If one were to pursue a career in mathematics, what would be the most "cutting edge" branch to specialize in? I was thinking topology, but there may be some interesting ones I haven't heard of. Thanks in advance for your opinions.
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I'm afraid that 'cutting edge' mathematics involves many ideas that you probably havent come across yet. One of the best places to check if you want some idea of 'cutting edge' research in maths, physics etc is the arxiv:
http://arxiv.org/
this is where papers go before they are published by journals.
http://arxiv.org/
this is where papers go before they are published by journals.
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If you are interested in physics, then the math that would be most appropriate would include algebraic topology and noncommutative geometry. These are the cornerstones of M-theory and quantum loop gravity which seem to be the hot topics in contemporary physics. Don't scrmp on Abstract Algebra since group theory is the language of symmetry.
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Pure mathematics tend to land jobs in the unemployment line...or a university if you're lucky.
Apply mathematics to engineering and you'll make some cash.
Apply mathematics to engineering and you'll make some cash.