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

许多真实世界的现象，如疾病的传播或股票价格的波动，都可以通过数学来理解和预测。这是通过建立一个用数学语言编写的模型来实现的，该模型简单易用，但包含描述所述系统所需的所有基本信息。例如，假设我们想要通过使用关于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、难以理解和无法使用的模型。然而，要建立一个成功的模型，只需利用这样一个事实，即在比单个颗粒大得多的沙丘长度上，沙子看起来像是一种连续的介质。在这个例子中，数学模型通过关注单个尺度，即各个沙丘的长度尺度来处理所讨论的现象。换句话说，系统在所讨论的尺度(沙丘的长度)上的基本行为并不是由更精细的尺度(颗粒的大小)上的行为决定的。然而，对于今天具有重大意义的许多现象，例如蛋白质折叠、催化过程或金属腐蚀，情况并非如此，在这些现象中，基本的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个典型的长骨，它的主要功能是在机械载荷下提供结构支撑。长骨中的一个特殊现象是，由于长时间施加的机械应力，它的成分可能会发生变化。这是通过重塑骨骼来形成对机械应力变化最具抵抗力和支持性的模式来实现的。这个过程始于机械负荷刺激蛋白质的翻译；然后形成的蛋白质帮助形成囊泡和转运蛋白分子，最终沉积形成骨骼的矿物质。上述解释的复杂性表明，为了成功地解决骨骼的动力学特性，数学模型必须包括跨越许多时间和长度尺度的分层行为。这说明了这样一个事实，即最简单的多尺度现象模型都极其复杂，太难使用了。我们面临的目标是从复杂的多尺度模型中提取有效的宏观模型。这就是‘多尺度分析’，特别是提案的用武之地。该项目将在材料科学的多尺度模型研究中开发新的工具，重点是一种被称为“超材料”的新型人造材料。这些工具的作用将是从这种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，以便为超材料制作简单而准确的模型。为了实现这一目标，我将用“操作者”的数学语言来讨论所讨论的多尺度模型，并通过使用操作者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操作者(以及他们的模型对应者)由于不同尺度的相互作用而产生的主导行为。该提案还将在齐化理论方面取得进展，齐化理论是一种数学方法，用于寻找多尺度系统的有效模型。该提案将利用均化理论的新进展来分析一大类为超材料建模的微分方程式。

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.