What Is Research?

January 3, 2008

Wikis –logic and magic

Before I had set out for the University of Chicago to pursue doctoral studies, one of my favourite intra-math hobbies was the development of my Group Theory Wiki, a place where I was aggregating and assorting ideas, facts and definitions in group theory, starting from the most basic. The development and organization was based on a property-theoretic paradigm that I had come up with long ago. As often happens with me, the wiki concept and its execution seemed just too good to be true, and the time I devoted to the group theory wiki was often not so much to learn the subject as to admire the magic of a wiki-style organization in aiding and abetting my sneaky thoughts in the subject.

The frustrating thing with magic is when I’m the only one to see it; it almost feels like I’m in my deluded world. My goal was to make the group theory wiki reach a stage where the magic could be seen by everybody. However, since I was almost single-handedly developing the wiki, and the vision of the wiki was too precarious to expose to large-scale public scrutiny, I was far away from that stage when leaving for Chicago. Of course, group theory is not a topic that unites the world’s masses, so there was in any case not much of a potential audience. But I wanted the tool to reach the extent that it serves well whatever audience it has.

When I left for the University of Chicago I realized that the wiki would have to be shelved for some time; large-scale structural and restructuring work on the wiki would not be possible with the course load at Chicago (I knew this, despite the fact that I significantly under-estimated the course load at Chicago). Thus, I decided to set aside the group theory wiki for good, and come back to it when in a position of greater strength.

Some time around May-June of last year, I had also started wikis on the same model as my group theory wiki, for subjects like differential geometry, topology, and commutative algebra. Most of these wikis had languished behind the group theory wiki, and I didn’t know whether, or when, I would pick them up again.

It was somewhere in the middle of October, when I was feeling particularly overwhelmed with my coursework in Chicago, that I decided on a somewhat novel addition to my approach for studying algebraic topology: I would contribute some articles to my topology wiki as I kept understanding the course material in algebraic topology, and I would keep improving the structure and organization to reflect my improved grasp of the relationships within algebraic topology. This was hard, because my understanding was very piecemeal, and it was intimidating to try writing a wiki page. The comforting thing was that nobody else was taking a look at these pages, so I could develop and move them around as I wished. Although writing on the wiki was only one of many tools that I used to help get to grips with algebraic topology (more significant ones being attending lectures, solving assignments, and discussing with fellow students) it was a tool that left the biggest imprint — the content I had put on the wiki continued to be just as accessible a month later, and I was quickly able to mould it and improve it over time. I even took a day off to review basic concepts from point-set topology, which led to further embellishments to the wiki. The Topology wiki now contains a reasonable amount of nontrivial matter, and it has reached a stage where it is self-organizing; where I am getting as much as, or more than, I give. It is, of course, still a long way from the point where it could be of use to all the people whose work involves or relates to topology.

My goal is that when somebody reads the page on normal subgroup or characteristic subgroup or Hausdorff space, something happens over and above just that person’s reading and understanding the definitions of the terms. A number of tidbits, that the person may have heard in class as random theorems or manipulations, suddenly start clicking. “Oh, that’s what’s happening!” is the exclamation that people should routinely make on reading the articles.

My hope and goal is that when a student struggling to solve an apparently unmotivated assignment problem tunes in to the relevant wiki, he/she not only immediately gets the solution (which is the primary requirement) through an effortless search, neatly presented, but also learns of all the secret things that were hidden behind the problem: the motivation and related ideas that the instructor had cleverly concealed (or tantalizingly revealed), the relation with other problems and ideas. For instance, a student who wants to check out the proof that an intersection of normal subgroups is normal or that a characteristic subgroup of a normal subgroup is normal should be rewarded with more than just aproof of the fact; the way that fact integrates with the rest of the subject should also be relevant.

As a person explores around the wiki, he or she should naturally develop a nose for what’s going on; every article should inspire questions like “okay, what about this variant?” Survey articles like varying Hausdorffness or ubiquity of normality can give the person a feel for the different perspectives and aspects to which a single notion can be scrutinized, as well as provide pathways to enter from the very basic definitions to a lot of advanced ideas in related subjects.

There is no great technological superiority being employed in the wikis at this stage; the main good feature is the ease with which linking, structuring and organization can be done. I personally feel that a lot of us spend a lot of time writing and reading books, solving homework problems, writing them up, writing exams, writing and looking for research papers. All these are valuable actions, but a lot of it, apart from the immediate value it gives, tends to get forgotten. Homework solutions written on paper find their way either to the trash can or to dusty drawers; homework solutions written up on machines have greater longevity, but not the same ease of access as a quick-to-search wiki page. The magic of wikis can lie in reducing recall and access time to the point where there is nothing to put the brake on a never-ending stream of ideas.

This may sound like an exaggeration of the role that wikis could play, or it may seem an infeasible model, which is why I want to prove, through action rather than words, the power of the wiki model. I will soon start working on developing further the commutative algebra wiki which has been languishing for some time. Efforts on the group theory and topology wikis will continue unabated, and I might soon pick up the differential geometry wiki as well.

It’s only a matter of time for the magic to unfold!

Eat pizza and a math speaker

Filed under: Chicago — vipulnaik @ 8:52 pm
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In an earlier post, I had described the system of student talks that I had initiated at Chennai Mathematical Institute, I’ve now passed out of CMI and the student talks are still continuing; in fact, they are flourishing better than they were in my time, thanks to the efforts of Swarnava and Kshitij.

Student talks at CMI were a small-town affair in my time: audience sizes ranged from 3 to 10, the speaker (usually I) would wait for all the people he/she knew would attend, before beginning, and the talk had no scheduled end time there was an estimated talk duration but nobody was accountable for it). Talks were schedule on arbitrary days, at the convenience of those who were interested in attending. The talks were usually delivered in a seminar hall, which had a seating capacity of around a 100 people, and I often used slideshows. With an audience of only 4-5 people in the huge hall, it was almost like a luxury event.

The student talks (dubbed Pizza Seminars) at the University of Chicago are a completely different affair. The first major thing is that the lure of mathematical knowledge is not the only incentive for attending: there’s free pizza too, and the talk is scheduled during the lunch hour, which, for the first-years, is the break-time between two lectures. Many of us, tired from the previous night’s assignment slog, used to grab pizza, finish it, and fall off to sleep.

(The pizzas are not funded by the speaker; they’re funded by the Math Department and the Physical Sciences Division of the University).

Secondly, the talks are held on a fixed schedule: once a week, at 12:30 p.m. on Wednesdays. This isn’t surprising; if pizzas are being offered, one can’t schedule an unlimited number of talks based on whims and fancies. Speakers have to start on time, break on time (for a second serving of pizza) and end on time.

Thirdly (and I wouldn’t say this is independent of the first reason) the talks are much better attended. The Barn, which can seat around around 50-60 people, is nearly full for every talk.

This leads to a lot of differences in the way the talks are conducted. In our undergraduate institution, people were often quite passionate on the topics they were talking about; here, the talks are largely viewed as a supplement to the pizza, and speakers, even the good ones, appear more indifferent to the impact they make on the audience. With a long line of speakers and only one talk per week, it’s very different from the situation in CMI where a talk slot was actually fixed based on the convenience of those who regularly attended. Finally, with such a huge audience, it’s much easier to get lost in what the speaker is saying, although there are a number of spirited interruptions to the Pizza Seminar in the first five minutes when the audience hasn’t given up hope.

An example can be the talks I myself gave on the same topic, one in CMI, the other in Chicago. The topic was extensible automorphisms. In the talk I gave at CMI, I spent a large amoun of time defining groups, inner automorphisms, extensible automorphisms, and developing machinery of representation theory as well as some leading ideas. The audience was much younger, and I’m not sure how much they understood of the talk, but there was a slow-world air about it. My talk in Chicago was a light-hearted but quick-styled affair; I jumped from here to there, throwing in some wry humour at various points, and trying to give a quick peek into the topic to a significantly smarter and more knowledgeable, but on the whole more preoccupied and less interested audience.

A more fundamental difference between the Pizza Seminars in Chicago and the student talks in CMI, however, is the backdrop. In our undergraduate institution, students often have little or no opportunities to teach or explain in a formal setting; graduate students in the University of Chicago, on the other hand, get to teach or assist in teaching regular courses. Thus, there is hardly that much novelty value in addressing a large audience. Further, there is not much one can do to “teach” graduate people in one hour. Student talks in CMI were actually teaching and learning opportunities, even if what was learned was eventually forgotten.

A few weeks ago, I asked the current coordinators of student talks at CMI, about the possible directions these talks might take if pizzas were offered at each talk. He said that attendance would certainly rise, because people cared more about pizzas than about math. Whether that is a good thing or a bad thing is debatable. The debate here is not about the health impact of pizzas, but rather about the issue of what the goal of talks is, who the target audience is, and what kind of value (over and above the pizza) is to be imparted to the audience.

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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