What Is Research?

August 30, 2007

To be set aside

Filed under: Regular updates — vipulnaik @ 7:37 am

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.


Setting off to Chicago

Filed under: Chicago,Places and events,Regular updates — vipulnaik @ 7:12 am

It’s a long time since I last posted on this blog. The last two months, since I returned from Paris, have largely been holiday time for me, and I’ve been doing some miscellaneous stuff to prepare myself for the next important phase of my life. On September 8, just ten days from now, I will board a plane to Chicago, to begin my five-year doctoral programme in mathematics at the University of Chicago.

The first year of the programme at the Universty of Chicago is mainly compulsory coursework. There are three quarters (each three months long) and three course sequences (Algebra, Analysis and Topology). In each quarter, there is one course from each sequence. So a total of nine courses for the first year.

Chicago differs in this respect from other graduate schools. In some graduate schools like Princeton, there is no well-defined framework of compulsory courses, rather students have to pick and choose their courses from a set of recommended courses and prepare themselves for examinations at the end of the first year. From what I can infer, the emphasis in places like Princeton is to get people started on research-like work from a fairly early stage. The pressure to publish thus begins in the first two years itself. In Chicago, on the other hand, there is no pressure to publish; the first few years are meant to strengthen the fundamentals in various areas of mathematics and research is intended for later years.

A couple of months ago, I received an email from Peter May, addressed to all the incoming graduate students, about coursework for the first year. I found that a lot of the material in the courses, particularly the Analysis sequence, was completely unheard of, and thought I should probably start reading up for it. However, I started reading up measure theory and analysis only recently, and am finding it somewhat hard for now. This is probably the consequence of not having done any courses in measure theory and not having a good analysis background. I hope that by contrast, my somewhat better background in algebra will prove an asset to me for the algebra courses, particularly the courses in representation theory and groups. Areas where I have a little, but not a very good, background, are algebraic topology, commutative algebra, and algebraic geometry. In these areas, I hope to keep reasonable pace with the coursework, though I probably will not find it too easy.

I think that the course-based structure for the first year at Chicago will definitely be a help to me so that I can get up to scratch in all important aspects of mathematics. More importantly, I will be able to overcome the fear and reluctance that I currently have with certain kinds of proof techniques and terminology (particularly that of analysis). Another advantage of such a structure is that I will automatically get an opportunity to interact with a number of Chicago faculty members in all branches of mathematics, something I may not be able to achieve of my own initiative. Further, I will also get to interact with my fellow graduate students in and outside the classroom.

In my second year at Chicago, I will be expected to write a paper in a topic of my choice, acquire a working knowledge of a language other than English (I’ll probably choose French, given that have already picked up some French) and submit a master’s thesis. During the second year, I will also be functioning as a Teaching Assistant for an undergraduate math course.

From the third year onwards, I will be expected to start doctoral work full-force, and simultaneously I will need to teach a course of Freshman Calculus. In Chicago, as in many American universities, all freshmen (incoming undergraduate students) need to study one calculus course, irrespective of their stream of specialization. The job of teaching these courses is assigned to doctoral students in the mathematics department.

Currently, it is too early for me to think of questions like what topic I will choose for my thesis, who my thesis advisor will be, how many years it will take me to complete my thesis work, and whether I want to continue to a post-doctoral position in mathematics after that. I do have some ideas and preferences on these counts, but it is only after I go to Chicago and observe the work environment there, and assess my own research abilities in that environment, that I can take the correct decisions. For now, my focus is to equip myself to get the best out of my first year, and to understand the temperament and qualities needed for research, through close observation.

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