权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Spaces of geometric structures via geometric transitions

通过几何过渡的几何结构空间

基本信息

批准号：
1510254
负责人：
Jeffrey Danciger
金额：
$ 16.91万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510254&HistoricalAwards=false
关键词：
Spaces geometric structures via transitions

项目摘要

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.

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。包含此类问题答案的数据称为模空间;它是一个拓扑空间，其点都是某种类型的可能几何流形，并且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。PI将通过进一步开发和应用一种新兴的新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。在这个项目中进行的研究编织在一起的想法从广泛的横截面的数学和物理学，包括几何，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，将有助于科学和工程领域许多创新所建立的基础知识的不断增长。PI和合作者最近的进展，在称为Margulis时空的完整仿射洛伦兹三维流形的设置中，提出了一个解决高维仿射几何难题的方法。通过进一步发展和推广用于研究Margulis时空作为弯曲时空极限的几何过渡技术，PI将解决围绕Auslander猜想的高维仿射几何中的重要问题。第二个主题是研究双曲和反德西特（AdS）几何在三维空间中的关系。PI在双曲线AdS过渡上的工作建立了两种几何形状之间明确而自然的联系，有可能加快两种设置中几何结构研究的进展。特别是，最近的联合工作提供了新的证据弯曲措施猜想凸AdS时空，对应瑟斯顿的猜想特征凸双曲三流形弯曲数据的凸核心。PI将通过进一步检查两种设置中凸结构之间的相互作用来解决这些问题。此外，PI将开始一个新的项目，以开发几何转换的应用程序，在设置的表示理论的双曲群到更高的秩李群，一个日益增长的主题被称为高等泰希米勒理论。