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I have seen that manuscripts published in the Proceedings of American mathematical society have "communicated by x" , just below the title of manuscript. I was thinking how to decide which name to write in place of x if we are working on some branch say topology. Is it the name of the referees, if yes then how to know a referee of any particular area?

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The “communicated by x” means x is the journal editor who was responsible for the paper. You do not write the “communicated by x” when you submit the paper. The journal will add that line after it is accepted for publication

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