Compute, compute, compute

The workshop gets over tomorrow and then a day later I'm going down to LA to spend about a week with my brother and his wife. It's been an interesting workshop. Of all the conference/workshops that I've gone for this was easily the most useful and productive one. What was different was the style and pace of the workshop. Most math conferences consist of a bunch of lectures, a few question-answer sessions and informal discussions with other mathematicians. Some of the discussions can lead to future work but it's mostly about networking and letting other people know about your work or existence.

This one was a very intense workshop with lectures in the morning on the computational aspects of number theory followed by coding sessions through the day. Different problems had been posed before the workshop and people worked in groups on the problem they were interested in. It was my first experience (since a couple of farcical undergraduate projects) of working in a group and trying to make presentations of everything that had been done in a short period of time. Over the last 10 years, as computers have become faster, a number of programs have been written which make computations of fairly sophisticated and abstract concepts much easier. The main purpose of this workshop is to integrate a lot of the existing programs into a free, open source language called SAGE. This will hopefully become the main resource for programming and building databases of examples and computations. As it's free and open source, changes and developments can be made by anyone using it. It's an ambitious project and it looks like it's gathering a lot of momentum.

For the past few years, I've worked in a very abstract and theoretical aspect of math (number theory - though it has nothing to do with numbers) and become used to working alone using simply pen and paper or a blackboard. Occasionally, about once a week or two, I used to meet my advisor for less than an hour to ask questions or show him what I'd done but that was about it. It takes months or close to a year to work out a reasonably advanced problem so one gets used to plodding away for weeks with nothing to show. There are no experiments to run, events to understand, societies or people to study or laws of nature to follow. Then, there's suddenly a flash of insight and a few lines of abstract nonsense can explain something very clearly. Sometimes subtle mistakes can be passed over since only a few people actually cared about the problem. What is daunting is coming across other mathematicians who are far more comfortable with the concepts one struggles with, and who seem to say things which can seem incredibly vague. When things make sense, it's a very enlightening moment.

Working on a computer with a group was completely different. There are instant results popping out all the time. This leads to a lot of quick thrills, high fiving and instant elation. Also, a lot of frustration, when there are errors or bugs in a program. Four of us sat around a computer and kept bouncing ideas off each other and writing down code. Since it was my first experience at programming there were more bugs and errors throughout but getting a program to work is pretty satisfying. A lot of the programming skills and structures that I saw and learnt are definitely useful. Writing a code to compute a series of examples of various cases actually helped to understand a lot of things compared to reading a paper or a book on the same topic. Especially as a slight error or bug is evident immediately. It also involves a completely different way of thinking compared to what I've done in the last few years and it's been an enjoyable experience.

Though the larger question did loom - was it really so useful and important? The main idea of the problem was to compute some very specific examples, so specific that it's hard to come up with a conjecture or construct a general theory around them. What surprised me was the large number of people who seem to do just this - compute, compute and compute. Words like interesting, cool and exciting were used a lot. A few conjectures kept floating around and since it's a high-speed game people tend to claim things too soon. I've always had a fairly low opinion of my own mathematical and research skills and in an objective discipline like mathematics it's easy to see how good/useful someone's paper is. My thesis was pretty average and at most reasonable. Seeing the way people were getting excited over small computations was puzzling. The prospect of publishing a paper out of all this is what seems to excite everyone, not the prospect of coming up with a useful idea. I'm pretty sure it's the same in other disciplines as well.

I have a long rant on how 80-90% of academic/research papers are complete nonsense (including my own thesis). Forget about them being useful to our daily life, they're not even useful in the narrow academic discipline they fall under. But that's something I'll touch on later when I have a lot more time to waste.