Symmetry and Self-Similar Structures in Geometry and Topology

对称在数学中是一个普遍的概念，它产生了美丽而有趣的几何对象，同时也为几何和拓扑学提供了重要的工具。传统上，研究集中在全局定义的对称性上，但最近人们意识到局部结构的对称性提供了更多的例子和与其他数学领域的联系。本项目将研究局部对称性，其在研究自相似几何对象和相关代数问题中的应用，以及最近发现的有效网络的几何结构（所谓的扩展图）。一个物体的几何形状和它的自同构群的代数结构之间的关系在过去已经得到了深入的研究，但在过去的十年里，在一个更丰富的对称性理论上取得了进展，这个理论只有在经过一个掩护后才会出现，即“隐藏”。该项目旨在研究这个概念在几种情况下，如黎曼几何（特别是非正曲率）和几何结构。此外，当对称性源于自相似时，即在自覆盖流形的情况下，特别是在几何兴趣的情况下，如Kaehler或仿射几何，预计会出现丰富的现象。最后，该项目旨在研究与群行为（所谓的弯曲锥）相关的粗糙几何对象，特别是在具有光谱间隙的行为的情况下，这些行为产生扩展图。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。

Symmetry is a pervasive concept in mathematics, giving rise to beautiful and interesting geometric objects, as well as providing an essential tool in geometry and topology. Classically, research has focused on globally defined symmetries, but recently it has been realized that symmetries of local structures offer vastly more examples and connections to other areas of mathematics. This project will study local symmetry, its applications to the study of self-similar geometric objects and related algebraic problems, and recently discovered geometric constructions of efficient networks (so-called expander graphs).The relationship between the geometry of an object and the algebraic structure of its group of automorphisms has been intensely studied in the past, but in the last decade, progress has been made on a much richer theory of symmetries that only appear after passing to a cover, i.e. are "hidden'" The project aims to study this notion in several contexts, such as Riemannian geometry (especially nonpositive curvature) and geometric structures. Further, rich phenomena are expected when the symmetry arises from self-similarity, i.e. in the case of self-covering manifolds, particularly in situations of geometric interest such as Kaehler or affine geometry. Finally, the project aims to study coarse geometric objects associated to group actions (so-called warped cones), especially in the case of actions with spectral gap, where these yield expander graphs.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.