Deformation Spaces of Geometric Structures

几何流形是一个抽象的数学对象，用来模拟我们生活的空间。这个概念还包括相对论中的时空概念，以及相空间、构型空间和物理学中其他有用的结构。虽然经验物理学致力于测量我们宇宙的精确特征（例如形状，大小等），但宇宙可能具有哪些特征的问题，受制于某些定律，是一个数学问题。构成这样一个问题的答案的数据称为模空间；它是一个拓扑空间，其点是某种类型的可能几何流形，其拓扑将这些几何流形组织成特征连续变化的族。这个项目解决了许多不同种类的低维几何流形及其模的重要问题。主要焦点是研究在真实射影几何和仿射几何上建模的流形，这两种经典几何具有恰到好处的对称性，允许具有神秘但可处理行为的大有趣模空间。这个项目将数学和物理的交叉领域的思想编织在一起，包括几何、拓扑、群论、动力学和相对论。最近引入的实射影一般线性群在射影空间上的离散子群的凸紧性的概念，扩展了经典凸紧Kleinian群和其他例子的定义，但它足以描述有限生成的离散群空间中的大开放区域，这些区域是其他方法没有探索过的。这个概念与labourrie的Anosov子群的概念之间存在联系，这里研究的凸紧子群涉及许多有趣的非双曲离散子群的变形空间。另一个研究方向是将马古利时空作为反德西特时空的几何极限来研究，从而对几何、拓扑和变形理论产生影响。在三维空间中双曲几何和反德西特几何之间的联系特别有前途。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。

A geometric manifold is an abstract mathematical object designed to model the space we live in. The concept also encompasses the notion of a space-time in the theory of relativity, as well as phase spaces, configuration spaces, and other useful structures in physics. While empirical physics is dedicated to measuring the precise features (e.g. shape, size, etc.) of our universe, the question of what possible features could the universe have, subject to certain laws, is a mathematical one. The data comprising the answer to such a question is called a moduli space; it is a topological space whose points are the possible geometric manifolds of a certain type and whose topology organizes those geometric manifolds into families whose features vary continuously. This project addresses important questions about many different classes of low-dimensional geometric manifolds and their moduli. The main focus is the study of manifolds modeled on real projective geometry and affine geometry, two classical geometries with just the right amount of symmetry to allow for large interesting moduli spaces with mysterious but tractable behavior.This project weaves together ideas from a cross-section of mathematics and physics, including geometry, topology, group theory, dynamics, and relativity. A recently introduced notion of convex cocompactness for discrete subgroups of the real projective general linear group acting on projective space extends the definition of classical convex cocompact Kleinian groups and other examples, but is general enough to describe large open regions in the space of finitely generated discrete groups that were left unexplored by other methods. A connection has been identified between this notion and Labourie's notion of Anosov subgroup, and the convex cocompact subgroups studied here involve many interesting deformation space for non-hyperbolic discrete subgroups. Another line of work studies Margulis spacetimes as geometric limits of anti de Sitter spacetimes, with consequences for geometry, topology, and deformation theory. Connections between hyperbolic and anti de Sitter geometry in dimension three are particularly promising.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.