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I recently heard about a branch called tropical geometry developed by Professor Bergman. I was wondering if there´s a newest but yet unexplored math branch, and by newest I mean developed in the last 60 years, and by unexplored I understand a branch without a huge amount of written articles, books, given seminars or courses in the top universities. I am just asking for pure math branch, and not applied math or mathematical physics. Thanks.

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    Apparently Shinichi Mochizuki has singlehandledly started a new branch of math in pursuit of the ABC theorem; he perhaps has a totally different approach to what numbers are. which is promising given the immense difficulties of number theory. It seems like it is going to take many mathematicians many years just to read what he has already done. Then maybe they'll give the field a name and people can work on it. Read about it here: http://projectwordsworth.com/the-paradox-of-the-proof/2013-12-22

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There's a lot of new and unexplored branches of math. I study finite subdivision rules, which were invented in the 80's and less than 50 papers discussing them explicitly. String theory in it's mathematical form is another.