What is Topology?

Topology is a branch of pure mathematics, related to Geometry. It unfortunately shares the name of an unrelated topic more commonly known as topography, that is, the study of the shape and nature of terrain (and sometimes more precisely, how it changes over time), but in our usage here, topology is not at all about terrain.

It is, however, about the shape of things, and in this way, it is a kind of geometry. The kinds of objects we study, however, are often fairly removed from our ordinary experience. Some of these things are four-dimensional, or higher-dimensional, and as such cannot truly exist in our everyday world. If some higher-dimensional being in a higher dimensional universe existed, they might be able to see these and the most difficult questions in this subject might be quite plain and commonplace to such a person. But in our usual three-dimensional world, we would have to turn to mathematics to understand these shapes.

Topology is the kind of geometry one would do if one were rather ignorant of the intricacies of the shape. It ignores issues like size and angle, which usually pervade our ordinary understanding of geometry. For instance, in high-school geometry, we examine squares, rectangles, parallelograms, trapezoids, and so on, giving them names and measuring their sides and angles. But in topology, we neglect the differences that have to do with distance, and so a square and a rectangle are topologically considered to be the same shape, and we disregard angle, so a rectangle and a parallelogram are considered to be the same shape. In fact, any quadrilateral is topologically the same.

Even if we shrink one of the sides to zero length, so that we have a triangle, we still consider this the same. Or if we introduce a bend so that we have more sides, this is still topologically the same.

So are all shapes the same? No. If we break open one of the sides and stretch it into a line segment, this is a different shape. The point is that this shape is *connected* differently. Topologically, a line segment and a square are different.

These objects are examples of curves in the plane. In some sense they are two dimensional since we draw them on a plane. In another sense, however, they are one dimensional since a creature living inside them would be only aware of one direction of motion. We might say that such shapes have extrinsic dimension 2 but intrinsic dimension 1.

To draw examples of shapes that have intrinsic dimension 2, it is best to look in our three-dimensional space. Imagine a basketball. The surface of the basketball is a shape of intrinsic dimension 2, as long as we agree that the basketball consists of the rubbery material (which we imagine is infinitely thin) and not the empty space inside. Topologically, we consider it to be the same shape even if we sit on it and thereby distort the shape, or partially deflate it so that it has all sorts of funny wobbles on it.

But imagine the surface of an inner tube. This is topologically different.

The notion of shapes like these can be generalized to higher dimensions, and such a shape is called a manifold. These manifolds are unrelated to the part you have in your car, and it's not even a very appropriate name. The term "manifold" is really the concept of "surface" but extended so that the dimension could be arbitrarily high.

The dimension we are talking about is often the intrinsic dimension, not the extrinsic dimension. Thus, a curve is a one-dimensional manifold, and a surface is a two-dimensional manifold.

One important question in topology is to classify manifolds. That is, write down a list of all manifolds, and provide a way of examining any manifold and recognizing which one on the list it is. Remember that these manifolds would not be drawn on a piece of paper, since they are quite high-dimensional. Rather they are described in funny ways, using mathematics.

The question of classifying manifolds is an unsolved one. The story is completely understood in dimensions zero, one, and two. The story is fairly satisfactorily understood in dimensions five and higher. But for manifolds of dimension three and four, we are largely in the dark.

If this seems paradoxical to you, it is. After all, in dimensions zero, one, and two, there is not much that can happen, and besides, we as three-dimensional creatures can visualize much of it easily. You might think that dimension three would be fine, too, but remember, the kind of dimension we are discussing is intrinsic dimension. To visualize it we would have to live in at least four dimensions. It turns out, however, that much of this visualization is irrelevant in the final analysis anyway, since you still need to mathematically prove your results, which is more demanding than simply drawing a picture and staring at it. But at the very least, the manifolds can become more and more strange as you increase in dimension. So the higher the dimension, the more difficult the situation might be.

But as we increase in dimension past dimension 5, we are suddenly able to understand the situation again. This is the paradox. The resolution to the paradox is that from dimension 5 and up, there is more room to do more fancy kinds of manipulation. There's a pretty neat move called the "Whitney Trick" that allows you to move complicated objects past each other and separate them out into understandable pieces.

This leaves dimensions 3 and 4. My research is in four-dimensional manifolds. We actually live in a four-dimensional manifold, if you count time, and if you disregard string theorists who wonder if we live in dimension 10 or so.

This does not help make this subject more applicable. But it does allow techniques that physicists have been working on for many years. On the early 1980s Simon Donaldson studied objects called "instantons" on four-dimensional manifolds and revolutionized our understanding of four-dimensional manifolds. Instantons are the sorts of things that physicists have been talking about since the 1970s in relation to the theory of subatomic particles and forces that they experience that are normally influential in our lives only to the extent that they hold the nucleus of the atom together.

This, then, is an application of physics to mathematics, instead of the other way around! Later, in 1994, breakthroughs in supersymmetry due to Nathan Seiberg and Ed Witten led to more techniques, and my research investigates what can be done with these new techniques.

What relevance does this have to our world? At this stage, the most important role this research plays is one of pure understanding. As a part of theoretical mathematics, we should strive to understand everything there is to understand. The more we understand, the more we will be able to deal with challenges that face us in the future. If we were to only focus on those problems which have direct application, we not only risk being able to address future problems, but we may end up looking at the problems we want to solve in the wrong way. The course of human history has shown that many great leaps of understanding come from a source not anticipated, and that basic research often bears fruit within perhaps a hundred years.

Of course, there is so much to study. Surely by blindly asking all questions we will be diluting our efforts too much. This is true, but that is not what theoretical mathematics does. Instead, it tries to examine those things that are "general", whose understanding will encompass many different areas of understanding at once. If we imagine a pre-mathematical being dealing with addition for the first time, we might imagine the creature making the discovery that two apples added to three apples make five apples, then having to make the discovery again when dealing with oranges or rocks or tennis shoes. But when the creature realizes there is a general truth that 2+3=5, the creature has made the first step in mathematics by generalizing this observation and talking in "abstract" concepts. This more "abstract" concept is more removed from the world since one cannot eat or throw or wear the concept "2", but it is at once more far-reaching in understanding what is true in the world, since it can apply to new objects that were previously unknown.

This is the aim of theoretical mathematics. Not to simply play games with objects that are irrelevant and imaginary, but to deepen our understanding of everything we can imagine, with the idea that this is the starting point in becoming a more enlightened species.

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