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I would love (in theory) to develop a little custom-made course called "let's make a publishable applied math problem out of that topic you love thinking about"
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Sure, if you're so inclined. As I'm sure you're aware, there's a journal out there for any paper you could imagine writing. So "publishable" is really just a synonym for "sufficiently completed and of sufficient quality that you're pleased to put it out there."
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