I'm an MS Mathematics student here in the Philippines, studying in a fairly well-known university. With the grace of a government scholarship, I luckily didn't have to be employed to support my studies. For a few semesters, I didn't think of anything but Math (a lie, I procrastinated a lot, god knows how many of those Japanese comics I've finished while I was doing graduate classes; our science library has a pretty decent internet connection), and doing well on my classes. During my third semester as a graduate student, I took a functional analysis course under a soft-spoken associate professor-- I did well in that class, and he'd soon become my thesis adviser. So with an adviser in tow, and leaving behind traditional classroom classes, I went in to my 4th semester as a research student.
For more than half a year, I lived off my scholarship stipend, all the while just reading and reviewing the current literature of my chosen field. I didn't have to show up in our institute anymore, save for the occasional meetings with my adviser. Good times, I guess. I could wake up anytime that I want, and study anytime that I want. My typical morning consists of walking through our science garden while thinking about math on my way to the library. I'd like to think I'm the type who could easily build habits, so the loss of structure didn't really affect me that much since I already made it a habit to study and read mathematics for the last few years. So even if there's no looming deadline to be seen, and no professor looking over me, I still got my reading done. Well, I did get my reading done, but at some point during that semester I kind of had too much fun and I ended up studying things that are not necessarily related to our research. There have been a few occasions that my adviser told me not to dwell on too much details, and that there are some proofs that I can skip. In other words, I should actually get some work done.
Between algebra, analysis, number theory and geometry, it seems like analysis was something I was most comfortable with, and also something I was the most interested in, it was actually when I took my first course in calculus (analysis and calculus are pretty much interchangeable) that mathematics intrigued me enough to consider actually majoring in it. I vividly remember trying to come up with reasons about where the formulas for derivates came from. It's nice to say that, years later, my curious self who only wanted to know why
My adviser specializes in a field of Mathematics called time-frequency analysis. It's a sub-field of harmonic analysis, and I won't go in depth about what we do, but we are basically trying to write functions in an infinite dimensional function space in terms of more "fundamental" functions. Time-frequency analysis seems to have a ton of real world applications, specifically in signal processing. All of these are magic to me though; I spent most of my time studying proofs, that is, the mathematics behind the mathematics, but not the real-world motivation behind them. I thought just being mathematically adept was enough, but it turns out that I lose a lot by being too rigid, not going out of my comfort zone and actually learning why some mathematical definitions are defined that way, and why we are interested in them. Of course being good with logic and proofs is something a mathematician should primarily strive for, my mistake was dismissing how much inspiration comes from intuition and real-world problems. Months in to my research, I feel like I have hardly progressed after this realization, I probably could have simplified a lot of things only if I knew how some of the concepts translated to the real world earlier.
I think most of my personal disappointment with research stems from the fact that I do not really like my current field. I think I'm still interested in analysis, but just in the more abstract and general aspects of it. I probably won't stay in my field, but I'm looking forward to studying more math-- right now I'm eyeing category theory and how it ties up pretty much all of mathematics. Anyway, hopefully I can graduate this semester, and continue on to do PhD in math; I plan to do a 'mathematical holiday' of some sorts next semester, I want to be able to freely study pretty much anything that will catch my interest.
