Diophantine Approximation and Aperiodic Order

该奖项旨在发展和收获一系列相互关联的结果，涵盖数论、动力系统和非周期秩序领域。一个中心主题是探索那些已知应用于被称为准晶体的物理材料的数学基础模型的问题。研究目标是利用数论、概率论、拓扑动力学和遍历理论的工具，证明一些在纯数学世界中具有重要意义的长期开放问题的新结果，这些结果也具有在现实世界中应用的潜力。首席研究员将指导与该奖项相关的研究生课题。准晶体是具有高度有序分子结构的物理材料，使它们产生纯点衍射图案，但也具有经典晶体学限制定理所禁止的旋转对称性。这些材料是由丹·谢赫特曼在20世纪80年代发现的，这一发现彻底改变了晶体学的世界，并为谢赫特曼赢得了诺贝尔奖。在数学世界中，准晶体的研究，甚至是准晶体的存在，都与欧几里得空间的非周期平铺理论密切相关。准晶体通常是通过切割和投影集来建模的，它们提供了大量这种拼接的例子，并且通常是具有“非周期性秩序”的系统的好例子。在过去的十年里，将数论问题与cut和project set问题联系起来的见解出现了爆炸式的增长。利用准晶体理论的思想，丢番图近似中许多未解决的问题被重新表述，有些问题已经得到解决。通过对逆连接的理解，也解决了关于非周期序集的主要问题。这个项目的研究是围绕着著名的开放问题，如Littlewood猜想和Pisot猜想，这是坐在这些学科的边界。它的目的是围绕它们发展理论，以便在纯数学和应用数学领域都取得进展。该奖项反映了美国国家科学基金会的法定使命，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。

This award aims to develop and harvest a collection of interrelated results spanning the fields of number theory, dynamical systems, and aperiodic order. A central theme is to explore problems which have known applications to the mathematics underpinning models for physical materials called quasicrystals. The research goals are to use tools from number theory, probability, topological dynamics, and ergodic theory, to prove new results about a number of long standing open problems with significance in the world of pure mathematics, which also have the potential for real world applications. The Principal Investigarot will mentor graduate students on topics related to this award.Quasicrystals are physical materials with highly ordered molecular structures, causing them to produce pure point diffraction patterns, but which also possess rotational symmetries that are forbidden by the classical Crystallographic Restriction Theorem. The discovery of these materials in the 1980's by Dan Shechtman revolutionized the world of crystallography and later earned Shechtman a Nobel prize. In the mathematical world the study, and even the existence, of quasicrystals is intimately related to the theory of aperiodic tilings of Euclidean space. Quasicrystals are often modeled by cut and project sets, which give an abundance of examples of such tilings and which, generically, are good examples of systems which possess `aperiodic order'. In the last decade there has been an explosion of insight linking problems in number theory with problems about cut and project sets. A number of unsolved problems in Diophantine approximation have been reformulated, and some have been solved, using ideas from the theory of mathematical quasicrystals. Major problems about sets with aperiodic order have also been solved by understanding the reverse connections. The research in this project is centered around well-known open problems such as the Littlewood Conjecture and the Pisot Conjecture, which are sitting on the border of these subjects. It aims to develop the theory around them in order to create progress in both the realms of pure and applied mathematics.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.