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My question is as follows: It is now the year 2013 as we know it, and I'm wondering what the "big topics" in mathematics are. What fields are of utmost interest and foundation in the modern era? How should an aspiring mathematician create variety in his/her mathematical thought? Simply put, what are the "big topics" in mathematics?

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In what context? Did you mean interesting topics and field for mathematics research ? –  srijan May 1 '13 at 17:40
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This question is way too broad. –  Asaf Karagila May 1 '13 at 18:00
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You can take a look at Smale's list of the mathematics problems for the 21 century. Here is the link. However I believe that each topic contains a number of hidden gems if you dig deep enough.

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Big topics in Math can be anything, it can be the things which haven't been proved for centuries, the applications of them in real world. Many conjectures of math still remains unproven(For instance, infinitely many twin primes conjecture). When Fermat gave his little theorem, he never thought it would be the key for RSA encryption.

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