Research Rollercoaster

I guess its been a while since I've done a "real" post. Sorry about that.
I've been busy, I've not had the energy to write when not busy, and I've not had anything to say when not busy and not too lethargic to write.

Now I'm still pretty busy, and not too energetic, but I do have some things to say, so here goes:

Most of my life right now is focused on trying to get all of the research done for my math dissertation. As I mentioned before, I'm not taking classes or teaching classes, and my role with Navs is greatly reduced. I'm getting better at being self-disciplined, organizing my time, and putting in time on research despite the absence of structure and a dearth or external motivation, but I'm still not where I want to be on those issues.

Where I'm really continuing to have a difficult time is with the research process itself now. Its a funny business. Trying to prove something that no-one has proved before is no picnic, but what really get to you is not knowing exactly what you are trying to prove. You have to try to get your head around the area you are looking at and try to get a sense for what might be true (and what might not be true - sometimes disproving something is as nice if not nicer than proving it).

Then once you have a strong suspicion that something might be true, you have to do a lit search to make sure that someone else hasn't already found out that what you think is true is actually true. If they have, its back to the drawing board. If they havn't, you see what similar things they have done, and whether anything they used to prove what they proved applies to your idea.

Usually, there are a few related papers you have to then either download, or get from the library and at least skim for content. Sometimes there are ideas that you can try to adapt, but usually, you just get a few ideas about why your initial suspicions are more or less likely to be true, and some vague ideas of what to try.

Then the hard part starts. You look at examples, try to prove special cases or preliminary related ideas and hope that things start to add up to an idea of how to prove what you REALLY want to prove. Often, ideas fizzle, or examples don't make sense, or you can't quite get the small preliminary results to work out like you think they should. When it gets really bad, you start running out of ideas and your brain goes numb and you spend an hour or two just staring blankly at a vast collection of papers, sketches, and random scrawlings on scratch paper hoping and praying that some coherent thought about how to proceed will come to mind.

After a few days (or weeks) of this, you usually end up scrapping the whole idea, or at least setting it aside temporarily, and you move on to another idea. Knowing how long to persist before doing this is one of the most difficult things to discern.

After repeating this cycle some indeterminate number of times, you sometimes actually succeed in proving at least part of what you set out to do. When this happens, you have to put together all of the details of your idea and make sure that you didn't make some drastic error in logic that invalidates your entire argument. If there is an error, you see if there is a way to fix it, and if not, its back to the drawing board. If there are no "unfixable" errors, then you can formally write up your result and show it to your advisor. Then you reach another fork in the road where you have to decide which, if any, of all the possible ways to extend this result are worth spending time working on, and the process begins again.

So that's a little window into the world of my research.

On a brighter note, after a couple of weeks of really struggling, I was able to prove most of a small result this afternoon, so I am now at the point described above where I have to check to see if there are any errors in my thinking, and hopefully soon afterwards write it up and start thinking about how to go on from there.

Thanks for listening...