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

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.