Comparison and Inverse Comparison Geometry

1827年，高斯发表了他现在著名的定理（theorma Egregium），大致翻译成英语是“非凡定理”（Remarkable Theorem）。“极限定理”建立了曲面在空间内弯曲的方式与其内在几何与欧几里得研究的平面经典几何的区别之间的关系。过分定理的一个直接结果是，不扭曲世界几何的某些关键方面，就不可能绘制世界地图。更令人吃惊的是，19世纪中叶，他的学生黎曼将高斯的思想推广到高维空间，由此诞生了现在被称为黎曼几何的学科。虽然黎曼几何及其推广在相对论、粒子物理学和数据科学等领域得到了广泛的应用，但黎曼几何中许多基本问题的答案一直是个谜。这个项目是对其中一些问题的双管齐下的攻击，通过所谓的比较几何和逆比较几何。比较几何是全局黎曼几何的一个分支，它通过将曲率受限的空间与曲率恒定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；也就是说，哪些空间允许黎曼几何满足给定的几何约束。该项目将采用这两门学科的方法，继续研究全局黎曼几何的核心问题。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。本文主要研究三个基本问题：微分同胚稳定性问题、正曲率下的夹持问题和几乎非负曲率流形的构造问题。微分同态稳定性问题研究的是如果一个列具有一致的下截面曲率界且不坍缩，那么该列的Gromov-Hausdorff收敛序列是否具有稳定的微分同态。Perelman的稳定性定理保证了这样的序列具有稳定的拓扑类型，Kuwae、Machigashira和Shioya的结果在适当意义上保证了在极限非奇异的情况下具有稳定的微分同态类型。一般极限有奇点。然而，PI和他的合作者Grove、Sill和Pro已经确定了某些奇异极限空间是异象稳定的。这包括Pro和PI最近的一个结果，表明在4维中，无论极限空间的奇异结构如何，微分同胚稳定性都是成立的。结合Grove-Petersen-Wu， Perelman和Kirby-Siebenmann先前的工作，这意味着Cheeger有限定理的结论在没有上曲率界假设的情况下成立。该项目旨在建立（与Chambers和Pro联合）分段线性稳定性在所有维度上都适用。Searle和Solórzano的项目将开发工具来证明一些极限空间不是微分同构稳定的。Searle, Solórzano，和PI也会用这些工具来证明在所有7维及更高的维度上存在几乎非负弯曲的奇异球体它们等于3 mod 4。正曲率的捏缩问题是研究当曲率被捏缩时，正弯曲流形的拓扑结构是如何受到约束的。在这个领域，定理和实例之间的差距是惊人的。与gujarro和Murphy合作的项目将通过利用gujarro和PI的Jacobi场比较引理来改进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。

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.