Here is a picture of a few of the books I am reading at the moment, all on the subject of “group theory” which is the area in which I am doing research for my MSc. thesis. Group theory is an area of mathematics, specifically of abstract algebra, and it is a particularly wonderful area indeed. Groups often naturally arise when dealing with the symmetries of other mathematical objects and so group theory is sometimes dubbed the “mathematics of symmetry”. James Newman summarised group theory as follows:
The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing.
This is, I think, probably more accurate than some would like to admit.
Group theory is going to be an integral part of my life this year, and probably for many years to come, so I think I will write several posts on the topic over the next while, teasing out my thoughts on the matter. I think it would be interesting to try to explain why I love group theory while avoiding, as much as possible, technical details. Is it possible, I wonder, the communicate my passion for the area to somebody who has no formal background in mathematics?
Well, why do I love group theory? The first reason, and the one that I shall offer today, is the simple elegance of the area. On the one hand group theory is a rather abstract area, one that seems to bear very little resemblance to anything concrete, to anything real. Still, as a mathematical system it is quite simple. It takes only a few minutes to explain the basics of group theory and yet out of such simple beginnings a rich, varied and surprising structure can be developed. Group theory has that sort of elegance where, as is also often the case with number theory, simple questions can have very difficult answers and seemingly unrelated things can turn out to be related in the most unexpected ways. It is also one of those areas where simple and elegant proofs can often be found, there is none of the long winded proofery of other areas.
So, if I were to offer one reason for why I chose group theory as the area I would like to study in, I would have to say that it is the simple elegance of the theory that initially draws me.
Now, if only I could get through those books…