Well, it's been a long time. So much has happened, but it is impossible to record it all here at once. It isn't healthy journal practice, anyway. Three advisors later, I think I finally really have found something worth doing and at the same time, not so over my head that it does not require continual pestering of the advisor (which would have been fine, were it not for those previous advisors being so reclusive and inaccessible, and in one case, I swear they had taken some form of sadistic pleasure in making me feel like I am stupid). I was starting to think that the only thing consistent in my bad advisor relationships was me! (SO CLASSIC, at least when applied to love. Speaking of which, though, wow, crazy things have happened there since the last update, but currently things are going well there also. But another time.)
Anyway, I've finally managed to reawaken my inner programmer that lay dormant for so long ever since starting grad school, and have moved in a much more applied direction--I actually get to compute things. Thank goodness, for I can actually see all these otherwise hopelessly abstract concepts. But the beautiful thing is, it is an excellent application of a lot of the theory I learned in real analysis so many years ago. All that weak solution/dual space/Lp space b0llocks actually is useful in real applications, aside from being mere curiosities. This method of solving things is very effective on cool curvy spaces (manifolds) as well, and it appears I finally have hit upon the correct theory for formulating all this on manifolds. There really is something to be said about learning things in context, because all the books seem to rehash the same old junk and blather on and on about exotic, weird identities that our exotic, weird objects satisfy, instead of, say, actually putting it to use and seeing how it works in (research) practice. But once you see it that way, it suddenly actually makes sense. I wonder if this is some kind of wall deliberately put up just to test budding mathematicians, and maybe, you know... kill off some competition. They weed out the so-called "less-capable" by making them lose patience. Nothing is more satisfying than making your colleagues feel like crap by testing them with a real bitch of a problem that could be so easily explained with the right tools, but instead, why don't we just hide them away, keep and it a secret, ok? You're smart, you figure it out! (Kidding. And sorry, ranting.)
Anyway, the concrete goal we have (and thank goodness for concrete goals... what a novel concept!) is to research, study, and extend evolution. No, Darwin hasn't been invited to my thesis, yet, but we are trying to study the different ways sufaces can evolve via prescribed laws (i.e. differential equations). It's a fruitful an interesting field. And, not only we want to study it, we also want to actually see this process in action by an actual program. More later..