Extremal metrics and K-stability
  We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it. 
G\'abor Sz\'ekelyhidi