Comparison and Inverse Comparison Geometry

比较和逆比较几何

• 批准号: 2203686
• 负责人: Frederick Wilhelm
• 金额: $ 36.49万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203686&HistoricalAwards=false
• 关键词: Comparison; Inverse; Geometry

In 1827, Gauss published his now famous Theorema Egregium, which roughly translates into English as Remarkable Theorem. The Theorema Egregium established a relationship between the way a surface curves inside of space and how its intrinsic geometry differs from that of the classical geometry of the plane, studied by Euclid. An immediate consequence of the Theorema Egregium is that it is impossible to make a map of the world without distorting some crucial aspects of its geometry. More startling was the generalization of Gauss’s ideas to higher-dimensional spaces by his student Riemann in the mid 19th century, which gave birth to the subject that is now known as Riemannian Geometry. While Riemannian Geometry and its generalizations have found wide-ranging applications, for example, in relativity, particle physics, and data science, answers to many basic, fundamental questions in the subject are enduring mysteries. This project is a two-pronged attack on some of these questions via what are known as Comparison Geometry and Inverse Comparison Geometry. Comparison Geometry is the branch of Global Riemannian Geometry that draws geometric and topological conclusions about a space with some constraint on its curvature by comparing it to a model space with constant curvature. Inverse Comparison Geometry concerns the opposite question; that is, which spaces admit Riemannian geometries that satisfy a given geometric constraint. The project will employ methods from both subjects to continue attacking problems that lie at the heart of Global Riemannian Geometry. The project also includes advising and mentoring of students, continued commitment to DEI initiatives and mathematical dissemination.This research centers on three basic problems: the Diffeomorphism Stability Question, the Pinching Problem in positive curvature, and the constructions of manifolds with almost nonnegative curvature. The Diffeomorphism Stability Question asks whether a Gromov-Hausdorff convergent sequence of Riemannian manifolds has a stable diffeomorphism type provided the sequence is noncollapsing and has a uniform lower sectional curvature bound. Perelman's stability theorem guarantees that such a sequence has a stable topological type, and a result of Kuwae, Machigashira, and Shioya guarantees a stable diffeomorphism type provided the limit is nonsingular, in the appropriate sense. A generic limit has singularities. Nevertheless, the PI and his collaborators Grove, Sill, and Pro, have established that certain singular limit spaces are di¤eomorphically stable. This includes a recent result by Pro and the PI showing that Diffeomorphism Stability holds in dimension 4, regardless of the singular structure of the limit space. Together with prior work by Grove-Petersen-Wu, Perelman, and Kirby-Siebenmann, this means that the conclusion of Cheeger's Finiteness Theorem holds without the hypothesis of the upper curvature bound. The project aims to establish (jointly with Chambers and Pro) that Piecewise Linear Stability holds in all dimensions. The project with Searle and Solórzano will develop tools to show that some limit spaces are not diffeomorphically stable. Searle, Solórzano, and the PI will also use these tools to show that there are almost nonnegatively curved exotic spheres in all dimensions 7 and higher that are congruent to 3 mod 4. The Pinching Problem in positive curvature asks how the topology of a positively curved manifold is constrained as its curvature is pinched. The gap between theorems and examples in this area is startling. The project in collaboration with Guijarro and Murphy will improve the Abresch-Meyer injectivity radius estimate by exploiting the Jacobi Field Comparison Lemma of Guijarro and the PI. The project also includes significant training and mentoring of students, mathematical dissemination and commitment to service and DEI initiatives.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.

1827年，高斯发表了他现在著名的Theorema Egregium，大致翻译成英语为卓越定理。Egregium定理建立了一种关系的方式曲面曲线内部的空间和它的内在几何如何不同于古典几何的平面，研究了欧几里得。Egregium定理的一个直接后果是，不可能在不扭曲几何学的某些关键方面的情况下绘制世界地图。更令人吃惊的是，在世纪中期，高斯的学生黎曼将他的思想推广到了高维空间，这就催生了现在被称为黎曼几何的学科。’虽然黎曼几何及其推广已经在相对论、粒子物理和数据科学等领域得到了广泛的应用，但该学科中许多基本问题的答案仍然是一个永恒的谜团。这个项目是通过所谓的比较几何和逆比较几何对其中一些问题的双管齐下的攻击。比较几何是整体黎曼几何的分支，它通过将一个曲率受到某种约束的空间与一个常曲率的模型空间进行比较，得出关于该空间的几何和拓扑结论。逆比较几何关注的是相反的问题，即哪些空间允许满足给定几何约束的黎曼几何。该项目将采用这两个主题的方法，继续攻击问题，在全球黎曼几何的核心。该项目还包括为学生提供咨询和指导，继续致力于DEI倡议和数学传播。该研究集中在三个基本问题上：非同态稳定性问题，正曲率的Pinching问题和几乎非负曲率流形的构造。复同态稳定性问题询问黎曼流形的Gromov-Hausdorff收敛序列是否具有稳定的复同态类型，只要该序列是非塌陷的并且具有一致的下截面曲率界。佩雷尔曼的稳定性定理保证了这样的序列有一个稳定的拓扑类型，并且在适当的意义上，如果极限是非奇异的，那么Joshie，Machigashira和Shioya的结果保证了一个稳定的非同态类型。一般极限有奇点。尽管如此，PI和他的合作者格罗夫、希尔和Pro已经建立了某些奇异极限空间是同胚稳定的。这包括Pro和PI最近的一个结果，该结果表明，无论极限空间的奇异结构如何，在4维空间中，同构稳定性都成立。结合Grove-Petersen-Wu、Perelman和Kirby-Siebenmann的先前工作，这意味着Cheeger的连续性定理的结论在没有曲率上界假设的情况下成立。该项目旨在建立（与钱伯斯和Pro联合）分段线性稳定性在所有维度上保持不变。Searle和Solórzano的项目将开发工具来证明某些极限空间不是同构稳定的。Searle，Solórzano和PI也将使用这些工具来证明在所有7维和更高维中存在几乎非负弯曲的奇异球，它们与3 mod 4全等。正曲率中的Pinching问题是关于正曲率流形的拓扑如何在其曲率被Pinching时受到约束。这一领域的理论和实例之间的差距令人吃惊。与Guijarro和Murphy合作的项目将通过利用Guijarro的Jacobi场比较引理和PI来改进Abresch-Meyer注入半径估计。该项目还包括对学生的重要培训和指导，数学传播以及对服务和DEI倡议的承诺。该奖项反映了NSF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。