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.

差异几何形状是一个广泛而活跃的主题，在现代数学，理论物理学和计算机科学中起着至关重要的作用。它研究了称为可区分的流形的空间，当放大时，看起来像熟悉的欧几里得空间的部分，而这是我们许多物理理论（例如一般相对论和弦理论）的正确模型。在现代的微分几何形状中，所谓的歌手猜想预测了在存在非阳性曲率的情况下，此类空间的几何形状和拓扑之间的连接。资助研究的一个主要目标是建立对此类问题的多学科研究，并探索与该领域其他重要问题的联系，例如Yau在标准化的Betti数字和HOPF问题上的问题。该提案的目的不仅是为解决这些问题的解决方案建立一个全面的计划，而且还提出了此类问题的扩展，以阐明它们的相互依存关系，并在差异几何学，复杂的代数几何形状和几何学拓扑中弥合与密切相关的问题的差距。此外，在几何形状之外，这些领域的进展可能会产生重大影响。实际上，这些问题与部分微分方程，几何群体理论以及数学理论物理学领域密切相关。本科生和研究生将通过参加拟议的活动进行培训，PI将继续组织研讨会和会议，并参与针对降低教育机会的学生的推出工作。更具体地说，该项目通过几何分析技术和Hodge理论解决了非物性弯曲空间上归一化贝蒂数和L2-BETTI数字的研究。 2017年，PI与Mark Stern一起开发了Riemannian歧管上谐波形式的价格不平等理论。 PI将探索和阐明谐波形式价格不平等的理论与紧凑型歧管的歌手猜想之间的联系。 PI还将研究具有价格不平等的非紧凑型有限体积负面弯曲的空间，他将将这些技术应用于更高的尺寸dehn填充歧管，较高的图形歧管和非积极性弯曲的环形紧凑型。此外，他还将研究带有负弯曲的riemannian歧管序列的共同体，从本杰米尼和施拉姆（Benjamini）和施拉姆（Schramm）的意义上讲，这些歧管的序列融合到了riemannian普遍的覆盖范围内。最后，在Kaehler设置的情况下，PI将把他的研究集中在光滑的不规则品种上。这将产生歌手在非球形歧管等类之外的原始猜想的扩展，并且还将开放与Yau关于归一化Betti数字问题有关的研究的新途径。这项奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子功能和广泛影响的评估来评估Criteria criteria criteria criteria criteria。