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One Introduction to Mathematical Research

Overly general introduction Few prospects are as daunting for a serious undergraduate math major as that of doing research. Mathematical research is harder than homework. You can't look for the answers to the chapter in the back of the book; you don't know which chapter, and you often don't even know which book, and it wouldn't be there anyway. You're not even sure what would constitute an acceptable answer. But in some ways, mathematical research is easier than homework. The time limit is flexible, and since you have chosen the question yourself, you are more likely to find it interesting. And, if you invent your own problem, but solve a different one, you can always say that it was what you were trying to do in the first place! The best research involves deep new ideas and you can't find all the secrets in any article. Nevertheless, there are a few tips that will put you in a more favorable position to start your research career. The most important thing you need to do is to adopt an active attitude towards the mathematics you learn. When you see something new, don't ask "Will this be on the test?", ask "How can I make this work for me?" Be bold. To paraphrase the 1960s revolutionary Franz Fanon: "Knowledge cannot be given, it must be taken." Learn as much mathematics as you can, in as many different areas as you can, including areas in which mathematics might be applied. Take hard classes. Go to as many mathematical lectures as you can, even if you won't understand it all. (Big secret revealed: most professors go to lectures and don't understand it all.) In my experience, research consists of question-asking, problem-solving, knowledge-finding and checking, checking, checking. There are many excellent books on problem-solving; start with How to Solve It by George P6lya. Knowledgefinding is extremely dependent on your location-always get to know your librarians. Asking big questions requires inspiration. The rest of this article is devoted to the tabletop craft of asking little questions.