Group Actions on Manifolds and Related Spaces: Regularity, Structure, and Complexity

流形是数学中的基本对象，它概括和系统化了我们所感知的空间，因此长期以来一直是数学家们非常感兴趣的对象。研究流形的一个卓有成效的方法是通过它们的对称性，这些对称性结合在一起形成一个称为群的代数对象。这个项目的重点是流形的结构之间的相互作用，既有局部的(近距离的)，也有全局的(作为系综)，以及它的对称性，或流形上的群作用。该项目旨在解决这一领域的一些基本问题，并调查一些新的现象。在特别感兴趣的问题中，有一些涉及到微积分意义上的群体行动的平稳性。其中一些问题的解决有可能洞察关于奇异球体的古老的悬而未决的问题，奇异球体是某些流形，可以变形成标准球体，但只能以一种非常奇异的方式。作用于流形上的群是一类受到广泛关注的对象，其理论近年来得到了迅速的发展。计划的更广泛的影响包括本科生的研究经验，会议组织，以及一本关于直角Artin群体的书。这个项目将在几个方面为这些进步做出贡献。该项目的一个重点领域是集体行动的关键规律性。为了计算一维直角Artin群的临界正则性，将对Kim以前的工作进行扩展，这将是第一类自然的非幂零群，其精确的一维临界正则性是有限的和已知的。此外，主要研究者打算在高维上发展临界正则性，目的是用一组有限生成的群来编码紧致流形的微分同胚型。这样的结果将为4维光滑庞加莱猜想提供一种代数方法。该项目的另一个重点是发展半单李群的薄子群的结构理论，目的是解决Shalom关于它们的公约子的离散性的猜想。该项目的最后一个主要关注点是在映射曲面类别组方面，PI概述了一个程序来证明映射类别组的非线性性，并通过研究曲面曲线图的模型理论将逻辑引入低维拓扑。该奖项反映了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。

Manifolds are fundamental objects in mathematics which generalize and systematize space as we perceive it, and as such have been of great interests to mathematicians for a long time. One fruitful way in which manifolds can be studied is through their symmetries, which taken together form an algebraic object called a group. The focus of this project is the interaction between the structure of a manifold, both local (up close) and global (as an ensemble), and its symmetries, or group actions on manifolds for short. The project aims to resolve some basic questions in this area and investigate some new phenomena. Among the problems of particular interest are those concerning smoothness of group actions, in the sense of calculus. A resolution of some of these problems has the potential to yield insight into old open questions about exotic spheres, which are certain manifolds which can be deformed into standard spheres, but only in a highly singular way. Groups acting on manifolds are a broad class of objects which have received a large amount of attention, and their theory has developed rapidly in recent years. Planned broader impacts include research experiences for undergraduates, conference organization, and a book on right-angled Artin groups.This project will contribute to these advances in several ways. One area of focus for the project is critical regularity of group actions. Previous work with Kim will be extended in order to compute the critical regularity in one dimension of right-angled Artin groups, which would then be the first natural class of non-nilpotent groups whose precise one-dimensional critical regularity is both finite and known. Moreover, the principal investigator intends to develop critical regularity in higher dimensions, with the goal of encoding the diffeomorphism type of a compact manifold by a set of finitely generated groups. Such a result would give an algebraic approach to the 4-dimensional smooth Poincare conjecture. Another focus of the project is to develop a structure theory for thin subgroups of semisimple Lie groups, with the goal of resolving a conjecture of Shalom about the discreteness of their commensurators. The final main focus of the project is in the area of mapping class groups of surfaces, where the PI has outlined a program to prove the nonlinearity of mapping class groups, and to introduce logic into low dimensional topology by investigating the model theory of the curve graph of a surface.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.