权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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，当放大时看起来像熟悉的欧几里得空间的碎片，并且惊人地是我们许多物理理论的正确模型，如广义相对论和弦论。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，并探讨其与该领域的其他重要问题，如丘的问题规范化贝蒂数和霍普夫问题的联系。这个建议的目标将不仅是建立一个全面的计划对这些问题的解决方案，而且还提出了这些问题的扩展，以澄清他们的相互依存关系，并弥合差距密切相关的问题，在微分几何，复代数几何，几何拓扑。此外，这些领域的进展可能会在几何学之外产生重大影响。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加拟议的活动接受培训，公共信息中心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，这个项目致力于研究规范化贝蒂数和L2-贝蒂数的非正弯曲空间的几何分析技术和霍奇理论。2017年，PI与Mark Stern一起开发了黎曼流形上调和形式的Price不等式理论。PI将探索和阐明调和形式的Price不等式理论与紧致流形的Singer猜想之间的联系。PI还将研究非紧有限体积负弯曲空间与价格不等式，他将这些技术应用于高维非球面德恩填充流形，更高的图形流形，和非正弯曲的环面紧化。此外，他将研究上同调序列的负弯曲黎曼流形收敛，在这个意义上的本杰明和施拉姆，他们的黎曼普遍覆盖。最后，在Kaehler设置PI将专注于他的研究顺利不规则品种。这将使Singer最初的猜想在非球面流形之外得到扩展，也将为与Yau关于规范化Betti数的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。