Operator asymptotics, a new approach to length-scale interactions in metamaterials.

许多现实世界的现象，如疾病的传播或股票价格的波动，都可以用数学来理解和预测。这是通过建立一个用数学语言编写的模型来实现的，该模型易于使用，但包含了描述所讨论的系统所需的所有基本信息。例如，假设我们想要利用每一粒沙子的信息来模拟沙漠中沙丘的形成。这将产生一个过于复杂的模型，难以理解，也无法使用。然而，要建立一个成功的模型，就必须充分利用这样一个事实：在一个比单个颗粒大得多的沙丘长度尺度上，沙子似乎表现得像一个连续的介质。在这个例子中，数学模型通过关注单个尺度，即单个沙丘的长度尺度来解决问题。换句话说，系统在问题尺度（沙丘的长度）上的基本行为并不是由更细尺度（颗粒的大小）上的行为决定的。然而，对于今天许多具有重要意义的现象，例如蛋白质折叠、催化过程或金属腐蚀，情况并非如此，这些现象的基本潜在行为跨越了广泛的长度和时间尺度。作为一个说明性的例子，让我们考虑身体中一根典型的长骨，它的主要功能是在机械负荷下提供结构支撑。在长骨内的一个特殊现象是，它的组成可以由于长时间施加的机械应力而改变。这是通过重塑骨骼以形成最能抵抗和支持机械应力变化的模式来实现的。这个过程始于刺激蛋白质翻译的机械负荷；形成的蛋白质帮助形成囊泡和转运分子，最终沉淀形成骨骼的矿物质。上述解释的复杂性表明，对于一个数学模型来说，要成功地解决骨的动力学特性，它必须包括跨越许多时间和长度尺度的分层行为。这说明了一个事实，即最简单的多尺度现象模型是难以置信的复杂和太难使用。我们面临的目标是从复杂的多尺度模型中推导出有效的宏观模型。这就是“多尺度分析”，特别是该提案的由来。该项目将开发用于材料科学多尺度模型研究的新工具，重点研究一类称为“超材料”的新型人工材料。这些工具的作用将是从这些材料的多尺度模型中提取宏观性质，同时保留有关微观尺度的关键信息，以便为超材料产生简单而准确的模型。为了实现这一目标，我将用“算子”的数学语言来描述所讨论的多尺度模型，并利用算子理论和多尺度分析的最新进展，提供一种新的分析方法来描述这些算子（以及它们的模型对应物）的主导行为，这些行为是由不同尺度的相互作用产生的。该提案还将在均质化理论方面取得进展，均质化理论是一种用于寻找多尺度系统有效模型的数学方法。该提案将利用均质化理论的新进展来分析一类模拟超材料的微分方程。

Many real-world phenomena, such as the spread of disease or the fluctuations of stock prices, can be understood and predicted by mathematics. This is achieved by building a model, written in mathematical language, that is simple to use yet contains all the essential information required to describe the system in question. For example, imagine we wanted to model the formation of sand dunes in a desert by using the information about each individual grain of sand. This would produce a model too complicated to understand and impossible to use. Yet, to build a successful model it suffices to exploit the fact that on the scale of a dune length that is much larger than an individual grain, sand appears to behave like a continuous medium. In this example, the mathematical model addresses the phenomenon in question by focusing on a single scale, the length scale of the individual dunes. To put it another way, the essential behaviour of the system on the scale in question (length of a dune) is not dictated by the behaviour on finer scale (size of a grain). This however is not the case for many phenomena of great significance today, for example protein folding, catalytic processes or corrosion of metals, where the essential underlying behaviour spans a wide range of length and time scales. As an illustrative example let us consider a typical Long bone in the body, whose primary function is to provide structural support under mechanical load. A particular phenomenon within Long bone is that its composition can change due to mechanical stresses applied over long time scales. This is achieved by the remodelling of bone to form patterns that are the most resistive and supportive to the changes in mechanical stress. This process begins with the mechanical load stimulating protein translation; the formed proteins then aid in the formation of vesicles and transporter molecules which finally deposit the minerals that form the bone. The complexity of the above explanation shows that for a mathematical model to successfully address the dynamical properties of bone it must include hierarchical behaviour that spans many time and length scales. This illustrates the fact that the simplest models of multi-scale phenomena are incredibly complicated and too difficult to use. The objective we face is to derive effective macroscopic models from complicated multi-scale models. This is where 'multi-scale analysis' and in particular the proposal comes in. The project will develop new tools in the study of multi-scale models in material sciences, with a focus on a new class of artificial materials called "metamaterials". The role of these tools will be to extract macroscopic properties from multi-scale models of such materials while preserving the key information about the microscale in order to produce simple and accurate models for metamaterials. To achieve this goal I will cast the multi-scale models in question in the mathematical language of "operators" and by using recent advances in operator theory and multi-scale analysis provide a new analytical approach to describing the dominant behaviour of these operators (and, in turn, their model counterparts) resulting from the interaction of the different scales. The proposal will also make advances in the theory of homogenisation, which is a mathematical method used to find effective models of multi-scale systems. The proposal will exploit new advances in homogenisation theory to analyse a broad class of differential equations that model metamaterials.