Comparison and Inverse Comparison Geometry

1827年，高斯发表了他现在著名的定理，这本书粗略地翻译成英语，称为显著定理。由欧几里德研究的埃格罗里厄姆定理建立了曲面在空间内的曲线方式与其固有几何与平面的经典几何的不同之间的关系。埃格瑞吉姆定理的一个直接后果是，如果不扭曲世界地图几何的某些关键方面，就不可能绘制出一幅世界地图。更令人震惊的是，19世纪中叶，高斯德·S的学生黎曼将其思想推广到更高维的空间，这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在相对论、粒子物理和数据科学中得到了广泛的应用，但该学科中许多基本、根本问题的答案仍然是永恒的谜团。这个项目是对其中一些问题的双管齐下的攻击，通过被称为比较几何和反向比较几何。比较几何是全局黎曼几何的一个分支，它通过将一个空间与一个常曲率的模型空间进行比较，得出关于其曲率受到某种约束的空间的几何和拓扑结论。逆比较几何涉及相反的问题；即，哪些空间允许满足给定几何约束的黎曼几何。该项目将使用这两个学科的方法来继续解决位于全球黎曼几何核心的问题。该项目还包括对学生的建议和指导，继续致力于DEI倡议和数学传播。这项研究集中在三个基本问题：微分稳定性问题，正曲率下的Pinching问题，以及具有几乎非负曲率的流形的构造。微分同胚稳定性问题是问黎曼流形的Gromov-Hausdorff收敛序列是否具有稳定的微分同胚型，只要该序列是非折叠的且具有一致的截面曲率下界。Perelman稳定性定理保证了这样的序列具有稳定的拓扑型，科威特、Machigashira和Shioya的结果保证了稳定的微分同态型，只要极限在适当的意义下是非奇异的。一般极限有奇点。然而，PI和他的合作者Grove、Sill和Pro已经建立了某些奇异极限空间是双同构稳定的。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，微分同构稳定性在4维都成立。再加上Grove-Petersen-Wu，Perelman和Kirby-Siebenmann先前的工作，这意味着在没有曲率上界的假设下，Cheeger的有限定理的结论是成立的。该项目的目的是建立(与钱伯斯和专业)分段线性稳定性在所有维度都成立。Searle和Solórzano的项目将开发工具来证明某些极限空间不是微分同胚稳定的。Searle，Solórzano和PI也将使用这些工具来证明在所有维度7和更高的维度上存在几乎非负弯曲的奇异球面，它们与3mod 4同余。正曲率下的收缩问题询问当正曲线流形的曲率被挤压时，如何约束其拓扑。这一领域的定理和例子之间的差距令人震惊。这项与Guijarro和Murphy合作的项目将利用Guijarro和PI的Jacobi场比较引理来改进Abresch-Meyer注入性半径估计。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智力优势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.