Research and the the way of the Academic

The Beauty of Mathematical Research

When you hear the word research what is it you think of? People in white coats? Beakers and chemicals and microscopes? That's what I remember thinking of when I was young. That would be medical research or possibly chemistry or biology. I've grown up a lot now so I know that there's a lot more to research than this. Even for those trapped in such experiments there's much more to the job. Some things to consider:

These questions are all fairly simple questions to understand. There are many more that we really don't consider when we think about research. Now that I'm in research I have a lot of these sorts of questions to deal with. Not only to figure them out but also to answer them. Before I continue showing the hardships I'd like to focus on the benefits of full time research.

When you have your own project to work on it becomes your baby. You care for it. You feed it, clean it when it gets messy and when it misbehaves you get frustrated. You ignore it for a while but it's always there and it does something cool and you back to being nice to it. The love is always there. Of course, it requires the right sort of person to do this sort of thing. The sort of person I am is a mathematician.

Mathematics is a form of poetry. With poetry, you take something in the world and try to use your art to elegantly describe it. Mathematics, however, transcends the world. What we see in the world in physics, sciences, engineering, economics and art is abstracted and given a pure and elegant form. An example of this is the well known theorem of Pythagoras, the sum of the squares of the shorter sides of a right-angled triangle is equal to the square of the length of remaining side, best known to the world as a2+b2=c2. Any time you have a right triangle you can use this unhesitatingly.

The inherent beauty of mathematics lies in elegant abstraction. A painting will take the beauty of form of something in the world and give it meaning of its own beyond where it came from. The beauty of the painting lies both in the abstraction and by the means chosen for presentation. The choice of emphasis and subject to catch the attention of the audience. Our art has the advantage that it may abstract on itself many times and still be beautiful.

The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

-- G. H. Hardy. A Mathematician's Apology (London 1941).

So hopefully you now understand that there is much motivation to do research. To add to the beauty in the world. To add to our understanding and perhaps our abilities. There is exhilaration when you unlock the mysteries of those who came before you. There are times in research when you know exactly where to go. You understand the problem and how to attack it. You know what needs to be done and you can feel you're near the end. It might last an hour, a day, a month or several depending on the problem. You want to find the answer and unlock its secrets.

So often in research you look at the mountain, its peak in the clouds. You climb. Sometimes it's easy, sometimes hard and sometimes you forget to aim for the top and can only think about the next place to stand. Sometimes the way becomes too steep and you have to try another way. Then you reach the top. You can see the whole mountain. The intricate details of the climb are forgotten and you view the gentle slope on the other side. This is both frustrating and rewarding because you struggled to get where you are but many results that you fought to find are now elegantly revealed and to return to the peak you need only remember how to start the easy route. Of course sometimes the rest of the mountain is steeper but then you just feel accomplished.

The ways of research are not always so rewarding. The baby screams late at night. Many a night on a problem, particularly when you're close, you are trying to sleep but cannot because you have an idea. They work as often as not but you're tired and want sleep. It's not worth trying. The baby screams loudly. There are always many difficulties in research, however rewarding it might be. As studious and clever as you may be, it's creativity that leads to progress. You might work a week and get nowhere. You might pick up an old problem and solve it in a heartbeat or pick up an old solution and puzzle over it for a day because you forgot how you solved it. Other difficulties encountered are not knowing how to approach a problem or not knowing what your next problem is. There is also the fact that there's a reason a problem remains unsolved.

The bane of the mathematical researcher is a pair of questions. They should never be asked of a mathematician in the same way that one should never attempt to use a painting for construction (they don't make good walls). How is this useful? What does this mean in the real world? It is difficult to explain how things are useful because the knowledge trickles down through many overlapping fields before it becomes easy to explain its use. Asking what something means in the real world is just plain rude. You might as well ask a sculptor to make his marble statue give a speech on management. Even asking about research is a dangerous thing. The reason for this is that just as many new parents delight in showing off their baby, so too, the researcher wishes to share. Unfortunately, whereas babies are well understood, research by its nature is open only to those with knowledge in the specific field.

Old Research

My original thesis research was on three topics:

  1. The W-spectrum of bounded linear operators between Banach spaces.
  2. Generalised spectra on Banach algebras.
  3. Applications of Banach spaces to differential equations.

The W-spectrum was an idea from my supervisor that led to some very beautiful theory. It could lead to a whole new world or just some nice toys for people but first I need to find an application for the theory. I'm hoping to reformulate some old definitions of differential equations so that they work with the W-spectrum. It may not be possible. The theory involved is very beautiful. I may post it here in future. I haven't the skill to complete it to make it publishable so maybe someone will find it useful.

About a decade ago, papers were published on what were called regularities. These regularities lead to generalised forms of the spectrum. However, with the advantage of some well established theory and a fresh point of view I have seen what is actually used in the theory and reformed the theory from this and it has lead to a very elegant generalisation of regularities. I may say this because it is a simpler definition, easier to work with, fully encompasses the work already done and generalises the results. Modesty aside, I am very proud of myself. This is my thesis. It's a bigger topic than this paragraph demonstrates.

I have always liked differential equations and in operator theory I can do some very entertaining things with them. I'm hoping to do something new but that can't be planned. Only time will tell if it works. Time has not been promising on this topic.