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CAREER: Locally Homogeneous Geometric Manifolds and Their Moduli Spaces

职业：局部齐次几何流形及其模空间

基本信息

批准号：
1945493
负责人：
Jeffrey Danciger
金额：
$ 40.01万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2026-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945493&HistoricalAwards=false
关键词：
CAREER Locally Homogeneous Geometric Manifolds

项目摘要

This project concerns locally homogeneous geometric manifolds. These are abstract mathematical objects designed to model the universe we live in. The term locally homogeneous refers to the presence of a high degree of local symmetry which is captured by a Lie group called the (local) symmetry group. It is this symmetry group which governs the geometry, in the following sense: the meaningful quantities we can measure in a geometric manifold (e.g. lengths or angles) are exactly those which are invariant under the symmetry group. There are many different possible symmetry groups which lead to different types of geometric manifolds useful in many contexts across mathematics and physics. There can also be many different geometric manifolds with the same local symmetry group. These all have the same local properties, but can look very different at large scales. The space of all such possibilities 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. While the precise features (e.g. shape, size, etc.) of our universe is a question for empirical physics, a moduli space is the mathematical answer to the question of what possible features could the universe have. The research component of this project studies geometric manifolds of many types, with focus on flat affine geometry, real projective geometry, and constant curvature pseudo-Riemannian geometry. These geometries have just the right amount of symmetry to allow for large interesting moduli spaces with mysterious but tractable behavior. Core elements of the educational component include training Ph.D. students, a new Texas Experimental Geometry Lab for undergraduate research, and the Texas Winter Workshop on Geometric Structures for early career mathematicians.A guiding philosophy in the PI's research program is that geometric manifolds of one type may be fruitfully studied by deforming, or transitioning, to a different type of geometry. For example, the PI’s work studying Margulis spacetimes as geometric limits of anti de Sitter (AdS) spacetimes, has yielded many results about the geometry, topology, and deformation theory of these affine Lorentzian three-manifolds. The PI will develop new tools, following the geometric limit point of view, for new affine geometry contexts in higher dimensions, in order to address questions surrounding important open problems such as the Auslander Conjecture. The PI will also pursue a broad program to study convex real projective structures on manifolds. One focus is to identify which three-manifolds admit such structures and to describe the moduli spaces, with applications to questions in low-dimensional topology. In a more general context, the PI will conduct a thorough investigation of a new notion of convex cocompactness in real projective geometry, generalizing the well-studied notion from the Kleinian groups setting. Educational goals of this project center on expanding and vertically integrating the research training program in Geometric Structures at UT Austin being developed by the PI. The Texas Experimental Geometry Lab (TXGL) will introduce undergraduates at UT Austin to research in geometry, topology, and/or dynamics through computational and experimental projects. TXGL will also provide opportunities for Ph.D. students and post-docs to gain experience mentoring and teaching undergraduates outside the traditional classroom setting. In addition, TXGL will produce visualizations that illustrate concepts from current mathematics research. The Texas Winter Workshops in Geometric Structures will bring together small groups of early career mathematicians in order to learn an emerging new topic at the intersection of different (but related) mathematical fields.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.

这个项目涉及局部齐次几何流形。这些是抽象的数学对象，用来模拟我们生活的宇宙。局部齐次一词指的是高度局部对称的存在，这种对称性是由李群捕获的，称为（局部）对称群。正是这个对称群支配着几何，在以下意义上：我们可以在几何流形中测量的有意义的量（例如长度或角度）正是那些在对称群下不变的量。有许多不同可能的对称群导致不同类型的几何流形在数学和物理的许多环境中都很有用。也可以有许多不同的几何流形具有相同的局部对称群。这些都具有相同的局部属性，但在大尺度上看起来非常不同。所有这些可能性的空间称为模空间；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。虽然我们宇宙的精确特征（例如形状，大小等）是经验物理学的问题，但模空间是宇宙可能具有什么特征的问题的数学答案。本课题研究的几何流形包括平面仿射几何、实射影几何和常曲率伪黎曼几何。这些几何形状具有适当的对称性，允许具有神秘但易于处理的行为的有趣的大模空间。教育部分的核心要素包括培养博士生，为本科生研究建立一个新的德克萨斯实验几何实验室，以及为早期职业数学家开设的德克萨斯几何结构冬季研讨会。PI研究计划的指导思想是，一种类型的几何流形可以通过变形或过渡到另一种类型的几何来进行富有成效的研究。例如，PI将马古利斯时空作为反德西特（AdS）时空的几何极限进行研究，得到了许多关于仿射洛伦兹三流形的几何、拓扑和变形理论的结果。PI将开发新的工具，遵循几何极限的观点，用于更高维度的新仿射几何背景，以解决围绕重要开放问题的问题，如奥斯兰德猜想。PI也将追求一个广泛的计划来研究流形上的凸实投影结构。一个重点是确定哪些三流形承认这样的结构，并描述模空间，并将其应用于低维拓扑中的问题。在更一般的背景下，PI将对真实射影几何中的凸紧性的新概念进行彻底的研究，推广Kleinian群设置中得到充分研究的概念。该项目的教育目标集中在扩展和垂直整合由PI开发的UT Austin几何结构研究培训计划。德克萨斯实验几何实验室（TXGL）将通过计算和实验项目，向德克萨斯大学奥斯汀分校的本科生介绍几何、拓扑和/或动力学方面的研究。TXGL还将为博士生和博士后提供机会，以获得在传统课堂环境之外指导和教学本科生的经验。此外，TXGL将制作可视化，说明当前数学研究中的概念。德克萨斯州冬季几何结构研讨会将汇集早期职业数学家的小组，以便在不同（但相关）数学领域的交叉点学习新兴的新主题。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。