Self-similarity: recursively definable objects in topology, analysis, category theory and algebra

把一个正方形水平切成两半，然后垂直切成两半，你会得到四个小正方形。从树上砍下一根分支，那根分支看起来就像一棵小树。取以零结尾的整数（10，20，30，...），这看起来像是所有整数（1，2，3，...）的展开版本。这些都是自相似性的例子，一个物体可以被切割成看起来像它自己的小副本。换句话说，我们有一个对象，看起来像几个自己的副本粘在一起。在更复杂的情况下，可能有两个或更多的对象：例如，一个对象X可能看起来像是三个X的副本粘在第二个对象Y的一个副本上，而Y可能看起来像是两个X的副本粘在四个Y的副本上。这就像学校数学中的联立方程（这里，X = 3X + Y和Y = 2X +4 Y）。自相似性出现在数学的不同部分，尽管人们并不总是很容易指出不同形式之间的确切联系。例子不仅包括众所周知的分形，还包括更平凡的物体，如圆，圆柱和球。我提出了一个广泛而深远的研究计划，以建立一个一般的自相似理论，并将其应用于几个领域，包括代数，几何，分析和理论计算机科学。这样一个方案的意义何在？数学上最伟大的进步是在那些看似不相关的现象被理解为一个单一的普遍现象的例子时，这些现象往往在看似南辕北辙的领域被观察到。(For例如，牛顿意识到板球的运动和行星绕太阳的轨道是由同一个力--重力--控制的，因此也是由同一个方程控制的。）这种不同观点的统一导致了极大的简化，通过类比提出了新的结果，并澄清了思维。我的目标是统一不同类型的自相似性。更具体地说，我相信我可以找到新的不变量。一个不变量是让你区分两个事物的东西。例如，你总是可以分辨出一件套头衫和一条裤子，即使你是在黑暗中穿着，你的衣服是由相同的宽松材料制成的：套头衫多了一个洞。这里的不变量是孔的数量;由于两个项目有不同的孔数，它们可以区分。现在，一些自相似物体最引人注目的例子是分形，它是由细丝和间隙组成的无限复杂的网络。由于大多数分形都有无限多的洞，这个不变量对于区分分形几乎是无用的。为了区分分形，我们需要一个更微妙的不变量。我想我可以定义一个。它是通过将通常的几何自相似性转化为代数自相似性（就像上面的联立方程组）而产生的，并且是称为欧拉特征线的不变量的扩展。我来这个项目的经验是找到以简单实用的方式描述不寻常的复杂结构的方法。这正是这里所需要的。像分形这样的物体可能看起来复杂得令人生畏，但我已经开始表明，它们可以用一种方式来描述，使困难的问题变得可以解决。为了实施这一计划，我需要其他领域专家的投入。为此，将有针对性地访问专家，并继续在不同地点为不同的受众举办大量研讨会，以交流想法。(This例如，一个月前，我在爱丁堡给代数学家开了一个研讨会，在利物浦给复杂动力学家开了另一个研讨会。）通过发展现有的合作，发起新的合作，并利用我自己的专业知识，我计划改变我们对自相似性的理解，并将其转化为一个非常实用的工具。

Cut a square in half once horizontally, then once vertically, and you get four small squares. Cut a branch off a tree, and that branch looks something like a small tree itself. Take the whole numbers ending in zero (10, 20, 30, ...), and that looks like a spread-out version of all the whole numbers (1, 2, 3, ...).These are all examples of self-similarity , where an object can be cut up in such a way that the pieces look like smaller copies of itself. Put another way, we have an object that looks like several copies of itself glued together. In more complicated situations there may be two or more objects: for instance, one object X may look like three copies of X stuck to one copy of a second object Y, and Y may look like two copies of X stuck to four copies of Y. This is like simultaneous equations from school mathematics (here, X = 3X + Y and Y = 2X + 4Y).Self-similarity occurs in remarkably diverse parts of mathematics, although it is not always easy to put one's finger on the exact connection between different forms of it. Examples include not only the well-known fractals , but also more mundane objects such as circles, cylinders and balls. I propose a broad and far-reaching research programme to set up a general theory of self-similarity and to apply it in several areas, including algebra, geometry, analysis, and theoretical computer science. What is the point of such a programme? The greatest advances in mathematics are made when apparently unrelated phenomena, often observed in areas that seem to be poles apart, are understood to be instances of a single, general phenomenon. (For example, Newton realized that the motion of a cricket ball and the orbits of the planets around the sun are governed by the same force - gravity - and therefore by the same equations.) This unification of disparate ideas leads to great simplification, suggests new results by analogy, and clarifies thinking. I aim to unify the different types of self-similarity.More specifically, I believe I can find new invariants . An invariant is what enables you to tell two things apart. For instance, you can always tell a jumper from a pair of trousers, even if you are dressing in the dark and your clothes are made of identical, baggy material: a jumper has one more hole. Here the invariant is the number of holes; since the two items have different numbers of holes, they can be distinguished. Now, some of the most striking examples of self-similar objects are fractals, which are infinitely intricate webs of filaments and gaps. Since most fractals have infinitely many holes, this invariant is almost useless for telling fractals apart. To distinguish between fractals we need a much more subtle invariant. I believe I can define one. It comes about by transforming self-similarity of the usual geometric kind into self-similarity of an algebraic kind (like the simultaneous equations above), and is an extension of the invariant known as Euler characteristic.I come to this project with experience in finding ways of describing unusual, complicated structures in a simple, practical way. This is exactly what is needed here. Objects such as fractals may appear forbiddingly complex, but I have begun to show that they can be described in such a way that difficult problems become approachable. To carry out this programme I will need the input of specialists in other fields. This will be achieved through targeted visits to experts and through continuing to give a large number of seminars to varied audiences at different locations, resulting in cross-fertilization of ideas. (This month, for instance, I am giving one seminar to algebraists in Edinburgh and another to complex dynamicists in Liverpool.) Through a combination of developing existing collaborations, initiating new ones, and using my own expertise, I plan to transform our understanding of self-similarity and turn it into a tool of great practical use.