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

归一化贝蒂数、非正曲率和辛格猜想

基本信息

批准号：
2104662
负责人：
Luca Fabrizio Di Cerbo
金额：
$ 20.59万
依托单位：
University of Florida
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104662&HistoricalAwards=false
关键词：
Normalized Betti Numbers Non Positive

项目摘要

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.

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发挥着至关重要的作用。它研究称为可微流形的空间，当放大时看起来就像熟悉的欧几里得空间的一部分，并且令人惊讶的是，它是我们许多物理理论（例如广义相对论和弦理论）的正确模型。在现代微分几何中，所谓的辛格猜想预测了在存在非正曲率的情况下，此类空间的几何和拓扑之间存在着令人着迷的联系。资助研究的主要目标是对这一问题进行多学科研究，并探索其与该领域其他重要问题的联系，例如丘关于标准化贝蒂数的问题和 Hopf 问题。该提案的目标不仅是建立一个解决这些问题的综合计划，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、复代数等密切相关问题的差距。几何和几何拓扑。此外，这些领域的进展可能会对几何学之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活动接受培训，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展工作。更具体地说，该项目致力于利用几何分析技术和 Hodge 理论研究非正弯曲空间上的归一化 Betti 数和 L2-Betti 数。 2017 年，PI 与 Mark Stern 一起开发了黎曼流形调和形式的 Price 不等式理论。 PI 将探索并阐明调和形式的 Price 不等式理论与紧流形的 Singer 猜想之间的联系。 PI 还将研究具有 Price 不等式的非紧有限体积负曲空间，并将这些技术应用于更高维非球面 Dehn 填充流形、更高图流形和非正曲环形紧致化。此外，他还将研究负弯曲黎曼流形序列的上同调，这些流形在本杰明和施拉姆的意义上收敛于其黎曼万有覆盖。最后，在凯勒环境中，PI 将把研究重点放在光滑的不规则品种上。这将产生辛格原始猜想在非球面流形类之外的扩展，并且也将为丘的归一化贝蒂数问题相关的研究开辟新的途径。该奖项反映了 NSF 的法定使命，并通过评估被认为值得支持利用基金会的智力优势和更广泛的影响审查标准。