In 1998,
Field Medalist Stephen Smale drew up a list of 18 unsolved problems in
mathematics as targets for the mathematicians of the
21st century, in the fashion of
Hilbert's
original 23 problems at the beginning of the 20th century.
It was also inspired by a suggestion by V I Arnold on behalf of the International Mathematical Union. Only one's been proved, so get cracking on the rest of them guys!
The 18 are as follows:
1) The Riemann Hypothesis.
2) The Poincaré Conjecture.
3) Does P=NP (are P-problems the same as NP-problems).
4) Integer zeros of a polynomial.
5) Height bounds for Diophantine curves.
6) Finiteness of the number of relative equilibria in celestial mechanics.
7) Distribution of points on the 2-sphere.
8) Introduction of dynamics into economic theory.
9) The linear programming problem.
10) The closing lemma.
11) Is 1-dimensional dynamics generally hyperbolic?
12) Centralizers of diffeomorphisms.
13) Develop a topology of real algebraic curves and surfaces. This was also Hilbert's 16th problem.
14) Is the dynamics of the ordinary differential equations of Lorenz that of the geometric Lorenz attractor of Williams, Guckenheimer, and Yorke?
This was proved to be 'Yes' by Tucker, 2002.
15) Navier-Stokes equations.
16) The Jacobian conjecture.
17) Solving polynomial equations.
18) Limits of intelligence.
Note that, like Hilbert's list, not all are simple problems to be proved. Some are more suggestions for future areas of research.
Disclaimer: I am not a mathematician, and do not pretend to be one. I have little idea what many of these problems actually mean, but I am interested in mathematics, and intending to do a degree in the same. So, if you have any questions, please save them for a few years.
Reference:
Mathworld: http://mathworld.wolfram.com/SmalesProblems.html