目的是研究Cayley图的扩展特性，并了解亲限和离散基团的代表理论特性。该项目涉及研究几何群体理论中心的对象，包括自由组和映射课程组的自动形态群体。调查人员计划追求以扩展器为中心的项目，这是一个研究领域，在过去的十年中经历了爆炸性的增长。 这项工作将涉及算术，群体理论和组合学之间互惠互利的相互作用。