Wednesday, August 23, 2006

Welcome to Antopology

What is this blog all about?

I plan to write a series of blog entries which are intended to be an introduction to topology and geometry, aimed at people with very basic mathematical backgrounds. To the cognisenti, I add the disclaimer that the eventual target of this blog is a user friendly explanation of concepts in differential, algebraic, geometric, and combinatorial topology,which are much more my cup of tea, but I shall start with the basics of point-set topology. To all the rest, have no fear, I plan to take my time and very slowly work through the basic concepts before reaching my goal. Now, let’s proceed to the show.

What is topology all about?

Before we start going into the deeper math, I’ll spend some time trying to show some of the basic questions asked in topology. Topology deals with the study and classification of abstract spaces. Roughly, topology deals with properties of spaces, which can’t be changed by stretching the space or squeezing it. This is best illustrated by the well known mathematical joke that to a topologist a coffee mug and a bagel are one and the same, since they both have one hole (I said that this was a well known joke, never said it was a good one). To a topologist, the fact that the mug has one hole, its handle, and the bagel has also only one hole, means that they are the same space topologically speaking.

Life as an Ant on a Line

The abstraction above is one of the hardest concepts to grasp, and therefore we will start with more concrete examples we can visualize or simply draw. Take a piece of paper and draw a line on it, which might curve around, but doesn’t intersect itself at any point. We want to study properties of this line. In order to dothis, we shall take the ant’s eye view. Imagine an ant living on this line, being unable to leave it or even look outside this line. It can go forwards or backwards, but that’s it. Now let’s give our ant a little car, and send her on a trip on the line. For the sake of our discussion, we refer to this line as the road from now on.

We can draw the road in many ways. We can make it straight and long, or straight and short. We can curve it nicely, or we can draw a very squiggly line. We can even give it“corners”, meaning we draw it with right angle turns in it, or any other angles we wish. As long as we never intersect ourselves while we draw, or lift our pencil from the paper, we are dealing with the same topological space. All we care about in topology is“relations between points”, meaning which point is near to which other point. Roughly, if our ant is standing at point A, and point B is“next” to point A (this is a gross over-simplification, since on our line there are an infinite number of points between A and B) any manipulation of the line which leaves its topology unchanged would have to leave B “next” to A.

Intuitively, we are much more geometric creatures, as is our ant. While topologically we can mess the road up as much as we want (again, as long as we don’t lift our pen or intersect ourselves), the ant will feel differently about each change in the road. Let us imagine that our ant lives on one end of the line, and frequents a pub at the other end. As long as the line is straight, the ant can get home even when it is drunk, since there is nowhere she can fall of the line. Now change the road to a curved road. The ant would have to drive much more carefully, since the road curves and she might have an accident along the way. Now we can do something nastier, by drawing the road with a right angle in it. The topology of this road is unchanged, but almost any attempt to drive on it will end in an accident. When our ant reaches the right angle, she’ll drive off the edge of her world. The poor ant has to come to a complete stop, turn the car to the right, and resume driving.

Basically, we already have some questions we can try to answer. How many topologically different roads can we have, and how do we know that those are all our options? What happens if we change the world that the ant lives on, to one that is two-dimensional (meaning that to the ant her world looks like the surface of a plane, much as the globe looks to humans, as has already been explored by E. Abbot in “Flatland”. See “The Annotated Flatland: A Romance of Many Dimensions” by E. Abbot and I. Stewart, 2001.) How many such worlds exist? How do we tell them apart?

The aim of topology is basically the classification of spaces. We will be exploring the techniques for this classification in detail.

The more established math blogger MarkCC of Good Math, Bad Math will be tag teaming with me on this subject.


Blogger punit said...

Just discovered your blog via GM/BM. Will be sure to follow as topology is something I have tried to understand a couple of times in the past but lacked either motivation or lucid enough explanation.

4:35 PM  

Post a Comment

<< Home