Normalized Betti Numbers, Non-Positive Curvature, and the Singer Conjecture

微分几何是一门广泛而活跃的学科，在现代数学、理论物理和计算机科学中起着至关重要的作用。它研究被称为可微流形的空间，当放大时，它看起来就像我们熟悉的欧几里得空间的片段，而且令人惊讶的是，它是我们许多物理理论的正确模型，比如广义相对论和弦理论。在现代微分几何中，所谓的辛格猜想预言了存在非正曲率的空间的几何和拓扑之间的迷人联系。资助研究的一个主要目标将是建立这样一个问题的多学科研究，并探索其与该领域其他重要问题的联系，如丘的标准化贝蒂数问题和霍普夫问题。本提案的目标不仅是建立一个解决这些问题的综合方案，而且还提出这些问题的扩展，澄清它们的相互依赖性，并弥合与微分几何，复杂代数几何和几何拓扑中密切相关的问题的差距。此外，这些领域的进展可能会对几何学以外的领域产生重大影响。事实上，这些问题与偏微分方程、几何群论以及数学理论物理领域的问题密切相关。本科生和研究生将通过参与拟议的活动而得到培训，PI将继续组织研讨会和会议，并参与针对受教育机会减少的学生的外联工作。更具体地说，本项目利用几何分析技术和Hodge理论研究非正弯曲空间上的规范化Betti数和L2-Betti数。2017年，PI与Mark Stern一起开发了黎曼流形谐波形式的价格不等式理论。PI将探索和阐明调和形式的价格不等式理论与紧流形的辛格猜想之间的联系。PI还将研究具有Price不等式的非紧致有限体积负弯曲空间，并将这些技术应用于高维非球面Dehn填充流形、高维图流形和非正弯曲环面紧化。此外，他将研究负弯曲黎曼流形序列的上同调，这些序列在Benjamini和Schramm的意义上收敛于它们的黎曼全覆盖。最后，在Kaehler设置下，PI将重点研究光滑的不规则品种。这将产生Singer的原始猜想在非球面流形类之外的扩展，也将为Yau关于规范化Betti数的问题开辟新的研究途径。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。

Differential geometry is a broad and active subject that plays a crucial role in modern mathematics, theoretical physics, and computer science. It studies spaces called differentiable manifolds that when zoomed in look like pieces of the familiar Euclidean space, and that strikingly are the correct model for many of our physical theories such as General Relativity and String Theory. In modern differential geometry, the so-called Singer conjecture predicts a fascinating connection between the geometry and topology of such spaces in the presence of non-positive curvature. A principal objective of the funded research will be to build a multidisciplinary study of such a problem, and to explore its connections with other important problems in the field such as Yau's question on normalized Betti numbers and the Hopf problem. The goal of this proposal will be not only to build a comprehensive program towards the solution of these problems, but also to propose extensions of such questions, to clarify their interdependence, and to bridge a gap with closely related problems in differential geometry, complex algebraic geometry, and geometric topology. Moreover, progress in these areas could have significant repercussions outside of geometry. Indeed, these questions are intimately connected to problems in partial differential equations, geometric group theory, as well to areas of mathematical theoretical physics. Undergraduate and graduate students will be trained through their participation in the proposed activities, and the PI will continue to organize seminars and conferences and to participate in outreach efforts targeting students who have experienced reduced access to education. More specifically, this project addresses the study of normalized Betti numbers and L2-Betti numbers on non-positively curved spaces with geometric analysis techniques and Hodge theory. In 2017, the PI together with Mark Stern developed the theory of Price inequalities for harmonic forms on Riemannian manifolds. The PI will explore and elucidate the connections between the theory of Price inequalities for harmonic forms and the Singer conjecture for compact manifolds. The PI will also study non-compact finite volume negatively curved spaces with Price inequalities, and he will apply these techniques to higher dimensional aspherical Dehn filled manifolds, higher graph manifolds, and non-positively curved toroidal compactifications. Also, he will study the cohomology of sequences of negatively curved Riemannian manifolds which converge, in the sense of Benjamini and Schramm, to their Riemannian universal cover. Finally, in the Kaehler setting the PI will focus his research on smooth irregular varieties. This will yield extensions of the original conjecture of Singer outside the class of aspherical manifolds, and it will also open new avenues of research related to Yau's question on normalized Betti numbers.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.