Is it possible to search the main site for questions which do not have a certain tag?
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This is doable with the existing search. Tags are indicated by square brackets, and a - sign indicates that the term which follows should not be present in the search results. Searching for "topology -[ag.algebraic-geometry]" (without the quotes) finds all questions containing the word "topology" and lacking the tag ag.algebraic-geometry.
Incidentally, to restrict to only questions with a certain tag, the same style of syntax works, so a search for "topology [ag.algebraic-geometry]" will find all questions containing "topology" and with the tag ag.algebraic-geometry.
