I’ve been interested in group theory for a long time (in fact, ever since twelfth standard, as I have documented here). Apart from learning the subject, I also explored organizational principles in group theory, particularly those centered around properties of groups and subgroups, and I made many observations regarding groups.
Some of these observations led me to the Extensible Automorphisms Problem as well as other questions in group theory. Other observations helped open the way for studying topics closely related to group theory such as representation theory, combinatorial group theory, and so on.
Often, many definitions and terms in group theory (specially those introduced in recent times) are hard to look up and understand because they are used only in obscure papers. I have strongly felt the need for a means for easy reference to definitions and proofs. I have also sought ways in which people can easily “look up” whether or not a certain fact is true without having to go through masses of literature, and also locate proof ideas that may be useful for a particular problem they are working on.
For some time, I have felt that with my understanding and perspective of group theory, I can work towards the creation of a knowledge repository in the subject that allows for quick lookup of facts and also encourages fruitful exploration.
Some time in mid-June and early July, during free time in between my Visiting Students’ Research Programme at TIFR, I started on an intensive programme of putting up articles on basic definitions in group theory on Wikipedia. Around that time, I decided that whenever I come across a new term or definition (that is reasonably standardized), I will put it up on Wikipedia. This way, Wikipedia serves as my canonical self-reference tool, while also providing a repository for others.
However, I realized that putting up material on Wikipedia has some limitations:
* I cannot organize the article content of definition articles the way I want
* I have very little control over the global structure, something which is very crucial to effective navigation and exploration.
* I cannot put up original work and original ideas
Thus, after some time, with the pressure of other work, I shelved aside my Wikipedia drive.
Some time in late October, with CMI’s shift to its new hostels imminent, the students of CMI decided to use a wiki for discussing the important shifting-related concerns. After some research, Shreevatsa started out the page here on editthis.info. Later, I decided to set up a wiki detailing the activities of CMI Spark. This differed from the shifting concerns as it was an ”entire wiki” as opposed to just a wiki page.
Towards the end of November, while working on the Extensible Automorphisms Problem, I got very confused about the whole lot of terminology that I was both using from standard sources and introducing on my own. I first thought of starting a wiki on extensible automorphisms, and even made a few pages. I realizd that the wiki required lots of definitions from group theory, so I decided to convert it to a group properties wiki, that I made here.
I first started work on this wiki in the end of November, just as I was leaving for home after the semester in CMI. After going home, I had to concentrate on wrapping up my applications, and so I worked only sporadically on the wiki. Towards the end of my vacations in December, I started putting huge spurts of effort for the wiki. After returning to CMI in January, I spent the weekend on the group properties wiki.
My initial attempt was to ”pour out” what I knew of group properties and develop the format of the articles through experience and actual practice with writing the articles. I also drew on the ideas of property theory. With each new batch of articles I wrote, I discovered some general ways in which the article structure could be made more effective. I then tried to do a back-cleaning of the articles as per these general ways.
In the beginning of January, once I started feeling that the wiki had grown to a reasonable size and had reasonable width of coverage, I shifted focus to creating policy guidelines for the wiki. This process has been continuing till yesterday.
Working on the group properties wiki taught me that a wiki could be a very effective tool for creating, formulating and sharing ideas. So far, the ”sharing” part has not been active.
Today, when I realized that I needed to brush up commutative algebra in order for studying certain areas of mathematics, I looked back at my commutative algebra texts. I realized that many term definitions that I had studied earlier failed to stay in my mind because I was unable to grasp the overall structure and linkages. Then, I thought about starting a wiki on commutative/non-commutative algebra and algebraic geometry. Unlike the group properties wiki, this would be on a topic where my knowledge and foundations were quite weak and where I did not have any ”new” ideas. On the positive side, though, I had my experience with the group properties wiki to provide me a rapid start.
I started the wiki today here and have already made decent progress.
I have also been trying to develop a theory of mine that I dubbed “APS theory”, and I am using this wiki to document the theory.
How these wikis shape up and whether they get used by people other than me, remains to be seen.