What Is Research?

January 3, 2008

One quarter down

Filed under: Regular updates — vipulnaik @ 8:13 pm

This is my first post since coming to the University of Chicago for the mathematics Ph.D. programme. I offer no excuses for the long delay — the main reason was that the past quarter at the University if Chicago was nothing short of hectic, and there were a lot of things I wanted to catch up with in the one-month winter break that followed (there’ll be more on that in subsequent posts). Let me try to give an idea here of the graduate work at the University of Chicago, and how it differs from my undergraduate experiences.

In the first year of the Doctoral Program at the University, we have absolutely no teaching duties; all mathematics students are on a University fellowship, and the main task of the first year is to get through the compulsory courses. There are three course sequences: Algebra, Analysis and Topology/Geometry, with one course of each sequence in each quarter. A “quarter” is an eleven-week term, and there are three quarters in the year (Fall, Winter and Spring). There’s also a summer quarter, which is the time for doing summer study, or freaking out.

Many universities in the United States run on the quarter system; examples other than Chicago are Northwestern University and California Institute of Technology. Others run on the semester system, which is closer to the system in India: two terms, each approximately 16 weeks. The quarter system means shorter, and more intense courses, and more frenzy, but it also means that you have to tolerate a course you don’t like for that much less time.

My first quarter at the University of Chicago was from September 24th to December 7th; the final week was examination week, so we effectively had ten weeks of instruction (with a short break for Thanksgiving). In this quarter, there were three compulsory courses. Each course had three lectures per week (three seems to be the favourite number here), one each on Monday, Wednesday and Friday. An assignment (whose solution was usually around 6-15 LaTeXed pages) was due in each subject each week, and the submission dates for the three subjects were Monday, Wednesday and Friday. Apart from coping with the material covered in lectures, the primary focus was thus getting through assignments.

The general expectation, from what I understood, is that everybody is expected to submit assignments on the due date, with complete solutions, and students are strongly encouraged to discuss solutions with others if they are not able to solve the problems themselves. The place where these discussions took place was a musty underground dungeon euphemistically called the first-year office, where all the first-year students had appropriated desks. Often, the day before assignments, the boards would be full of solutions or key ideas for solutions, with people hopping around and explaining the solutions to each other. It was not uncommon for students to actually take down notes on their laptops as solutions were being explained. The first-year “office” would usually be up and running till late at night, and usually till early morning, before the submission of the assignment.

Assignments filled up so much of our mindshare that keeping track of what was taught in the classes was often a secondary, or even irrelevant, concern. Nonetheless, well-designed assignments forced us to go back to material covered in class and thus led to a high probability that we assimilated well the topics that were covered in class, as well as those that were glossed over or given short shrift.

The Algebra course was taught by Professor Victor Ginzburg, a well-known person who works in (as far as I understand) a broad gamut of noncommutative geometry and representation theory. Algebra was my preferred subject when I came to Chicago, and the first few weeks in Professor Ginzburg’s course were quite pleasant, although his teaching style was different from what I was accustomed to. Towards the later part of the course, Professor Ginzburg switched to noncommutative algebra, a topic which was largely new to me, and I felt increasingly frustrated at my inability to take out time to study the topic and having to cope with the assignments one at a time rather than getting a bigger picture of what’s happening. It was finally around the time of Thanksgiving that I decided to channel my frustrations into something positive, and started preparing notes titled “A Flavour of Noncommutative algebra”. These notes were such fun to write that I actually started enjoying the beauty of noncommutative algebra, and many of the pieces which Professor Ginzburg had mentioned in class or given us in the assignments, started fitting together. I enjoyed them so much that I even wrote up a Part 2 following the first part, and passed on the notes link to my batchmates, some of whom gave a number of useful comments that helped me augment the notes. For those who’re interested, here are Part 1 and Part 2 respectively.

On balance, algebra could have been a more enjoyable experience for me than it was. One of the reasons why I found it hard to enjoy or appreciate was that I had a lot of previous notions and ideas in algebra, and so whenever something was covered, I would always feel that it would have been better viewed in this way, or that some essential point was missed. This led me to resent the subject in a way that was unnecessary, and I would probably have done better to get started with preparing my notes and trying to see the new perspectives and ideas earlier on, rather than wait for Thanksgiving time to catch up.

Analysis was taught by Professor Gregory Lawler. Being a probabilist by profession, Professor Lawler mingled in a lot of probability with the measure theory and analysis, which made it more interesting as well as harder. Although the level of material we covered in the course was not conceptually too hard, the assignments were demanding, specially for me, since I had no prior experience of solving analysis problems. Analysis was the subject where I gleaned the most from the first-year office, and had the least amount of insight from within, and it is probably the subject where I will need to put in the maximum effort to get the rhythm in future quarters.

One of the mistakes I made in analysis (and which I hopefully will not make in future quarters) was that because the first few weeks were light, I did the assignments and didn’t think about the subject further; I didn’t take the opportunity to familiarize myself with subject material in later chapters. I should have realized that given my poor grounding in analysis (compared to many batchmates who had done a graduate course in real analysis earlier), reading ahead would be profitable. But then, one lives and learns (hopefully!).

Algebraic topology, taught by Professor Madhav Nori, was for me the most fascinating new thing. In the first couple of weeks, I was not following too much, and moreover, I was making a lot of careless errors with the subject; long exact sequences of pairs, Mayer-Vietoris, and the like seemed mumbo-jumbo and I’d often mix the pair with the subspace. Again, it was somewhere around the fourth or fifth week that I decided to take some time off and study the subject properly. It was just a single day off; which cost me significantly in terms of assignments, but which gave me added confidence in the subject and helped me to then build a love for the subject, and a reasonable degree of intuition. It was also an occasion for me to revisit point-set topology, and to appreciate many of the subtle points in point-set topology that I had started exploring long ago, and then abandoned.

This first quarter at Chicago was extremely different from the academic atmosphere I was used to in my undergraduate institution. In most of the courses there, we had little homework; there were plenty of courses with no assignments, and even those courses which had assignments didn’t have more than three. There was only one course where we had weekly assignments, and this course was taught by Dr. Amritanshu Prasad, who did his Ph.D. at the University of Chicago. But even his assignments were much shorter than the average assignment we got here, and we had to do three in a week here!

What I learned from this first quarter most was the importance of taking time off in different senses. Firstly, there’s taking time off from studies altogether, which is the obvious meaning, but secondly, there’s taking time off from assignment work and trying to get back and view the broader picture. In some sense, my undergraduate education almost entirely consisted of taking time off, so I never had to put in an effort for it; but here, taking time off requires a deliberate effort.

It is hard to debate (and I have too little experience to even offer consolidated opinions) on whether a system with a lot of homework, as in the University of Chicago, prevents people from taking time off and getting the bigger picture. However, I must say that most people, even in the absence of a heavy workload, find it hard to stand back and get the bigger picture. In my undergraduate institution, the lack of homework pressure, while ideal for people who wanted to take time for their own study and exploration, also meant that people could simply goof off and treat the semester as vacation time: something which is rampant in our undergraduate institution, at least for people who are smart enough to get by the exams with a reasonable amount of effort. Secondly, one can take time off and get the bigger picture only after toiling on all, or at least some of, the little details, and well-constructed assignments give this opportunity.

It falls on me to integrate the various approaches and try to evolve a method for handling my studies which is not only most efficient, but also most stress-free and enjoyable.

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