Three years and one month ago, I was at the start of a phase of my academic life: undergraduate study. When I joined Chennai Mathematical Institute for undergraduate study, I was very sure that my future lay in doing study and research in mathematics. My confidence had been boosted by fairly successful performance at the International Mathematical Olympiad (two silver medals). I had also eagerly started reading higher mathematics books, including two books in abstract algebra, and two books in topology. I was eager to learn as much as I could in the coming three years at CMI, and to pave the way for further studies in mathematics.
Now that I have completed my three years at CMI, and am about to start off with a doctoral programme in mathematics at Chicago, I can probably declare myself successful in what I set out to do. In these three years, I have learnt a lot of mathematics, although there are many branches of the subject where my knowledge is below par, and I intend to focus on these over the coming year. Yet, in some ways, my path of learning mathematics was not the way I had envisaged. My term-time and holiday-time learning in mathematics was very different from the way I had studied subjects in school, different from Olympiad mathematics as well. For school study, all I typically did was to sit quietly, read a book, make a few notes, solve a few problems, and I would have “studied” a particular topic.
But after I joined CMI, I found that I had little patience, and little need, for that kind of approach. In fact, a whole lot of college life and hostel life was so full of distractions and other things around that I could not really sit quietly, make a few notes, solve a few problems and move on. Rather, my study of mathematics was disorganized and fragmented — read a bit here, a bit there, juggle it in the mind some other time, and keep failing to understand, but moving on.
My ways of learning the subject were unbalanced and I was not following any particular book, so there were lots of things I would miss out, simple things I would forget, and complex things I would grasp. I also found myself undervaluing, and ignoring, the importance of being “bright” and “sharp” and being a good problem-solver. Instead I was more and more fascinated by the idea of reading a bit here, a bit there, building a grand and beautiful picture, most of it on very elementary proofs but big in complexity because of the large number of building blocks needed.
This eccentric approach towards learning, this obsession with doing things my own way, led me after some time to severely doubt whether I was really enjoying and doing well with mathematics. I found that, looking back on my past few months of work, I could not list too many new things I had learnt. I also started doubting whether mathematics was really as enjoyable and worthwhile as I had initially thought it to be. Further, I started doubting both the utility of mathematics and my own ability to pursue it with a sense of discipline. I feared that I would not have the discipline to study mathematics in an ordered, systematic way.
These doubts gripped me very strongly in the beginnig of my second year. There was also the problem of some courses that I did not enjoy — I skipped classes in those courses, could not bring myself to study those subjects.
Towards the later half of my second year, however, I decided that the study of mathematics is the best hope I have, and that it is the only thing so far that I have learnt both to be good at and to enjoy. Thus, I started re-engineering my life in my fourth semester, and this included determinedly attending all courses (even if I felt too tired) and trying to read up and study new aspects of mathematics.
At the beginning of my third year, after a lot of dithering, I decided to apply abroad for after completion of undergraduate studies. Surprisingly, I found that my sense of discipline was very much there — I was able to study for the general GRE verbal and essay part, the TOEFL and the subject GRE in the same ordered and systematic manner as I used to study in my school days. I also found that I was able to juggle that well with my coursework. In the final semester, I expored the creation of a group theory wiki, and did many other experiments, while reading up and learning new mathematics and taking up a number of extra courses.
Now, as I am on the brink of a new phase of my life, I feel confident that I have the ability and the strength to study and do research in mathematics. However, I am still far from sure that mathematics is the correct choice of long-term career. In some sense, I realize that my youthful confidence that mathematics is my destiny was rooted in ignorance and idealism. Rather, I have replaced it with a more wary attitude where I try, at each stage, to equip myself best for the present and the immediate future while gathering knowledge and resources that will help me in the farther future.
For now, it is time to set aside the past, and get ready for what awaits me.