Geometry and groups: Structure and complexity

几何和群：结构和复杂性

• 批准号: 1408458
• 负责人: David McReynolds
• 金额: $ 16.94万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1408458&HistoricalAwards=false
• 关键词: Geometry; groups; Structure; complexity

The Principal Investigator will investigate different notions of complexity and structure in mathematics. These topics are relevant to several active mathematical research areas. Specifically, the goals of this proposal are connecting complexity with structure and the characterization/recognition of highly symmetric spaces from from natural data associated to the spaces. Aside from the direct appeal of these questions, this research hopes to establish new connections between areas in mathematics in an effort to foster cross-discipline interactions. Such interactions have been central to the progress of mathematics and more broadly science. This research project investigates the interplay between geometry, topology, and group theory in three broad, distinct projects. First, complexity functions associated to decision problems on groups. The main purpose is the interplay between the behavior of the complexity functions and the structure of the group. One specific focus for this project is on linear representations as both a tool and a conclusion. There are direct ties to the algorithmic complexity for decision problems on groups that has the potential to connect to areas beyond mathematics. Second, a group theoretic take on some classical work of Thom on representing homology classes. This leads to the study of the possible homological dimensions of fundamental groups of smooth manifolds. Part of this work displays prominently the interaction between geometry, group theory, dynamics, analysis, and topology. The potential results are quite exciting as are these rich connections. Third, the analogy between primitive geodesics on a negatively curved manifold and prime ideals in the ring of integers of a number field. The investigation centers around arithmetic progressions and a weaker version of progressions in the set of primitive geodesic lengths. This project has two main possible results, one that characterizes arithmetic manifolds, another that resolves an old conjecture in spectral rigidity problems. The weak notion of progressions has the potential to have impact in areas beyond geodesic geometry. These projects have the potential to have impact beyond the subjects they directly address. Indeed, part of the motivation for this work is the production of rich connections between distinct mathematical fields.

首席研究员将研究数学中复杂性和结构的不同概念。这些主题与几个活跃的数学研究领域有关。具体来说，这个建议的目标是连接复杂性与结构和表征/识别的高度对称的空间从自然数据相关联的空间。除了这些问题的直接吸引力之外，这项研究还希望在数学领域之间建立新的联系，以促进跨学科的互动。这种相互作用对数学和更广泛的科学的进步至关重要。本研究计画探讨几何学、拓扑学与群论在三个广泛而不同的计画中的相互影响。第一，与群决策问题相关的复杂性函数。主要目的是复杂性函数的行为和群的结构之间的相互作用。这个项目的一个具体重点是线性表示作为一种工具和结论。群体决策问题的算法复杂性与数学之外的领域有着直接的联系。第二，一组理论采取了一些经典的工作，托姆代表同调类。这导致了光滑流形的基本群的可能的同调维数的研究。这项工作的一部分突出显示几何之间的相互作用，群论，动力学，分析和拓扑结构。潜在的结果是相当令人兴奋的，因为这些丰富的连接。第三，负曲流形上的本原测地线与数域整数环中的素理想之间的类比。调查中心周围的算术级数和一个较弱版本的原始测地线长度的集合中的级数。这个项目有两个主要可能的结果，一个是算术流形的特征，另一个解决了谱刚性问题中的一个老猜想。弱概念的进展有可能在测地线几何以外的领域产生影响。这些项目有可能产生超出其直接处理的主题的影响。事实上，这项工作的部分动机是在不同的数学领域之间产生丰富的联系。