What Is Research?

March 28, 2007

Talks: how important are they?

Filed under: Uncategorized — vipulnaik @ 4:32 pm

Last semester, I took the initiative to formally start a system of Student Talks in CMI. The student talks started off well but then due to some technical glitches with transportation (this was before CMI got its hostel), the frequency of talks went down and there were in fact only four talks in the second half of the August – November semester.

In January, CMI shifted to its hostel, and we started off Student Talks with a fresh slate, scheduling the talks in the evening when all students were free. Soon the logistic issues with the talks got settled and it was now only a question of finding a steady stream of people who could give talks. This proved a challenge for a variety of reasons.

For one, a large number of people who were in principle interested in giving talks, either didn’t have a topic they wanted to talk about or didn’t feel they had enough knowledge in the topic to given a talk on. Some people who actually came up with talk plans realized that they don’t have enough time to prepare a talk in proper depth.

Towards the fourth month, there have at last started coming some other people who plan to give student talks. However, most of the talks were delivered by me (mathematics), Ramprasad (computer science), Anirbit (mathematics and physics). Sometimes I wonder if my overenthusiasm to give student talks myself is leading to a misuse of my power as student talk coordinator, but since there are hardly any others who compete with me for student talk slots, I don’t see any reason to feel guilty about giving too many talks myself.

This whole business of student talks really makes me wonder: what really is the goal/achievement of a student talk/student presentation? How does it help that particular student? And how does it help students to listen to talks from other students when they already have so many courses to attend and so many seminars by far more qualified people to go to?

I explore some answers based on my own experiences with student talks.

First, how the process of giving a student talk helps me. I have so far given ten student talks, some of them being on subjects that I already know or that are “pet” topics, while others were on topics which I didn’t really know but wanted to learn.

The very act of trying to express what I have to say in slides helps me sharpen my thoughts. For one, the total space available in a slide is limited, so I have to break all my ideas into small enough chunks so that each chunk carries a central theme that can be put into a slide. I often don’t do a very great job of this (as per the criteria of each slide having a central theme) but at least keeping that in mind helps me better understand the subject.

Also, while giving a talk, it is important to create a story, a build-up to the subject, that I may cut out on completely while learning myself. For instance, while I read up on Fourier series myself, I just picked up a collection of separate isolated facts: Fourier series, Fourier transforms, dual groups, Fourier transforms on reals, Fourier transforms on finite fields. While giving a student talk, I had to organize all these ideas into a particular thread so that they seemed to flow naturally. Similarly, my talk on approximating solutions to equations was based on many tidbits that I already knew, but while giving the talk, I had to formally understand each part and I also ended up doing experimentation by writing the codes in Haskell.

Another aspect is that the very act of delivery of a talk often makes one feel good, particularly when one is able to share certain insights (however trivial) with others and feel the joy of their understanding the stuff. For me at least, student talks have been an important way of getting me to feel a greater sense of enjoyment in studying the subject. This is a bit like the cook for whom having the food eaten by others gives an altogether different quality of satisfaction from simply eating the food oneself.

I also think that being able to talk on a technical topic to an audience composed of peers (rather than junior people) is an exercise in developing the confidence to express and present oneself. Many people are (rightly, perhaps) reluctant to give student talks on account of not having much to say, or not being sure whether what they will say will be useful to others. However, those who do end up giving talks realize that once you come up on the stage to say something, you can usually say it. And I suppose that since mathematics involves a lot of teaching and learning, student talks are a genuinely good preparation for a later life of mathematics.

Coming to the other half of the question: what are the advantages of attending a talk by a fellow student?

I have observed that by and large, I tend to feel less sleepy while attending talks by other students as compared to attending regular lectures or seminars or expository lectures by more senior people. One reason for this possibly is that in the student talks, I tend to feel more involved with what is being discussed, and feel more of a commitment to try to follow what is being said. This is because, firstly, students usually make a sincere effort to ensure that what they are saying makes sense to other students, and secondly, they may be a better judge of how their fellow students are understanding what they are saying.

Another interesting thing about the student talks is that they are generally more relaxed. This is probably beacuse student speakers don’t have the same baggage of expectations in terms of rigour, speed and correctness that more senior speakers might. For instance, I myself, while giving my talks, use prepared slides but I don’t try to rush through any slide or finish each side in a particular allotted time; I also avoid “leaving things as exercises” and I feel free to go into digressions and tangent if they will help my cause.

In more formal talks, the speaker has to keep in mind that the audience comprises people who are already well-versed with the basics and that if he/she spends too much time on basic stuff, the others may lose interest. Also, many of the formal talks are actually a way for the speaker to present some original work or ideas, and the contents may be used by others to evaluate the rigour of the ideas. Hence, there is more pressure on the speaker to go fast, be more rigourous at times when rigour compromises understanding, and not go into tangents.

Student talks do fill an important niche, but despite this, participation in student talks is rather low. This is where there could lie a possible disadvantage with an event that involves only the students: student do not feel any kind of compulsion/moral obligation to attend student talks, and therefore may choose to not attend if they are feeling tired, etc. Personally I don’t think this is a serious and important problem, because, after all, if a person doesn’t feel that he/she will get much from the talk, then it doesn’t make sense for the person to come all the way to attend the talk and fatigue himself/herself. Nonetheless, it is important for the phenomenon of student talks to create enough of a reputation for itself that people feel more enthused to attend a student talk and more confident of the irreplacable value/enjoyment they’ll get out of it.

I’ve also been wondering what will happen to the student talks initiative once I leave CMI (this is my final semester at CMI). More than 50% of the talks are being given by me and Ramprasad, and once we both pass out, it may be difficult to sustain the momentum of student talks. This actually brings about the more important question — how important is an individual to a community initiative, and what kind of steps should and can be taken to make sure that community activities continue even when the individuals change?

I’ll explore these issues in a later post.

A summer in France

Filed under: ENS,Places and events,Regular updates — vipulnaik @ 2:06 pm

To set the context for this blog post, I am among three students from Chennai Mathematical Institute who will be visiting Ecole Normale Superieure, Paris for an exchange programme. This exchange programme happens every year, with the top three students from the passing-out undergraduate batch going for the two summer months (May-June) to the ENS.

I came to know about this some time in the month of November (of course, I unofficially knew about it long ago). For some time, the upcoming visit has been filling me with a mix of hopeful anticipation and a sense of dread.

Among the more basic issues are the issues of food, living etc. it seems there may be some adjustments needed on that front, but on the whole, it should be manageable. Then there’s the fact that I am visiting Paris, which is supposed to be one of the best cities in a variety of ways (I don’t really know much about these things, but I’ve been told this so I am looking forward to seeing the place for myself).

On the academic front, I need to find myself a guide (I’m not sure about the need to part but I guess that to make my stay academically useful it’s best to work under a guide) and then to follow up on reading some stuff under that guide. From what I understand, I’ll have to give a presentation at the end of it.

The great thing is that from what I have fathered, the ENS is among the best research institutes for mathematics across the world, and the academic environment there is likely to be good. The mathematics department is much larger than that of CMI, and contains some big names. And I have interacted with some people from the ENS who have come to CMI and I’m definitely keen to meet more of them.

On the flip side, I have the memories of my one-month stay at TIFR, Mumbai where I went for the Visiting Students’ Research Programme. I had a nice time there (Navy Nagar, Mumbai is a nice place and TIFR, situated right on the seashore, is particularly nice) and moreover I got computer access and I used that to do all the things I usually do at home and in CMI and more. And there were some other people who had come to the VSRP with whom I occasionally used to have intense discussions. And I got to meet some fine professors.

But for the paper that I had been assigned to do (viz ”Lie Group Representations of Polynomial Rings”) opccupied very little of my time — in fact, there were days on end when I hardly even touched the paper. True, I did put up a few intense spells on it, but I wonder whether these few intense spells were all that the academic component of TIFR was.

There were also the illuminating sessions with my guide, Professor Dipendra Prasad, but unfortunately, because I was not making good enough progress on the paper, these sessions could not be too frequent. Had I worked more on the paper, and perhaps on some other related things, I may have been able to extract more.

I’m wondering whether the ENS, France will be something like that.

The further complication is that, out there at the ENS, I’ll be in a foreign place where I may not know that much about how to interact with the people and what their social conventions may be. While I don’t think that this will lead to any major social gaffes, it could definitely hamper my comfort level in approaching people and in seeking them out. Also, since I’ll be in a far-off country, the number of sources of amusement, reassurance and comfort (if things aren’t working out) will be fewer.

Here at TIFR, for instance, even when the academics was getting too bad, I could always pass my time on the computer, or talk to the other VSRP fellows, or go out in the streets and in general meet up with other people. In Paris, I’m not sure how easy it’ll be for me to do such things.

On the other hand, I do have the advantage of more maturity, and also, I know that, like TIFR, I can enjoy and have fun and do a bit of work — I don’t have to do a lot of work just because I am going there.

A new phase of my life in the offing

Filed under: Regular updates — vipulnaik @ 11:01 am

Yesterday night, I was feeling extremely fatigued and worn, and when I stopped to think of what was causing this, I realized that I’ve just been going on and on for days on end without stopping and thinking of where I’m heading.

This semester has been rather light for me (in terms of course load: only five courses) but I have offset that my taking on a number of ambitious and tricky tasks, most notably the Group theory wiki. In addition, I’ve been involved with a teeny-weeny bit of work on things like CMI Spark, CMI Online Programming Contest, Olympiad teaching and training.

I’ve also been giving quite a few Student talks and have been trying to prepare lecture notes for some of my courses such as Lie-theoretic methods in analysis (I haven’t worked on that for a long time, but that’s a different story).

In addition, I’ve been trying to get started on some other wikis, particularly a Differential geometry wiki. And I also plan to get started on my Commutative algebra wiki. I’ve also been trying to work on developing a theory (which again hasn’t been working out, called APS theory, on a wiki devoted to that.

So in the midst of all this (attempted, and so far largely unsuccessful) activity, I have so far forgotten to face up to and think about the new life that lies ahead of me.

To set some background.

I am currently in the third and final year of my undergraduate programme at Chennai Mathematical Institute. Once I complete my B.Sc., I plan to join into a Graduate School (viz a place for Ph.D. in mathematics). Once in Graduate School, I hope and expect to work on research in some important problems in mathematics, probably make some original contributions that’ll earn me a Ph.D., and then, continue with mathematical research.

I’ll be starting on Graduate School probably some time in August-September, and I need to begin the emotional journey towards that. I need to figure out how it’s going to be different from the undergraduate life, what the new pressures and responsibilities will be, and how I can do well in Graduate School.

I also need to look at my more long-term plans and see how Graduate School will fit into them.

Unfortunately, I’ve been too caught up with the things I am currently doing to be able to stand back and stare.

But now, as I write this blog post, I am forcing myself to think hard.

One of the basic differences between my current undergraduate life and life in Graduate School, will be that in Graduate School, I’ll actually be expected to produce results. As in, there’ll be papers I have to read, results I have to create afresh or extend.

Right now, whatever little of papers I read or research work that I try acquainting myself with, I do of my own accord. And I always have the freedom to stop reading when I get bored or when the stuff gets too complicated or when I realize that this is not the right order in which to read things. So, because I don’t really need to get anything done, my efforts to acquaint myself with new research are sporadic and ad hoc.

On the other hand, getting a doctorate, as far as I understand, is not something that can merely be accomplished by knowing a bit about everything, reading a bit here and a bit there. One needs to take a specific problem and make definite progress on it that is worthy of publication somewhere. Trying to make definite progress that hasn’t been made by any person so far seems a daunting task. I mean, what guarantee is it that, however smart I am, I’ll actually be able to find something new and significant in the five or so years I am allotted for my Ph.D.?

From what I have observed of the research world and the way it operates, a crucial component of research is reading and thoroughly studying research papers. Unfortunately, I don’t think I have yet gotten the hang of it — also somehow I don’t seem to enjoy it (of course, it may be too early for me, considering that I am still an undergraduate, but the sinking feeling I get on seeing a long series of pages with abstruse symbols and claims and proofs does depress me). This is notwithstanding a post I myself put on this blog on
How to read research papers.

This is one of the things I need to look into to overcome. So far, I have looked at research papers, but it’s usually for something specific — a definition, a little claim, a particular theorem, etc. The first big attempt I made to study a research paper was under Professor Dipendra Prasad at the Visiting Students Research Programme at TIFR, Mumbai. This was the paper on Lie Group Representations of Polynomial Rings.

I got off on the paper with a good but not too good start — and it was really slow progress. By the end of the programme, I had managed to read and understand the proofs for only twenty of the eighty pages. Of course, that was my first experience, and there were many learnings I got from it which I feel will help me in future experiences (some of which are documented in earlier blog posts). Somehow, what I didn’t learn was to like deciphering a paper from start to end.

Another aspect of research and graduate life that frightens me as of now is the need to collaborate with others and work under people. So far, I have been working largely of my own initative, and most of my interaction with teachers is for learning specific subject material or clarifying doubts or general discussions. The notion of actually going through a paper with a guide, where I have to meet him/her on a regular basis and report progress, sort of scares me. The problem here is that it’s not just a commitment to myself or a personal choice, it’s based on and largely driven by the other person (particularly since as the guide, he/she is the senior and the one deciding the directions).

Of course, there are all kinds of guides and all possible arrangements one could have with the guide, and in fact, many research students have told me that the guides try to guide as little as possible and let the student decide the course.

Among other things that frighten me about my Ph.D. is that if I get too emotionally involved with trying to solve a particular problem, and if it doesn’t work out, then what do I do? And if I don’t come close to getting any new results within five years of getting started (or whatever the number of years is within which I need to complete my Ph.D.) How desparate will I get?

What happens if I have to abandon my favourite subject for a subject where I can get a result faster? What happens if the areas where I want to work turn out to be duds?

The other thing that’s bothering me about my Ph.D. is that it seems in some sense a final step — after that, I’ll stop being a student, I’ll be at the giving rather than the receiving end of knowledge (while doing a Ph.D. one is sort of at the interface between the two, one is trying to create knowledge while simultaneously attempting to acquire it).

So I want to utilize the time I spend in my Ph.D., to prepare myself as thoroughly as possible, for the road ahead. This is not just in terms of the mathematics content, but also the way mathematics and research is done, the various politics and administrative issues involved with research, the key to teaching and learning good mathematics, the various issues facing the mathematical and research community and the way they interface with other disciplines.

I’d ideally like to have a good network of people doing various parts of mathematics at various places, with whom I can interact and interface so that I can experience the feeling of doing mathematics in a community.

On the whole, it’s going to be an interesting and challenging phase, and the things I can do now (or perhaps over the next few months) are try to talk more to the people who are already doing it, and try to get over the mental block that I have (or think I have) with reading research papers.

I also need to wrap up, or reach a sort of logical point, with all the many ambitious projects that I have started and gotten underway right now. I don’t know how much time I’ll get to pursue these once I start out Graduate School, but I think there’s a fairly good chance it’s going to be far less than what I have right now. Which means I should go to Graduate School with these shelved aside (that doesn’t mean I won’t work on them — it just means that I won’t expect a regular output on them from my own side).

Let’s see how things unfold…

March 20, 2007

The matter of scalability

Filed under: Teaching and learning: dissemination and assimilation — vipulnaik @ 3:14 am

Recently, I was looking at the various costs involved in different research activities. I was thinking of questions like: what is the intrinsic cost of doing mathematical research? Why do research institutes require such high/lavish research funding when all that research basically needs is smart minds? Are all the grand architecture, good food, high pays, transport facilities etc. provided to researchers reflected in better research output?

For instance, in a time when land is scarce, when travel is expensive, are the huge amounts paid to have people from all across the world attend a research conference really justified? In a time when printing and publishing costs are high, are the huge costs that institutes spend on having well-stocked libraries worth it in terms of increased research output. Given the high costs involved in transportation, is it worth it for an institute to offer regular transportation
facilities to its students and faculties? Given the high costs involved with food and accommodation, why should institutes sponsor part of the costs?

These questions are particularly relevant in an era where many of these expenses could be cut down on significantly through the use of technology (at least in principle). A hundred years ago, communication across countries invariably meant physically going here and meeting; now, communication can happen over Internet, the phone, and voice/text chatting and conferencing. Of course, needless to say, these things don’t carry the same romantic flavour as meeting up physically and discussing problems over tea or coffee, but is the slight advantage
gained thus worth the tremendous extra costs incurred?

Really, how much is an atmosphere worth for research? How much are we willing to pay for things that “feel better” even though they involve greater costs in terms of time, money, and organizational power — like going out there and meeting feels better than communicating over a long distance?

Interestingly, I think it is still worth the high costs, the reason being, interestingly, that being at a place and in a right atmosphere is very critical to getting one started along the right path of thinking. Living in a ghetto even with a distant access to the greatest resources in the world, one may not develop the same ways and patterns of thinking that are needed to utilize those resources. Further, being in a place means that you and the others are “forcefully exposed” to each other, to an environment, to an atmosphere where “serendipity” occurs. The use of the word “serendipity” stems here from a write-up on the onsite library at the University of Chicago, where all volumes are kept onsite with the hope and belief that serendipity occur. Interestingly, analysis of book usage patterns at the University of Chicago also indicates that those who use Internet resources more also tend to use the print books in the library more. In other words, greater use of the new technology correlates positively with the good old print books.

However, the main issue (problem) with the old way of doing things (viz going nad meeting, holding physical conferences) is that it doesn’t scale. That is, it may be worth it to invite the top 40 people to an international conference, and they may genuinely get a
lot of benefit that justifies the costs and efforts. But one can’t invite the top 4000 people for the same: there are decreasing returns to scale.

On the other hand, the newer technologies and the approach of “each to his/her own” without grand institute funding, can scale effectively. To take an example, consider that an organization wants
to spread greater awareness in a particular area of cryptography. The organization could consider the following approaches:

  • Start an organization/department that conducts research in this area: Great returns, but a lot of investment and hassles, high risk, and a long gestation-period.
  • Hold a conference, workshop, or summer school: This can be done with less effort, and with more immediate returns. But a conference or workshop or summer school means travel expenses for the participants, accommodation and lodging expenses, and it requires people to shell
    out a huge chunk of their time. Further, one cannot call a very large number of people to a conference.
  • Encourage colleges/universities to start courses in those areas: Here, the scaling is more. The investment from the side of the organization is less. The focussed return on investment is also less — it is possible that many of those who benefit may benefit too
    little or too late or may never “pay back” for the benefits.
  • Prepare a series of books/lecture notes in the subject and throw them into the market: Again, this requires effort, but the advantage is again that the effort is far more distributed, the focussed investment may be less, and the reach is wider. Again, the effect may be diluted: after all, how many people will actually read the book?
  • Put a free online resource on the subject: This again requires effort, which may in many ways be much less than the effort needed to prepare a book. The advantage of a free online resource is that the reach is much greater, the costs per person reached may be much less. On the other hand, the focussed impact may also be much less. Online resources, however, scale. It’s much easier to take an online resource from 40 to 4000 people than to take a conference size from 40 people
    to 4000 people.

There are all sorts of approaches at different scales, with different kinds of immediate and long-term impacts, with different degrees of investment and different degrees of return to investment. Some of these, being newer, have not been tried so far, and so people may feel
reluctant to invest in them.

For instance, many people in the research profession find themselves shuttling from place to place, attending one conference after that, giving one lecture after another, following time-tested practices of knowledge dissemination and sharing. They may spend hours correcting
and solving homework problems. Yet, the amount of time they may want to shell out towards writing a book or creating a definition resource for general consumption could be much less.

Since individuals are invariable overworked and have to meet performance criteria, it is up to the institutes to say :do we want to scale up and reach out? If so, should we focus on creating resources that can be used far and wide, so that people are attracted towards our work? Or should we concentrate on financing only those things that are tried and tested, those that will give us immediate returns in the short run?

Why should scaling up be important? After all, people argue, research is not a commercial enterprise where we want to attract a huge number of customers and sell out to all of them. How does scaling up affect the few people who are currently doing a great job of it? Will it “dilute” the quality of research?

First, scaling up within a particular sub-discipline of mathematics may simply mean letting people from other sub-disciplines access the results of that sub-discipline with greater ease. It may help clear up the “confusions” and “misunderstandings” and “intellectual barriers” between sub-disciplines of mathematics. Thus, when the differential geometer wants to understand that little bit of algebraic geometry, he/she can use a free and easy-to-access algebraic geometry resource along with a couple of standard texts in algebraic geometry rather than hunting around for algebraic geometers, fixing appointments with them, and trying to attend conferences/seminars in the subject (of course, he/she could do these later — but is not forced to do it for a minor point).

In other words, people come to use the bigger and more precious resources (like personal discussions, conferences, lectures etc.) only after exhausting the smaller ones — which also means they come better prepared to gain value from the bigger resources.

Second, there are people in other professions/disciplines who need to use and understand mathematics, and not all of them may be interested in taking mathematics courses to get clear on a couple of definitions. Even those who are keen may not be fortunate enough to take formal courses or have personal sessions with people who can clarify their doubts. For such people, a bok can be a great help. An online resource designed for all people along with a book can be an
even greater help.

For instance, a physics student in my institute wrote in his blog that mathematics teachers urging him to understand the whole thing mathematically, was a principal factor in putting him off mathematics in the beginning.

Third, mathematics is not done only by people in the best of institutes who can meet each other. True, these may be the only people “producing new mathematics” but there are many others who are studying mathematics, either professionally or as a hobby pursuit. There are young student who are thinking of pursuing mathematics. There are people who could not pursue their dreams of research in mathematics and are stuck in more mundane careers, but would like to stay in touch
with the subject.

March 10, 2007

Wikis 2

It’s been over a month since I put my last post on wikis, and at that time, I had just started on my group properties wiki. Much content has flown into the wiki since then, and with the passage of time, I have become more and more convinced that wikis could be a powerful aid to learning and sharing knowledge of mathematics — perhaps even a tool for the active creation of mathematics.

When I first started out in CMI, I was full of a lot of curiosity regarding groups, and I would try to satisfy this curiosity in different ways. I read the Wikipedia articles on the subject, I would surf arbitrarily for terms related to group theory, locate some papers and try reading a few pages and gathering a few definitions from the papers. This was all really exciting and I got a number of ideas and organizational principles on group theory.
ideas were a mesh of complicated stuff. I had this idea here, that idea there. I had some frameworks for my ideas, and I tried writing up the ideas here and there, but it was way too much effort — developing an idea in a full article required a whole lot of effort, and further, there was so much constant changing and updating that was needed that it tended to become a pain.

As a result, many of the ideas and organizational principles that I had remained undocumented. Even the ones that did get documented were usually hard to retrieve/read from the documentation because they were split across multiple files, some of their definitions got duplicated, some lemmas got re-referenced and kept changing in their wording, and so on. So it was a mess. I didn’t know of any way in which I could put the entire web of thoughts I had, somewhere down in writing.

Given the lack of a medium, I shelved aside my work on this front, and tried working on other fronts. In most cases, though, whether I was working with original ideas or trying to grasp existing ones, the structure of the ideas was highly interwoven and nonlinear and articles/write-ups didn’t seem to capture the understanding properly. Still, I used these to whatever extent possible so that at least whatever was covered in the lectures, got properly documented.

Some time in June 2006, I started on an ambitious project of putting up important group theory definitions on Wikipedia. I in fact did put up a whole lot of new articles; however, I soon realized that much of my effort was poorly directed and poorly organized, and moreover, that a lot of it may be undone because there were many other people who felt the articles and material should be organized differently.

It was towards the end of 2006 that I seriously started considering the possibility of starting my own wiki in group theory, and by early December 2006, I had gotten started.

Now for the experience with making the wiki so far.

My initial goal was to use the wiki as a Pensieve (one with easy retrieval) for all the various ideas and facts that were literally taking up too much space in my head. Thus, to begin with, I just kept introducing/writing random articles in nearly random formats. However, as I went along, a certain format/pattern started emerging (in fact I had been subconsciously following this pattern earlier). I also realized that this pattern could be significantly improved upon and thus I designed some generic layouts and formats for various kinds of articles. (I have documented some of these at Groupprops:Article)

Initially, I would manually insert the categories for each article. But then I realized that a more efficient and nicer way would be to create a ”template” for the article type (refer Groupprops:Templates). The advantage of this is that apart from including in a category, it could also print a nice message on screen telling us something about the term (which causes it to be included in the relevant category).

I have also tried to keep a few of the articles always in tune with the latest style. For instance, I have tried to make sure that the articles on normal subgroup, characteristic subgroup, and some others, have the latest format so that they can be used as reference points.

During the month of December and part of January, I was focussing on putting in definitions. Probably that’s because definitions are the thing that fascinate me the most. In January, i started experimenting with articles describing facts and their proofs — I prepared a format for these and started churning out proofs of important statements.

Currently, I am trying to work on improving the depth of the wiki in various themes, for instance, the Classification of finite simple groups, the Extensible Automorphisms problem, combinatorial/geometric gorup theory, and linear representation theory. I am also working on writing more survey articles and expository articles (like the ones on conjugacy class-representation duality, varying on the subgroup property of normality, and varying on the subgroup property of simplicity. These help tie in a lot of definitions and proofs, thus leading the wiki to offer more value than just an organized set of definitions and proofs.

Somehow, though I have put up a lot of content on the wiki and am rapidly adding more, I am still not clear about where this wiki will finally be headed. For instance, I am not clear about the question: do I currently want other people to join into the collaborative effort, or would I prefer to make the wiki fairly robust before I get started on that? Also, do I want other people to come and read it — if so, how and where should I publicize the wiki?

I’ve already circulated the link among a few of my friends here in CMI and a few people outside, but so far I don’t think it has picked up.

(On the other hand, how much can a wiki in group theory pick up?)

Another question I want to answer is: how much effort am I willing to sustainedly put into the group theory wiki? As in, will I continue working on it once I join Graduate School? If the wiki does indeed grow bigger with more people participating and getting involved, am I willing to take on the additional responsibility of coordinating and maintaining it? What end will that serve?

In answer to that, I think I can use the wiki very effectively to document and clarify my ideas in group theory, if only to myself. Hence if I choose group theory as my dissertation subject, the wiki should be very useful in helping me formulate and refine my ideas.

I’ve also been thinking beyond the group theory wiki, towards a general culture of having a wiki per topic (or rather, a huge mass of wikis such that every subject is covered by at least one) — possibly each with different organizational principles or paradigms. For instance, others who don’t like the style or organization of my group theory wiki, but like the idea of a wiki, can start another wiki of their own. Multiple wikis means more competition and more pressure to produce quality material. It also means greater variety for the end-user, each user can choose the wiki whose style more suits his/her personal learning style.

Of course, wikis would not replace the traditional tools of learning, but they could supplement, and provide new inputs. I definitely find learning in front of a computer, with the freedom to click on whatever links I want, and the freedom to search for any term, far more exciting than reading from a textbook. And it often points me to interesting texts to read that I wouldn’t have heard of otherwise.

So for now, I guess, it’s: keep working at the wiki till I get an idea of where it’s headed.

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