Translational Tilings and Orthogonal Systems of Exponentials

这个项目涉及两个重要且密切相关的数学对象的结构：平移平铺和正交系。平移平铺是指使用某些构建块的翻译副本来覆盖空间，这种平铺被称为“平铺”，没有重叠。这个项目的重点是该地区的主要问题：哪些是空间瓷砖的可能方法。多年来，许多数学家对瓷砖的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，以及业余数学家罗伯特·安曼和莫里茨·科内利斯·埃舍尔。这项数学研究在其他科学中发现了许多重要的应用。特别是，它导致了物理学中最重大的突破之一：发现了天然准晶，即原子排列不是周期性的物理固体。由于非周期瓦片(即不能通过它们在空间中任意大但有界的部分的外观来“确定”的空间铺设)是准晶的数学模型，丹·谢克特曼(Dan Shechtman)在自然界中发现了这些非周期形式，这是基于非周期瓦片的数学发现。正交系对数论、代数几何和分析等数学分支都很重要。它们的优点之一是它们可以促进将功能(例如，描述光波或声波)顺从地分解成具有高度相互独立性的更易于管理的部分。单独研究这些部分可以深入了解系统的物理属性，如主频。这项研究(与合作者合作)包括揭示与一系列不同数学领域的联系，并在这些领域过去成果的基础上，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翻译拼接的结构。主要目的是解决著名的周期平铺猜想，该猜想断言欧氏空间的任何有界可测平铺必至少包含一个周期平铺。众所周知，这一猜想适用于实线的平铺，在更高的维度上也有一些与之相关的部分结果。然而，周期平铺猜想在二维和更高的维度上还没有得到解决。随着时间的推移，很明显，在许多方面，平移瓦片的行为类似于其希尔伯特空间(即，该域上所支持的二次可积函数的空间)允许指数函数的正交基的域。因此，第二个重点领域是在两种不同的设置下的正交指数系统的频率集的结构：在时频空间(Gabor基)中和在某些域的Hilbert空间中。该项目的这一部分旨在为后面提到的正交系统研究找到新的方法，在数论、代数几何和傅立叶分析之间建立新的桥梁。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.