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Translational Tilings and Orthogonal Systems of Exponentials

平移平铺和正交指数系统

基本信息

批准号：
2242871
负责人：
Rachel Greenfeld
金额：
$ 11.71万
依托单位：
Institute For Advanced Study
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2242871&HistoricalAwards=false
关键词：
Translational Tilings Orthogonal Systems Exponentials

项目摘要

This project concerns the structure of two important and closely related mathematical objects: translational tilings and orthogonal systems. A translational tiling is a covering of a space using translated copies of certain building blocks, called the “tiles”, without overlaps. The focus of this project is on the major question in the area: which are the possible ways that a space can be tiled. The structure of tilings has been extensively studied over the years by many mathematicians; among the most famous and significant contributors are the mathematical physicist Roger Penrose and the amateur mathematicians Robert Ammann and Maurits Cornelis Escher. This mathematical study has found many important applications to other sciences. In particular, it led to one of the most significant breakthroughs in physical science: the discovery of natural quasicrystals, physical solids whose atoms’ arrangement is not periodic. Since aperiodic tilings (i.e., pavings of space which cannot be “determined” by how they look in an arbitrarily large but bounded part of the space) serve as the mathematical models of quasicrystals, the discovery of those aperiodic forms in nature by Dan Shechtman (for which he won the Nobel prize), was based on the mathematical discovery of aperiodic tilings. Orthogonal systems are of importance to several branches of mathematics, including number theory, algebraic geometry and analysis. One of their advantageous properties is that they can facilitate an amenable decomposition of a function (e.g., describing light or sound waves) into more manageable pieces with a high level of mutual independence. Studying these pieces separately can yield insight into the physical properties of a system such as dominant frequencies. The research carried out (in conjunction with collaborators) consists of revealing connections to a range of different areas of mathematics and building on past results in those areas, as well as developing novel methods and techniques.The first part of this project will be devoted to the study of the structure of translational tilings. The main goal is solving the well-known periodic tiling conjecture, which asserts that any bounded measurable tile of the Euclidean space must admit at least one periodic tiling. This conjecture is known to hold for tilings of the real line, and there are some partial results towards it in higher dimensions. However, the periodic tiling conjecture has not been settled yet in dimensions two and higher. Over time it has become apparent that in many respects translational tiles “behave like” domains whose Hilbert space (i.e., the space of quadratically integrable functions that are supported on the domain) admits an orthogonal basis of exponential functions. Thus, a second area of focus is on the structure of frequency sets of orthogonal systems of exponentials in two different settings: in time-frequency spaces (Gabor bases) and in Hilbert spaces of certain domains. This part of the project aims to find new approaches to the latter mentioned studies of orthogonal systems, building new bridges between number theory, algebraic geometry and Fourier analysis.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.

这个项目涉及两个重要的和密切相关的数学对象的结构：平移平铺和正交系统。平移平铺是使用某些构建块的平移副本（称为“瓦片”）来覆盖空间，而不重叠。这个项目的重点是在该地区的主要问题：这是一个空间可以瓷砖的可能方式。多年来，许多数学家对平铺结构进行了广泛的研究，其中最著名和最重要的贡献者是数学物理学家罗杰·彭罗斯（Roger Penrose）和业余数学家罗伯特·阿曼（Robert Ammann）和莫里茨·科内利斯·埃塞尔（Maurits Cornelis Escherer）。这种数学研究在其他科学中有许多重要的应用。特别是，它导致了物理科学中最重要的突破之一：天然准晶体的发现，这种物理固体的原子排列不是周期性的。由于非周期性平铺（即，虽然准晶体是一种非周期性的晶体（空间的铺层不能由它们在空间的任意大但有界的部分中的外观来“确定”），但丹·谢赫特曼（Dan Shechtman）发现自然界中的非周期性形式（他因此获得诺贝尔奖），是基于非周期性铺层的数学发现。正交系对数学的几个分支都很重要，包括数论、代数几何和分析。它们的有利特性之一是它们可以促进函数的顺从分解（例如，描述光或声波）分解成具有高度相互独立性的更易管理的片段。单独研究这些部分可以深入了解系统的物理特性，例如主频率。该研究（与合作者一起）包括揭示与一系列不同数学领域的联系，并在这些领域的过去成果的基础上，以及开发新的方法和技术。该项目的第一部分将致力于研究平移拼接的结构。主要目标是解决著名的周期瓦片猜想，该猜想断言欧几里得空间的任何有界可测瓦片必须至少允许一个周期瓦片。这个猜想是已知的，以保持平铺的真实的线，并有一些部分结果，它在更高的层面。然而，在二维及更高维度上，周期平铺猜想尚未得到解决。随着时间的推移，已经变得明显的是，在许多方面，平移瓦片“表现得像”其希尔伯特空间（即，域上支持的二次可积函数的空间）允许指数函数的正交基。因此，第二个领域的重点是在两个不同的设置：在时间-频率空间（伽柏基地）和希尔伯特空间的某些领域的指数正交系统的频率集的结构。该项目的这一部分旨在为后面提到的正交系统的研究找到新的方法，在数论、代数几何和傅立叶分析之间建立新的桥梁。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。