Geometry and groups: Structure and complexity

首席调查员将调查数学中关于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相关。具体地说，这项提议的目标是将复杂性与结构联系起来，并从与空间相关的自然数据中表征/识别高度对称的空间。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，努力促进跨学科的互动。这种相互作用一直是数学乃至更广泛的科学进步的核心。这项研究项目在三个广泛、不同的项目中调查几何、拓扑学和群论之间的相互作用。第一，与群上的决策问题相关的复杂性函数。其主要目的是复杂功能的行为和群体结构之间的相互作用。这个项目的一个具体焦点是将线性表示作为一种工具和结论。这与决策问题的算法复杂性有直接联系，这些问题涉及到可能与数学以外的领域相联系的群体。其次，对Thom关于表示同调类的一些经典工作进行了群论研究。这就引出了光滑流形基本群的可能的同调维度的研究。这部作品的一部分突出地展示了几何学、群论、动力学、分析和拓扑学之间的相互作用。潜在的结果是相当令人兴奋的，这些丰富的联系也是如此。第三，负曲流形上的本原测地线与数域的整数环上的素理想之间的类比。研究的中心是算术级数和原始测地线长度集合中的较弱形式的级数。这个项目有两个主要可能的结果，一个刻画了算术流形，另一个解决了谱刚性问题中的一个古老的猜想。弱的级数概念有可能对测地线几何以外的领域产生影响。这些项目有可能产生影响，超出它们直接涉及的主题。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。

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.