THE ART OF RESEARCH

1. Survey method: What is known about ...? You try and gather the known facts about a topic and put them in some order and coherence. This usually reveals what is not known. Thus, clarify what you know so that what you don't know becomes revealed. For example, survey the topic of counting finite groups.
   
   
2. Observe and Conjecture method: every science is based on accurate observations. We observe, we experiment and notice a pattern. Then we make a conjecture about the general pattern. Such results can be published in Mathematics of Computation. The Birch---Swinnerton-Dyer Conjecture is a good example of this kind of research. Conjectures are nice for two reasons. One, we can look at special cases and try to prove them. Two, we can look at some of the consequences and try to prove those independently. For example, if we assume GRH, what can we get? If we don't, what can we prove? Can we average? You can look at the universe through one of these conjectures, like ABC, or Schaunel's conjectures, Selberg's conjectures and so forth.
   
   
3. Re-interpretation method: what is the theorem really saying? My favourite example of this is the Kummer congruences proved in 1859 in the same year that Riemann analytically continued the zeta function. The real meaning of Kummer did not come out till 1964 when Kubota and Leopoldt saw its p-adic significance.
   
   
4. Analogy method: we can always ask about the function field analogue. Emil Artin discovered the zeta function of function fields in this way and paved the way for the Weil conjectures that revolutionized number theory and algebraic geometry in the last century. Another good question is: what about the p-adic analogue? For example, even though Euler discovered the Gamma function around 1700, the p-adic version was discovered only in the 1960's. Then there are q-analogues versus t-analogues that recur in the theory of zeta functions.
   
   
5. Transfer method: can I use this theorem anywhere else? A good example is the Chebotarev density theorem and its ubiquitous appearance.
   
   
6. Induction method: what made the proof click? This can also be called the Bourbacki method of generalization. What were the essential ingredients in the proof? Can we isolate them and put them in a general context?
   
   
7. Converse method: what about the converse? For example, the converse of Fermat's Little Theorem leads to the notion of pseudoprimes and has applications in probabilistic primality testing.
   
   
   Here are some habits that stimulate research.
   
   
   
   1. Browsing: one must browse through jounals to keep some awareness of the literature. Often taking an old volume of your favourite journal and looking through the papers till you find one you're interested in is very helpful. This stimulates the serendipity factor. You never know what you will find. Looking at Russian Math Surveys, or Sugako Expositions, or even the Notices of the AMS or Math. Intelligencer is a good beginning.
      
      
   2. Memorization: memorize the proof of a theorem. Then only can you savour the subtle and key ingredients. The techniques enter the subconscious and will rise up when the occasion demands it. But they have to be put in memory first.
      
      
   3. Writing: find a time every day when you can write non-stop for about 2 hours. A moderate knowledge of a book can be gained in about a month's time this way. In fact, 60 hours is more than 36 hours devoted for a formal term course.
      
      
   4. Taking Notes: when you go to a lecture, take notes and ask a few questions. That will help engage the mind in concentration. Ask yourself, can I take home an idea from this talk? Richard Feynman used to have at least half a dozen research problems at the back of his mind and during lectures his one question (to himself) would be: can I use what this guy is blabbing about in my work?
      
      
   5. Writing Down and Writing Up: when you prove a theorem you should write down the proof and check for mistakes. Explain it to a confidante and that way you can find any hidden errors. Once it is certified right, write it up for publication. Make sure there is a good introduction briefly indicating the history of the problem and where this result sits in the larger mathematical landscape. It is good to have time limits. After one month of intense work on a problem, you have to ask yourself if you got anywhere and write down whatever you have. After three months of intense work on a problem, you should write up whatever you have for publication. Three papers is a good average to maintain to secure tenure.
      
      
   6. Giving Talks: take every opportunity to give talks either to learn new material (like in a graduate student seminar) or present your work in conferences. Know the difference between a class talk, a seminar talk and a colloquium talk. A talk should strive to convey ideas more than technical details. A colloquium talk should never be encumbered with technical details. The usual recipe for a colloquium talk is to have the first quarter accessible to undegraduates, the second quarter to graduate students, the third quarter to faculty and the fourth quarter to the local experts (if any).
      
      
      
      SUGGESTED READING
      
      
      1. J. Hadamard, The Psychology of Invention, Gauthier-Villars, 1908.
      2. P. Halmos, I want to be a mathematician, Springer, 1985.
      3. P. Medawar, Advice to a Young Sicientis, New York, Harper and Row, 1973.
      4. H. Poincare, Mathematical creation, reprinted in J. R. Newman's The World of Mathematics, Simon and Schuster, New Your, 1956.
      5. G. -C. Rota, Indiscrete thoughts, Birkhauser Boston, 1997.