There is a general feeling that the internet and our new “connectedness,” have tremendous power to foster new communities.  However, sometimes the internet can feel like a very lonely place (in the same way, I suppose, that New York City can be the loneliest crowded place on earth).  I wanted to point the way towards two amazing communities centered around mathematics.  They are both long-running and large enough to have developed a “community feeling,” and have enough dedicated members to keep that feeling positive and continuous.  They are both “question and answer” type sites, but the questions and answers often spark additional commentary and sometimes lively debate.  The target audiences are quite different.

The first, math.stackexchange.com, is aimed at people studying mathematics at any level, from high school to graduate work.  Can’t figure out how to factor that polynomial?  Want some broad context on the challenges of integration?  Can’t prove that theorem in group theory?  Post your question here.  Homework questions are welcome, if asked honestly and explained properly, but don’t expect to post your assignment and return to find all the answers — for the most part, there is a nice culture of explanation, encouragement, and providing hints.  Some sample questions:

• Factor the binomial 
• Rotating two vectors to point in the same direction
• Prove: If and and are nilpotent, then is nilpotent.

The second, mathoverflow.net,  is aimed purely at research-level mathematicians.  A number of world-class mathematicians, including several fields medalists, post regularly on this site. Working through a famous paper, and don’t understand a key step in a proof?  Want to know if the hypotheses of a given theorem are really necessary?  Have a question, and not sure if it’s been studied before?  Studying something out of your area of focus and want some information on background, relevance, or sources?  This is a great place to ask.  Beware, the “research-level” condition is strictly enforced, and askers are expected to put a little effort into seeing if an answer exists on the internet already.  Violators are gently directed to math.stackexchange.com (referred to as “math.SE” on the site).

• Is there a natural random process that is rigorously known to produce Zipf’s law?
• How to memorize (and understand) Nakayama’s lemma and its corollaries?
• There are few “big-list” type questions (as opposed to focussed questions with a single answer) — these questions are not really encouraged, but past instances have provided some very rich reading, including:
  A single paper everyone should read