Spaces of geometric structures via geometric transitions

几何流形是一种抽象的数学对象，旨在为我们生活的空间建模。这个概念还包括相对论中的时空概念，以及相空间、构型空间和其他物理上有用的结构。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，根据某些定律，宇宙可能具有哪些可能的特征，这是一个数学问题。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，它的点都是某种类型的可能的几何流形，它的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型的低维几何流形及其模的重要问题。PI将通过进一步开发和应用一种新的数学框架来研究这些问题，该框架通过一种称为几何过渡的机制来描述不同类型几何流形的模空间之间的相互作用。这个项目中将要进行的研究将来自数学和物理的广泛领域的想法编织在一起，包括几何学、拓扑学、群论、动力学和相对论。这些问题的进展将对这些学科的许多研究人员具有重要意义，更广泛地说，将有助于不断增长的基础知识，许多科学和工程创新都建立在这些基础上。PI和合作者最近在完全仿射洛伦兹三维流形(称为Marguis时空)的背景下取得的进展表明，在高维仿射几何中有一个解决难题的准则。通过进一步发展和推广用于研究Marguis时空的几何转换技术作为弯曲时空的极限，PI将解决围绕Auslander猜想的高维仿射几何中的重要问题。第二个主题是研究三维空间中双曲与反de Sitter(ADS)几何之间的关系。PI关于双曲线-ADS过渡的工作在两个几何之间建立了明确和自然的联系，有可能加快在两个背景下的几何结构研究的进展。特别是，最近的联合工作为凸ADS时空的弯曲度量猜想提供了新的证据，这与瑟斯顿的猜想相对应，该猜想根据凸核上的弯曲数据刻画了凸双曲三维流形。PI将通过进一步研究两种环境中凸起结构之间的相互作用来解决这些猜想。此外，PI将开始一个新的项目，开发几何转换在双曲群到高阶李群的表示理论中的应用，这是一个日益增长的学科，被称为高阶Teichmüler理论。

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 all 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 PI will study these questions by further developing and applying an emerging new mathematical framework which describes interaction between moduli spaces of different types of geometric manifolds through a mechanism called geometric transition. The research to be conducted in this project weaves together ideas from a wide cross-section of mathematics and physics, including geometry, topology, group theory, dynamics, and relativity. Progress on these problems will be of significance to many researchers across these disciplines and, more broadly, will contribute to the growing base of foundational knowledge on which many innovations in science and engineering are built.Recent progress by the PI and collaborators, in the setting of complete affine Lorentzian three-manifolds called Margulis spacetimes, suggests a rubric to approach difficult questions in higher dimensional affine geometry. By further developing and generalizing the geometric transition technology used to study Margulis spacetimes as limits of curved spacetimes, the PI will address important questions in higher dimensional affine geometry surrounding the Auslander Conjecture. A second main theme is the study of the relationship between hyperbolic and anti de Sitter (AdS) geometry in dimension three. Work of the PI on the hyperbolic-AdS transition establishes an explicit and natural connection between the two geometries with the potential to expedite progress in the study of geometric structures in both settings. In particular, recent joint work gives new evidence for the bending measure conjecture for convex AdS spacetimes, a counterpart to Thurston's conjecture characterizing convex hyperbolic three-manifolds in terms of bending data on the convex core. The PI will work toward the resolution of these conjectures by further examining the interaction between convex structures in both settings. Additionally, the PI will begin a new project to develop applications of geometric transitions in the setting of representation theory of hyperbolic groups into higher rank Lie groups, a growing subject known as higher Teichmüller theory.