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Interaction of Algebraic, Algorithmic and Asymptotic Properties in Finitely Generated Groups

有限生成群中代数、算法和渐近性质的相互作用

基本信息

批准号：
1901976
负责人：
Mark Sapir
金额：
$ 53.99万
依托单位：
Vanderbilt University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2023-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901976&HistoricalAwards=false
关键词：
Interaction Algebraic Algorithmic Asymptotic Properties

项目摘要

Modern group theory is a very fast developing area of mathematics that uses methods from algebra, geometry, combinatorics, topology, probability, logic, and computer science. Applications of group theory are ubiquitous throughout science, from physics and chemistry to cyber security. This award supports research in the area of asymptotic methods in group theory, an area that has been extremely active since seminal papers by Gromov on hyperbolic groups and asymptotic invariants of groups. This project will significantly advance knowledge of the area by solving several open problems. The broader impact of the project will affect students at all levels, and professional investigators. The principal investigators will continue running Nashville Math Club for K-12 students, engaging undergraduate students in research projects, and advising graduate students. In this research the principal investigators will work on several fundamental problems together with ideas of their solutions related to Burnside, asymptotic and algorithmic problems of finitely generated groups. The problems are intrinsically connected by the proposed methods of solutions and by the tools used in the proposed approaches in their solutions. The project for a definition of a proper analogue of asymptotic cones for Golod-Shafarevich groups has a potential impact far beyond group theory. Other problems concern the existence of an infinite finitely presented torsion group for which a detailed approach for a positive solution is provided, the existence of amenable groups with non-trivial Morse boundary, and the construction of a finitely presented group with maximal possible (continuum) different asymptotic cones (assuming the Continuum Hypothesis).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

现代群论是一个发展非常迅速的数学领域，它使用了代数、几何、组合学、拓扑学、概率论、逻辑学和计算机科学的方法。从物理和化学到网络安全，群论的应用在整个科学中无处不在。该奖项支持群论中渐近方法领域的研究，这一领域自格罗莫夫关于双曲群和群的渐近不变量的开创性论文以来一直非常活跃。该项目将通过解决几个悬而未决的问题，极大地促进对该地区的了解。该项目的更广泛影响将影响到各级学生和专业调查人员。主要调查人员将继续为K-12学生开办纳什维尔数学俱乐部，吸引本科生参与研究项目，并为研究生提供建议。在这项研究中，主要研究人员将致力于与Burnside、有限生成群的渐近和算法问题有关的几个基本问题及其解决方案的想法。这些问题通过拟议的解决方法及其解决办法中所使用的工具内在地联系在一起。定义Golod-Shafarevich群的渐近圆锥的适当类比的项目具有远远超出群论的潜在影响。其他问题涉及无限有限表示扭群的存在，其中提供了一个正解的详细方法，具有非平凡Morse边界的从属群的存在，以及具有最大可能(连续)不同渐近圆锥的有限表示群的构造(假设连续统假设)。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。