Deformation Spaces of Geometric Structures

几何流形是一种抽象的数学对象，旨在模拟我们生活的空间。这个概念也包括相对论中的时空概念，以及相空间，配置空间和其他物理学中有用的结构。而经验物理学致力于测量精确的特征（例如形状，大小等）。在我们的宇宙中，宇宙在服从某些定律的情况下可能具有什么样的特征，这是一个数学问题。包含这个问题的答案的数据被称为模空间;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类别的低维几何流形及其模的重要问题。主要的重点是研究流形上模仿真实的射影几何和仿射几何，两个经典的几何与恰到好处的对称性，以允许大有趣的模空间与神秘的，但易于处理的行为。这个项目编织在一起的想法，从一个横截面的数学和物理，包括几何，拓扑，群论，动力学和相对论。最近引入的作用在射影空间上的真实的射影一般线性群的离散子群的凸余紧性的概念扩展了经典凸余紧Kleinian群和其他例子的定义，但足够一般地描述了在其他方法未探索的离散群空间中的大的开放区域。 这一概念与Labourie的Anosov子群概念之间的联系已经被确定，这里研究的凸余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斯时空作为反德西特时空的几何极限，并对几何学、拓扑学和形变理论产生影响。 三维空间中双曲线和反德西特几何之间的联系特别有前途。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.