Generating research, interest

I have taken Stephen Semmes‘s topics course for each of the past six semesters, and think he has a fantastic teaching style, partly because I could never see myself using the same one.  During a typical class, he will make a definition or prove a theorem, and then ask “Does anyone have anything to say about this?”  The path the class takes is then largely controlled by answers to this question.  I am constantly impressed by how many answers to this question (often early in the morning!) Dr. Semmes and the students in the class have.  More specifically, I am interested in the process of formulating such answers.

This is, of course, a huge topic (that is to say, how are scientific ideas/good research questions generated).  Just to put down a few observations of good questions to ask:

Statement of a definition (“this is an X“):

  • An example of the object/phenomena being described (“x is an X”)
  • A characterization of the object/phenomena (“x is an X if and only if x is a Y,Z,…”
  • The object/phenomena’s relationship to previously defined objects (“every object that is X is also Y” or “every object that is Y is also X“)

Statement of a theorem (“If A, then B“):

  • Is the converse true? (“If B, then A“)
  • What is true with weaker hypotheses? (“If a, then b“)
  • What is true with stronger hypotheses? (“If A, then B“)
  • Is the converse true with stronger hypotheses? (“If B and C, then A“)
  • Is there a stereotypical example of the situation described? (“X is A, so X is also B“)
  • Why is this surprising or interesting? (“If almost A, then not B“)
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