What problems are there left to solve? From the ancient Greek mathematicians (Archimedes, Pythagoras) before Christ to Issac Newton to George Birkhoff, these mathematicians have made huge strides in mathematics, developing theorems and even entire mathematical systems. But these theorems and systems were created with a objective in mind, a problem to be solved. Nowadays, asides from a few theorems to be proved (such as the Millennium Problems) it seems that there are no problems which haven't been solved or haven't had an algorithm for solving the problem developed. 
While I consider Mathematics a noble and virtuous pursuit, comparable to Physics or Medicine, where can modern mathematics go? What problems are there to solve? What physical or abstract problems have been left unsolved?
 A: I have come to think of mathematics as an ever growing tree. The leaves are the open questions, and the branches are the already existing theorems; the more important and older the theorem, the thickest the branch. The thing is, we have been doing math for quite some time so the tree is pretty big ; therefore it takes effort and time to reach the top and understand enough to make it to the open questions and research level. 
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To mention a few perspectives for further developments with real life applications : by definition almost all of "applied mathematics", including statistics, a good part of probability theory, PDE's, combinatorics and discrete mathematics, and the so-called "maths for biology".
A: Regarding real-world applications:
I think it's fair to say that a vast majority of the mathematicians who have developed mathematics in the centuries that have passed had no idea how useful we have found their progress to be today. While mathematical theorems and problems solved today may seem to be completely impractical, people may feel very differently a few centuries from now.
