Exact Results in Aperiodic Systems

越来越真实的是，我们的现代世界运行在解决受约束的最优化问题上。例如，公共交通的调度、电力分配和互联网流量。凝聚态物理学--研究大量粒子相互作用时产生的物质--是一个重要的例子：单个粒子受相互作用和局域环境的限制，将其能量降至最低。对产生的物质的理解使得运行我们生活的技术成为可能。二聚体模型对这种约束和涌现的相互作用进行了优雅的描述。二聚体最初被认为是降落在材料表面的双原子分子。二聚体在表面的堆积密度取决于表面原子的排列和连接它们的键。理解这个过程--吸附--从未像现在这样迫切。这是因为它是许多碳捕获技术(在这种情况下，“二聚体”是二氧化碳分子，它包含三个位于一条直线上的原子)的基本组成部分。吸附也是催化的基础，催化是减少进行化学反应所需能量的过程：如果一个分子暂时粘在一个表面上，它与第二个被粘住的分子反应的机会就会增加。催化每年被用于产生价值超过1万亿美元的产品。因此，对二聚体模型的研究是工业和环境领域迫切关注的问题。二聚体的数学研究在统计物理学中有着崇高的历史，在那里它导致了一些深刻的结果。例如，共形不变性是一种出现在各种物理模型中的性质，从量子重力到压力下的沸腾-但统计物理学中共形不变性的第一个正式证明是最近才在二聚体模型中发现的。二聚体被用来证明有12,988,816种方法可以用来用多米诺骨牌来铺棋盘，并可以用来证明下面这句不同寻常的话：如果你把一副牌分成13堆，每堆四张，总是可以从每一堆牌中选择一张，以便从每个数字中选择一张。Flicker博士最近在一个全新的背景下开始了对二聚体模型的研究：准晶。1982年，这些了不起的材料的发现导致了诺贝尔奖，并迫使人们重新定义什么是晶体。在此之前，所有的固体被认为要么在原子尺度上是无序的，要么是它们的原子以规则的周期性结构排列，就像棋盘的正方形一样。准晶没有周期结构，但它们也不是无序的；相反，它们具有显著而美丽的对称性：它们的结构可以数学地描述为穿过高维晶体的片状。虽然现在许多准晶都是人工生长的，但只有三个自然产生的准晶被发现，它们都在同一颗西伯利亚陨石中。准晶在吸附方面有潜在的优势：它们很脆，这意味着它们可以分解成直径只有几百个原子的颗粒，从而使它们的表面积最大化；它们在高温下保持固态，有助于提高反应速度；它们的表面原子具有相邻原子键之间的广泛角度，可以被降落在这些表面上的弯曲分子利用。这个项目的目的是在二聚体模型和其他约束优化的统计模型中建立关于准晶和相关结构的精确数学结果。与项目合作伙伴-实验物理学家、机器学习初创公司、视觉艺术家和作曲家-合作，这些成果将被用于实际和工业用途，并将把这些耐人寻味的问题固有的审美吸引力带给更广泛的受众。

It is increasingly true that our modern world runs on solving problems of optimization subject to constraints. Examples include the scheduling of public transport, electrical power distribution, and the flow of internet traffic. Condensed matter physics - the study of that which emerges when huge numbers of particles interact - is an essential example: individual particles minimise their energies subject to the constraints of their mutual interactions and local environments. Understanding the resulting matter has enabled the technologies which run our lives.An elegant description of this interplay of constraints and emergence is provided by dimer models. Dimers were originally conceived of as two-atom molecules landing on the surfaces of materials. How densely the dimers can pack onto the surface depends on the arrangement of surface atoms and the bonds connecting them. Understanding this process -- adsorption -- has never been more urgent. That is because it is an essential component in many carbon capture technologies (`dimers' in this case being carbon dioxide molecules, which contain three atoms lying in a straight line). Adsorption also forms the basis of catalysis, the process of reducing the energy required to carry out chemical reactions: if a molecule temporarily sticks to a surface the chance of it reacting with a second stuck molecule can be increased. Catalysis is used in the generation of over a trillion dollars' worth of products annually. The study of dimer models is therefore of pressing industrial and environmental concern. The mathematical study of dimers has an exalted history within statistical physics, where it has led to some profound results. For example, 'conformal invariance' is a property appearing in diverse physical models from quantum gravity to water boiling under pressure --- but the first formal proof of conformal invariance in statistical physics was found only recently, in a dimer model. Dimers were used to prove that there are exactly 12,988,816 ways to tile a chessboard with dominoes, and can be used to prove the following remarkable statement: if you deal a pack of cards into thirteen piles of four, it is always possible to choose one card from each pile so as to select one of each number.Dr Flicker recently initiated the study of dimer models in a fundamentally new context: quasicrystals. The discovery of these remarkable materials in 1982 led to a Nobel prize, and forced a redefinition of what it means for something to be a crystal. Until then, all solids were believed to be either disordered on the atomic scale, or to have their atoms lined up in regular periodic structures, like the squares of a chessboard. Quasicrystals lack periodic structure, but nor are they disordered; instead they feature remarkable and beautiful symmetries: their structures can be described mathematically as slices through higher-dimensional crystals. While many quasicrystals have now been grown artificially, only three naturally occurring quasicrystals have ever been found, all in the same Siberian meteorite. Quasicrystals have potential advantages for adsorption: they are brittle, meaning they can be broken into particles only a few hundred atoms across, maximising their surface area; they remain solid at high temperatures, facilitating increased reaction rates; and their surface atoms feature a wide range of angles between neighbouring atomic bonds, which could be exploited by bendy molecules landing on these surfaces.The aim of this project is to establish exact mathematical results in dimer models, and other statistical models of constrained optimization, on quasicrystals and related structures. Working with project partners - experimental physicists, machine-learning startups, and visual artists and composers - these results will be put to practical and industrial use, and will bring the inherent aesthetic appeal of these intriguing problems to a wider audience.