Research

This is the homepage for the Research module of the Spring 2010 quarter of Mathematical Systems.  More information will be distributed in class (Wednesdays 12-1 in Sem2 C3107) than will appear here, so come to class!

General requirements are that you:

  • come to class every week,
  • work at least 10 hours per week on this subject, and
  • keep a research log (with time, date, and a summary of what you did in each work session; this may include, but is not required to include, the actual work you do, which is to say, it may also be your research notebook, but it can be a separate object from the paper on which you do your research).

Here’s a quick summary of the current and past assigned work for this part of the program:

  • Week 1 (due Week 2): read this packet (and work some of it, as much as you like and have time for) to get a sense of that author’s perspective on how to ask good questions; find at least 3 problems you’ve seen and can solve (from past homework, workshops, exams, Putnams, recreational math problems, whatever) that you might be interested in expanding into your research area for the quarter.
  • Week 2 (due Week 3): For each of the 3 problems you identified in your work for Week 1, develop at least 20 related exploratory questions (for a total of at least 60 in all).  The questions should span the biggest range of difficulty that you can manage – all the way from very easy problems you could probably solve in half an hour to questions that are so hard and/or big that you might never be able to solve or answer them definitively.  Try to find a useful way to organize your questions.  Bring your 60+ questions on paper to turn in next week.  If this work doesn’t take you at least 10 hours, spend the rest of your time for this subject this week working on some of the problems/questions you came up with.
  • Week 3 (due Week 4): Do the items in the list below. Also, remember that we’re meeting at 4pm next Monday (Week 4) in Sem2 C3107. 
    1. Choose the problem/area (from among the 3 you’ve developed over the last two weeks) that you’ll be working on for the next 7 weeks (if you want to choose something other than one of the 3 you’ve already been thinking about, you need to talk to me about it).
    2. Develop/choose 1-3 clear “context-providing” questions for your work: these are big, probably very general questions that seem difficult enough that you could spend years working on them without giving an exhaustive answer.  You won’t work on these, but they will provide the context for your work and clear background for your audience when you talk about your work.
    3. Choose 3-5 questions that you don’t know the answers to but that seem like reasonable goals for your work for this quarter.  These should be at a range of difficulties, but a rather small range – not too easy and not too hard.  (These are not the questions you’ll start with, so be sure you don’t choose questions that are too easy.)
    4. The bulk of your work for this week: Choose a doable problem from your list (or one that you add to the list) and do it!  If that goes well, choose another problem and do it.  If that doesn’t go well, try working on another one.  Come to class next Wednesday with at least one clear solution to a problem from your list; you’ll give a short presentation (roughly 5 minutes) of your work in class next Wednesday.
  • Week 4 (due Week 5): Keep working on your problem.  Next Wednesday we’ll check in and talk about how things are going.
  • Week 5 (due Week 6): Write up (which means type) a short document containing the following: (1) the question that started your investigation; (2) the high-level framing questions that you developed previously for your work; (3) a summary of your work so far; and (4) a proof of at least one of your results so far.  Think of this as a brief report on your work, and structure it accordingly.  Also, come to class prepared to give a short presentation on your work and to turn in your report.  I’ll collect your research log as well.
  • LaTeX Stuff: Here are a couple of files that might interest you if you’re working with LaTeX:
    • texsample.tex, the sample TeX file we made in class
    • format.sty, my own standard formatting file
    • combfinal.tex, the Winter 2010 Combinatorics final (of possible interest for the formatting commands at the very beginning, as well as for the Combinatorics formatting commands)
  • Week 6 (due Week 8): Write up a second research report.  We’ll be on retreat in Week 7, so this is due in Week 8.  You’ll also give another short presentation in Week 8.
  • Week 8 (due Week 9):  Keep working!  We’ll just check in in Week 9.
  • End of Quarter: You’ll give a final research presentation, up to 30 minutes long, on the Monday of Evaluation Week.  You’ll also submit a final research report, which should include the following sections (and any others you deem appropriate): 
    • Introduction & Overview
    • Sections about your work
    • Open Questions / Directions for further work
    • Reflections (This should be a detailed reflection on the research work you did this quarter.  What did you learn from the experience?  How did it transform you?  What can you take from this as you go forward?)