When you say "hartest", I assume you mean "hardest". There are a total of 7 Millenium problems, which one of them has been solved by Gregori Perelman. These problems are so intense that, whoever gives a correct answer (or proof) will be awarded 1 million USD. The topics are: P versus NP The Hodge Conjecture Yang-Mills Theory The Poincare Conjecture Riemman Hypothesis Navier-Stokes Birch and Swinnerton-Dyer Conjecture