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，你就得到了四个小正方形。从一棵树上剪下一根树枝，那根树枝本身看起来就像一棵小树。取以0结尾的整数（10,20,30，…），它看起来像是所有整数（1,2,3，…）的展开版本。这些都是自相似的例子，一个物体可以被切割成碎片，看起来就像它自己的小复制品。换句话说，我们有一个物体看起来就像自己的几个副本粘在一起。在更复杂的情况下，可能存在两个或更多对象：例如，一个对象X可能看起来像三个X的副本粘在第二个对象Y的副本上，而Y可能看起来像两个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 + 4Y）。自相似性出现在数学的各个领域，尽管要指出不同形式的自相似性之间的确切联系并不总是那么容易。例子不仅包括众所周知的分形，还包括更普通的物体，如圆圈、圆柱体和球。我提出了一个广泛而深远的研究计划，以建立一个自相似的一般理论，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。这样一个计划的意义是什么？数学上最大的进步是当看似不相关的现象被理解为一个单一的、普遍的现象的实例时取得的，这些现象往往是在看似截然不同的领域观察到的。（例如，牛顿意识到板球的运动和行星围绕太阳的轨道是由同样的力——重力——控制的，因此也由同样的方程控制。）这种不同思想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。更具体地说，我相信我能找到新的不变量。不变量使您能够区分两个事物。例如，你总是能分辨出一件毛衣和一条裤子，即使你是在黑暗中穿着，而且你的衣服是由相同的宽松材料制成的：毛衣多了一个洞。这里的不变量是洞的数量；由于这两个项目的孔数不同，所以可以区分。现在，自相似物体的一些最引人注目的例子是分形，它是由细丝和缝隙组成的无限复杂的网络。由于大多数分形有无限多的孔，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。我相信我可以定义一个。它是通过将通常的几何类型的自相似性转化为代数类型的自相似性（如上面的联立方程）而产生的，并且是被称为欧拉特征的不变量的扩展。我带着用简单实用的方式描述不寻常的、复杂的结构的经验来到这个项目。这正是这里所需要的。像分形这样的对象可能看起来令人生畏地复杂，但我已经开始展示它们可以用这样一种方式来描述，使困难的问题变得容易接近。为了执行这个计划，我将需要其他领域的专家的投入。这将通过有针对性地访问专家和继续在不同地点为不同的听众举办大量讨论会来实现，从而实现思想的交流。（例如，这个月，我将在爱丁堡为代数学家开一个研讨会，在利物浦为复杂动力学家开另一个研讨会。）通过发展现有的合作，发起新的合作，并利用自己的专业知识，我计划将我们对自相似性的理解转化为一种非常实用的工具。

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.