Pure mathematics

[VJLQj4hQs2Q] Interview with Terence Tao.

Doing research math feels a bit like starting to watch an infinitely long tv series at the 1000th episode. Initially you don’t know who are the main characters, who’re the good guys, who the bad, what the overarching plots are. But over time you start to pick up things.

http://web.math.princeton.edu/generals/ “Graduate Students’ Guide to Generals: A database of the general examinations taken by graduate students in the Princeton University Mathematics Department.”

I was reading some of these, including Terence Tao’s. It occured to me that in my math studies I never quite got to understand how you decide that some line of research in pure mathematics is important. In applied mathematics, including computer science, it’s kind of easy because you can imagine the technologies the discovery or invention enables. I liked studying most kinds of math, but now that I think it always felt a bit like “studying the classics” in the sense of literature. Sure fourier analysis forms the basis of tons of applications, but the way I got to study the mathematic details of it, it wasn’t clear at all knowing them would be useful. Could I use them to prove theorems in other related fields? Are there still major discoveries to be made in Fourier analysis? Who knows.

It felt like I was studying that stuff because I had to study something, so might as well pick the fun-looking and important-considered stuff. But as it turns out, now that I no longer really do maths, it didn’t matter what details or even what fields of math I studied, just that I did and built general math and problem solving skills. It feels somewhat of a waste to have that detailed knowledge fade away, because for example if I instead knew details of computer science it might help me with all the programming I do daily.

Regarding how to decide direction of research in pure mathematics, I can make some guesses: Problems like Poincaré conjecture are important because researchers have already found results and directions of research that hinge on the assumption of truth of a conjecture, so proving what they suspect to be true “unlocks” further research.

Second is the known trend of generalizations to eventually prove useful. Something like real analysis is obviously useful by itself, for solving physical problems etc, and mathematicians are able to see “straightforward” ways to generalize the results. Similarly, by intuition mathematicians are able to spot missing analogies between fields and go on to discover them. I think many mathematical results are valued simply because the elegance of their existence satisfies other mathematicians. They feel like a missing piece has been found.

In theory you can generate infinite amounts of mathematics by starting from new assumptions no one else uses, but my understanding is that in general you can usual “fit” such research on to the existing similar theories, so anything too out-there will be rejected as ugly.

But still I think from the practical perspective of being a fresh graduate student it might be really intimidating to know what to research. As I understand one of your supervisor’s most important duties is to guide you to the “interesting” problems. But because math is such an old and deep field, as a new graduate student you might not even be close to the edge of current research and you have to wander in almost blind. There, studying new papers I imagine it might be quite difficult for a novice to understand why (or if) a paper’s result is considered to be interesting. Probably everyone can see ways to extend the results of any of the papers, given sufficient work, but usually that would result in “trivial” advances and rejection of the papers because you didn’t really understand where the research is trying to go.

By the way, I hate the only scientific paper I have authored because it felt so trivial to me in this sense. In my view I just marginally improved some numerical results without providing any kind of insight into the topic. I think maybe it’s not so shameful to begin your career from somewhere low like that — by the time the paper was complete I had about 10 months of experience with computer vision, or actual computer science as whole. But I feel like people who are suited to a particular line or research are able to provide that “insight” from the very beginning of their careers, and I wasn’t one of them. At the very least you need to see the insight in the major results by others and be excited about them. I wasn’t.