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!