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

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等方法。从物理、化学到网络安全，群论在科学领域的应用无处不在。该奖项支持群论中渐近方法领域的研究，自Gromov关于双曲群和群的渐近不变量的开创性论文以来，该领域一直非常活跃。该项目将通过解决几个尚未解决的问题，显著推进对该领域的认识。该项目的广泛影响将影响各个层次的学生和专业研究人员。主要研究人员将继续为K-12学生运营纳什维尔数学俱乐部，让本科生参与研究项目，并为研究生提供建议。在这项研究中，主要研究人员将研究几个基本问题，以及与有限生成群的Burnside，渐近和算法问题相关的解决方案。这些问题通过所提出的解决方法和所提出的方法在其解决方案中使用的工具内在地联系在一起。关于gold - shafarevich群的渐近锥的适当类似的定义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呈现扭转群的存在性，并提供了其正解的详细方法，具有非平凡莫尔斯边界的可服从群的存在性，以及具有最大可能（连续统）不同渐近锥的有限呈现群的构造（假设连续统假设）。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.