To prove theorems I need a particular kind of intense concentration, sustained for hours, that I don't need for programming, fiction writing, guest blogging, or anything else I've ever done. This kind of concentration seems to come naturally to some researchers, but it never has to me. So over the past four years, I've been keeping a list of what in my physical environment and state of mind facilitates the proving of STOC/FOCS-type results. Although this list is personal and idiosyncratic (and even a bit embarrassing), I offer it in the hope that its very specificity will inspire you to add your own ingredients. Feel free to do so in the comments section.
- Lots of light.
- Adequate sleep the night before (duh).
- Freedom from buzzing insects, screaming babies, ringing phones, slamming doors, and car alarms. I'll never know what I could have proved if not for these things.
- A well-ventilated room with fresh, non-oxygen-depleted air at about room temperature. (Bug screens allow the last two ingredients simultaneously.)
- Caffeine or other stimulants.
- A comfortable swivel chair, or else a couch or bed to sprawl across.
- Long deserted halls or outdoor walkways. (Pacing around in tight circles is no good.)
- Hours and hours of concentration with no end in sight. I've never been able to set aside (say) two hours for serious work, in between other commitments. That's why I work at night.
- Lack of awareness of how much time has elapsed with no new ideas. Before starting to work I take off my watch and hide the Windows taskbar so I can't see the little clock in the corner.
- Comfortable clothes. I've never proved a publishable result wearing a shirt with a too-tight collar.
- Black erasable pens, unruled paper (the backs of printouts serve nicely), Scientific Workplace for TeX, and (don't laugh) MS-DOS QBasic for quick calculations. Substitute your own favorite tools.
- No tempting distractions. Train rides are good: plenty of room to spread out papers and a laptop, but no Internet access (something I hope doesn't change soon).
- No people around toward whom I have strong unresolved feelings (attraction being only one example).
- Freedom from bodily annoyances and pains. Advil, cold medicine, lip balm, a nail clipper, and a glasses cleaning cloth are important weapons in my theory arsenal. Also, I can't do serious work until about half an hour after a meal.
- A positive attitude, which is fostered by a calm, uneventful week in my life.
- Colleagues to talk to. People able to shoot down wrong proofs are ideal, but even "write-only black boxes" are invaluable as sounding boards. Of course I try to reciprocate both services.
- A problem that I consider "mine" -- either because I posed the problem, I've had recent successes on subproblems or related problems, the problem is important for one of my research goals (or even better, two goals), or I'm (rightly or wrongly) seen as the world expert on the problem.
- A problem that others are eager to see solved. It's easier to let myself down than to let others down.
- Conference deadlines. They motivate me to work, but then if I miss them (as I do), my "research GPA" doesn't suffer: there's always the next conference.
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