Research (according to Wikipedia)is often described as an active, diligent, and systematic process of inquiry aimed at discovering, interpreting and revising facts. The “discovery” part of research has always fascinated me because of its connotations of “charting out new territory” and “exploring the unknown”. I want to know: how does one determine the right direction to proceed in order to discover new stuff?
Research activity was classified by WT Gowers into problem solving and theory building. The approach of researchers often varies between these two “ends” of the spectrum. Compulsive problem solvers, like Paul Erdos, feed themselves with an unending supply of problems, and churn out the solutions just as quickly. Erdos was legendary not only for his ability to quickly solve and move between problems, but also for his ability to identify, and set rewards for, the problems which required more effort. The apotheosis of the theory-building extreme is Grothendieck, who spent a concentrated thirteen years creating the “foundations” of algebraic geometry.
“Erdos style problem solving” has a more childlike appeal to it than the grave “Grothendieck style theory building”. Also, problem solving has a higher “discovery component” compared to theory building, which also involves interpreting and revising existing knowledge. For a person (like me) who’s just entering the research world, the problem solving route seems more attainable.
Hence the question: How does one select problems and go about solving them?
What is an open problem? It is a problem that hasn’t yet been solved. Open problems could be:
(i) Conjecture: A conjecture is an assertion which the “community” believes in, but which has been neither proved nor disproved. For instance, the Riemann hypothesis, the Poincare conjecture, and the Goldbach conjecture.
(ii) Gamble: A gamble is a yes/no open problem where the “community” does not believe in either a yes or a no.
(iii) Loosely formulated problem: This is an open problem that has not been formulated precisely. For instance, in classification problems, the meaning of the term “classification” is usually ambiguous. A loose problem formulation often indicates that the solution requires more of “theory building” than “problem solving”.
Should a researcher always pick open problems for research? An open problem is not the only option for researching on. A researcher can work instead on “interpreting” and “revising” existing knowledge by giving yet another proof of a known result or patching up an existing proof.
But an open problem carries with it the whiff of mystery, and solving it means expanding the body of knowledge and uncovering new territory. It is one thing to say “I found yet another proof of the Fundamental Theorem of Algebra” and another to say “I proved the Fundamental Theorem of Algebra”. It is one thing to say “I classified all non Abelian geometries” and another to say “I have recompiled the classification of all non Abelian geometries”.
So the next question: How should a researcher pick his/her open problem? Picking an open problem has two parts: picking the problem, and picking the stand on the problem. Picking the stand is particularly important. A researcher cannot just say “I am going to solve the Goldbach conjecture.” He/she says: “I am going to prove the Goldbach conjecture” or “I am going to disprove the Goldbach conjecture.” In fact, he/she has to go further and say “I believe in the Goldbach conjecture because of this-and-this and I hope and plan to prove it using such-and-such.”
A researcher needs to believe in what he/she is setting out to solve. I don’t think it pays to say “A lot of people want to prove the Riemann Hypothesis, so let me do that”, or worse, “A lot of people want to prove the Riemann Hypothesis, so let me disprove it and become really famous”. Belief, both in the goal and the approach, is crucial for the researcher to carve the path towards the goal.
When I say believe in the goal, I don’t mean that a researcher pick a vague topic of his/her own choice, work on it in isolation, and come back to publish it. Rather, I mean that the researcher picks a problem which is both important to the community and lovable to him/her.
But how is it that some people struggle to find a problem they can fall in love with, while others go around solving problems left, right and center? The question puzzles me a lot, and I hope that the comments and subsequent posts will throw some light.
Do check out how these views tie in with the You and Your Research page.