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How can Mathematicians in Isolation Collaborate? 
      Collaboration from an isolated location is relatively easy today, using tools like Skype for direct communication and "dropboxes" for sharing text and computer code. Regularly scheduled meetings are particularly helpful to keep a project at the top of everyone's list.
        The difficulty is in beginning the collaboration. This is most easily done in person, so one of these methods might work:
          If you have an idea and a possible collaborator:
            -- Before a professional society meeting, contact a person you are interesting in collaborating with and ask if you can meet at a specific time at the meeting to discuss a possible collaboration.
            -- Contact the person by email and ask if you can have call to talk about the idea.
            If you have a possible collaborator but no specific idea:
            -- Invite the person to give a talk at your institution (if you have the funds).
            -- Ask if you can give a talk at her/his institution (and offer to pay your own way).
          In either case, mention your hope that you can set up a collaboration.

        Christina Sormani
          Long distance collaboration does not necessarily require live communication.   In fact, I find communication by email to be more effective even when collaborating with someone located nearby.  This allows me to think things over more carefully before suggesting or rejecting an idea.   The way I've collaborated long distance has been to keep a running tex file, which is exchanged between the coauthors with one owner at a time.   The file can start with just a list of conjectures.  Then eventually we start filling in lemma statements which would turn the conjectures into theorems.   Slowly the lemmas are proven or restated.   At every stage, the file is sent back and forth with comments in blue or in capitals pointing out what has changed since the prior version.  With this form of collaboration, the paper is written up already as soon as the proofs are completed, so that one needs only write an introduction and the paper is ready to go.   This has been a very effective means of coauthoring with two coauthors of mine (Wei and Shen) and we never met once in person or called each other for the entire collaboration.  With Shankar, we completed a collaboration after only one meeting in this manner.   With my doctoral students, I have them keep a running tex file in a similar style even though we do meet once a week.  It forces them to keep what they've done precise.  One of my doctoral students who recently graduated from a nearby university, stopped needing the weekly meetings and completed his thesis via texed email exchanges alone.

        Natasha Dobrinen:
          In addition to the above methods of collaborating in isolation, I'd also like to mention that research visits can be very motivating and fruitful. Find a week or two or a month when you can visit a collaborator, or the collaborator can visit you. These times can jump-start progress on a project. Apply for grants to cover this research. Ask the host institution if it can cover some of the travel/lodging expenses. Some seminars have funds at their disposal to pay for visitors. This can be utilized to pay for part of a research visit. Agree to attend the same conference and stay extra days before or after the conference to work together. In-person visits act as a catalyst for the continued work done in isolation between visits.

      • See also Stephanie Alexander's response to IIIK about Research in Pairs programs.