What is Algebraic Topology?
Thursday, October 25th, 2012 | Author: Konrad Voelkel
This short article is intended to be read by non-mathematicians who don't quite remember what the term "matrix" or "polynomial" refers to. I'll try to give you an intuitive idea of what a PhD student working on "Algebraic Topology" studies nonetheless.
First of all, studying or researching mathematics is not about remembering formula or calculating some really large numbers. That is a part of mathematics, but it's not what drives it, it is merely a tool that is more and more handed to computer systems.
So, what is mathematics instead? I am not competent to give an answer (and there have been many many different answers to that question in the past) but I can explain you my view on pure mathematics:
Pure Mathematics is the study (by any means) of the statements of which truth can be derived syntactically, i.e. by some computational process. (Yes, I'm quite a syntactic thinker).
To give you a concrete example (which you might as well skip), think of a table and playing cards on it. This is a collection of objects, and you can put cards aside other cards horizontally as well as vertically. If you have four cards, you can put pairs of two of them aside horizontally first, and then put the pairs together vertically; or you first put them aside vertically, and then align the pairs horizontally. The end-result is the same (provided you didn't mix up the order of the cards). This is an example of a mathematical structure: a collection of objects (an object is some arrangement of multiple cards, or just a single card) with two operations on it (putting them aside horizontally and vertically). Now the mathematicians job is to study this structure.
Enhancing on the example, let's stretch our imagination a little bit more, and think of any empty space between two cards as an "object", the "empty object". Then putting something aside the empty object is possible, you just leave some space, more or less. We don't want to distinguish, however, between more or less larger empty space, so we consider two cards with some empty space between them to be "equal" to the same arrangement of two cards with less or more empty space between them (or even none!). In math-speak, we just added a neutral element to our structure, the "empty object". For a mathematician, we now have the structure of a set (the objects) with two monoidal operations (arranging vertically and horizontally) that share a neutral element (the empty space) and distribute over each other (doesn't matter whether you put four cards aside horizontally first, then vertically, or the other way around). By a lucky accident, in this structure one can prove (and it is one of my favorite proofs in mathematics) that the two operations are to be considered "equal" and "commutative" in the mathematical sense. This is the same sense in which we considered more or less empty space to be the same "empty object". The word "commutative" means, we have to consider the card arrangement "A, B" to be equal to "B, A". The good thing about this mathematical reasoning is, whenever we encounter a structure like this, we can apply the theorems we proved. The proofs are just like computer programs, but we are used to a very abbreviated language which is unreadable to today's computers.
Now to the "Algebraic Topology" in scare quotes. The parts "algebraic" and "topology" ought to be described individually, and then the whole means more-or-less: "Algebra applied to problems in Topology, and Topology applied to problems in Algebra". Historically, it was definitely the application of Algebra to Topology, but nowadays we see a lot of interesting stuff in the other direction, too.
Algebra, as a sub-field of mathematics, came out of Number Theory, the study of properties of numbers. Most of the time, one studies solutions to equations of the kind "x²+2x+4=0", but not of the kind "f'(x)=3x²". Many of these equations have certain symmetries, for example "x²-2x+4=0" is equivalent to "(x+2)(x-2)=0", and the solutions are -2 and +2. The symmetry is that one solution is the inverse to the other, like a mirror. More complicated equations have more complicated symmetries, of course.
Topology came out of Geometry, the old sub-field of Mathematics already researched by the Greeks (and probably much earlier as well). Topology is often described as rubber-band geometry, since you don't care so much about the idea of "this thing here is one meter fifty away from that other thing" but more about "this thing is nearer to the other thing than to something else", so more of a vague idea of proximity instead of precise measurements. This blurred vision seems harder than precise analysis of a situation, but sometimes the full information is just too much and it is easier to take blurred glasses first. It also happens often that we can derive geometric information from topological information, as in "these two objects can not be joined in a straight path of one meter length, since there is a big, big hole between them".
To come to an end, I want to explain heuristically an old Topologist's joke:
A doughnut and a coffee cup are the same.
Why should a Topologist think that? Well, an Algebraic Topologist can resort to the following: Model the doughnut and the coffee cup with small triangles made of cardboard, glued at the edges (with any angle allowed). If you take many, many such triangles, build a really large model and then walk away to look at it from a distance then the models should look like the real thing. Okay, I didn't mean "model it" when I said that, I just wanted you to imagine it (that's what we mathematicians do all the time, and then sometimes we still have to build models, if this is possible). Okay, now you just count the number of vertices (V), edges (E) and faces (F) of your models (all of them!). The funny thing is, the number V-E+F does not depend on how you build your model. It doesn't even differ in the two models! I can tell you directly, it will be just 0 in both cases. If you do the same with a sphere (say, an orange), then the number you get out of V-E+F is not 0, it will be 2 instead. We call this number the Euler characteristics, and in some way it contains an answer to the question "How many holes has the object". An orange doesn't contain any holes, but a doughnut and a coffee cup do contain a hole and a handle, which is arguably some kind of hole. The formula V-E+F and the resulting number lie in the domain of Algebra, and the original object is geometric in nature, but we look at it with Topologist's eyes.
In contemporary physics, one asks question very similar to "How many holes has our universe", and Algebraic Topology is really needed there, and there is still a lot of research going on. A PhD student in Algebraic Topology will most likely focus on a very specific problem which is not directly applied somewhere in physics or even the real world. To explain exactly what she/he does will be a lengthy task, in general.
Let me hear your opinions in the comments!