Wednesday, February 10, 2010

Approximating Functions of Shapes

Every now and then, I end up writing about my research at a not-too-technical level. Here's my latest attempt:

The branch of mathematics known as topology is concerned with the study of shapes. Whereas shapes in geometry are fairly rigid objects, shapes in topology are much more flexible. If one flexible shape can be flexed and twisted and not-too-drastically mangled into another flexible shape, topology deems them to be the same. It becomes much more difficult, then, to tell if two shapes are different. The question of topology is, ``Given two flexible shapes, how can we tell if they are the same?''

In geometry, much time is spent studying shapes like triangles and quadrilaterals. If you draw one triangle on one sheet of paper, and another triangle on another sheet of paper, then there is an easy way to tell if those two triangles are the same. Namely, shift the papers around, possibly flip one or the other over, rotate them, and see if you can get the triangles to line up. If you can, then the triangles are ``the same'' (congruent). If you can't line up the triangles (and have put forth enough effort), then the triangles are not the same. Similarly, you could draw a square on one paper, a triangle on the other, and try to line those shapes up. Of course, you will never succeed, because the two shapes will always have a different number of sides.

In topology, we might consider the same scenario of triangles and squares, but instead of using ordinary paper, we would draw our shapes on a sheet of flexible rubber (an inflatable balloon, say). Indeed, topology is often referred to as ``rubber-sheet geometry.'' Now that the surface is flexible, by pulling it in the appropriate directions one can get the square to look like a triangle. Or either shape to look like a circle. To a topologist, this means that all of these shapes are ``the same'' (homeomorphic). The topologist still has some rules in place, though. You are not allowed to cut holes in the balloon, or glue parts of the balloon together.

With all of the additional flexibility, we have a harder time telling two shapes apart. We can no longer rely on the number of sides, or the lengths of sides. Other methods must be found.

It turns out that a good way to get a sense of how similar two shapes are is to understand how the shapes are related to yet another shape. Suppose we have two flexible shapes, called Shape 1 and Shape 2. One might try to obtain information about Shape 1 in isolation, and compare that to information obtained about Shape 2 in isolation. However, as the shapes are flexible, there's only so much we can tell by looking at each shape independently. Instead, we introduce another flexible shape, Shape 3. Generally Shape 3 is a shape that has been much studied, like a circle. Now we ask, ``How many ways can Shape 3 be put into Shape 1?'' and, ``How many ways can Shape 3 be put into Shape 2?'' We call a single way of putting Shape 3 into Shape 1, if it is reasonably well-behaved, an ``embedding.'' If the embeddings of Shape 3 into either of the two original Shapes are similar, we might decide that the two shapes themselves are fairly similar. We can then replace Shape 3 with Shape 4, repeat the question, and get an even finer appreciation for how similar Shapes 1 and 2 are.

The entire collection of embeddings of Shape 3 into Shape 1 can all be gathered up and reasonably interpreted as making up yet another shape. However, it can be a fairly difficult shape to get a good sense of. Instead, one might ask if there is a way to approximate this shape. Perhaps an initial approximation can be attained, and then that approximation can be refined, and refined further, giving a better and better understanding of the complicated shape we originally sought. Using geometry as an example, one might approximate a circle by a square, and then refine the approximation to an octagon, and carry on the process of adding more sides to the approximating figure. This was the method used, historically, for calculating the area of a circle.

But the idea of approximating the shape of the collection of embeddings can be further generalized. If we have a strong handle on a particular shape, say S, then our first step in understanding any new shape that comes along might be to find all the ways S can be put into the new shape. We have, then, described a function. Given a shape, X, we can ask for the shape of ways that S can be embedded in X. We might denote this function by E(X), and call it the embedding function for S. Now instead of understanding a particular value, E(X), of this function, we might ask about understanding the function E as a whole. While the function itself is likely complicated, we can try to approximate it, as before.

In my research, I study a family of embeddings functions. The functions are based on the ``Euclidean spaces'' of various dimensions: a point, a line, a plane, three-dimensional space, and on to higher dimensions. From a topologist's perspective, these are the easiest shapes there are, but they can be used to build up nearly all of the other shapes worth studying, so are of fundamental importance. Instead of just thinking about one such space as the basis for my function, I might make a collection of them; say, 3 lines and 2 planes. I only ask about E(X) when X is, itself, a single Euclidean space.

It might seem that by restricting my study to particularly nice shapes S and X, and particularly nice ways to put S into X, I have eliminated any hope of doing anything useful. However, this is not the case. An important class of shapes, called manifolds, are built up from Euclidean spaces by gluing them together. By obtaining a particularly nice description of embedding functions using my restricted shapes S, there are known ways to assemble them into the embedding functions for more general manifolds.

Previous work has found nice ways to describe embedding functions for some of the cases I am considering. If the base space, S, is a finite collection of points, then spaces of embeddings are what are known as ``Configuration Spaces,'' and these detect one type of behavior for embeddings. Alternatively, if S has just a single space, but of any dimension, the spaces of embeddings are detecting a different sort of behavior. Nice descriptions for the embedding functions in both of these cases are known. It has been my task to unify the two descriptions, allowing for an understanding of embedding functions when the known space, S, is a collection of any number of Euclidean spaces.

Update 20100225: There's a somewhat different version of this on my other blog.

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