Ok so a new maximal interval has passed since I last updated, even along with several Canonical New Quarter Musings (although I may make backdated entries to restore the order of things). Sorry about that. Life in grad school has been a very rocky one... despite the good time I had up in Berkeley reading a lot and not having to TA, it was not well-structured enough for me to read and understand the current research papers in the field at any sufficient speed... well, at a speed sufficient to satisfy the prospective advisor I had back then. Anyway, I did get a new advisor, and some semblance of order is restored now. I'm making some progress, although I'm still put off by some long estimates and calculations in the high-powered analysis used in a lot of proofs. My "field" of math, I would say now, is conformal and spectral geometry. Overall, it is still geometric analysis.
We'll see what happens. Hopefully I'll keep the journal up semi-regularly. I think weekly might be a good thing to shoot for. I can't do it daily, because it then becomes all-consuming and excessive. I'm obsessive like that. And hopefully it won't be just about progress in math, but perhaps also progress in social life, something I have also been working on. And perhaps also about a new girl I like ;-) But more seriously, half the advisor problems came from a lack of communication and, you know, that terrrible shyness that I have (yes, that's with 3 r's). (But I may chronicle progress and insights in learning concepts in the math friends list).
So why didn't I benefit so much from spending 2 quarters at Berkeley? First off, research-level analysis pays inordinate attention to estimates, long calculations, and exotic techniques in the theory of PDEs, and was completely unfamiliar to me. The focus is profoundly different from what was introduced in coursework. Plus my then prospective advisors were highly distracted with their own work, such as writing books, so it was difficult to schedule meetings with them and really establish a connection. If I had been in better communication with them, perhaps they could have provided some sources and tips from their own experience to assist in the transition to research-level analysis.
Next, the very fact I had not been present in SD during the crucial 3rd year sort of threw off the schedule and kept me out of communication with my peers who were going through similar things. For example, perhaps a bunch of us could have gotten together and given talks on more intricate analytic concepts. Plus of course it just helps to have the moral support. I missed some important department meetings arranged specifically to help grad students understand the advisor/student relationship. The upshot: although all my fellow grad students thought me extraordinarily lucky to have had such a 3rd year opportunity, it is not all a blessing, and could do more harm than good. It was fun and interesting, to be sure, but profoundly disruptive.
Anyway such inability to read papers further exacerbated tensions between me and the prospective advisor which started with my not doing so well on the Complex Analysis Qualifying Exam. It was of course unfair that he made it all 3rd quarter material, in direct contradiction to what he'd said previously, but no matter. What counts is that I was a disappointment and caused him to be generally leery about anything I did. It didn't help that his philosophy seems to be "drop you in the deep end, and learn to swim. with the sharks." The last sequence of mathematical posts has basically chronicled my attempts at really getting the material down and regaining his trust. But that was not enough. I actually asked to work on a problem, and he gave me one. It involved reading up on the problem that won a mathematician the Fields medal. I say this not to brag (after all, it's about my failings, not my success), but rather to show that this, um, might not be as easy a read as my advisor would have liked it? I pretty much threw my hands up in frustration at the use of the Schauder regularity theory. God, that stuff is ridiculously hard. I was pretty much left to just "figure it out." with no guidance as to what were the most important results in the theory, which proofs i could afford to skip, why the theory was invented in the first place, and why it is superior to other methods of solving PDEs. As a consequence, this advisor reported to the vice chair his lack of confidence in my ability to be advanced to Ph.D. candidacy on time, and further more, I would lose funding for the Winter Quarter. To make matters worse, once I did find a new advisor, my funding would still not be restored (fair system, isn't it?) Which brings us to....
Finding a new advisor. I actually am extremely fortunate, for another professor during my 2nd year had recognized my interest in analysis and PDEs, and was willing to take me on. Without that, I'd probably be out on the street now. Ok just kidding, leaving graduate school with a Master's probably would not have been bad, if I could get over the pure disappointment in myself. Anyway this new prospective advisor still trusted my mathematical ability (but it does kind of help that there was no qualifying exam in that subject..... hehehe). She is more experienced in advising and a more of a guide. Anyway she helped me get up to speed, and I did some interesting calculations, good enough to get me advanced to candidacy on time, after all (although they still didn't give me funding for that quarter, blarrrgh).
Oh yes, I'm also very happy about that election. As chronicled in this entry from *gasp* February of 2007, I'd already read The Audacity of Hope and have been cheering for the guy for nearly 2 years now. Whoever has that bumper sticker must be very proud. [UPDATE: backdated entry on 11/04 added]