Making use of noncommutative algebraic geometry (an area of mathematics introduced by Manin and his students in the eighties), I will explain how to prove Grothendieck's standard conjectures in some new cases.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/6826