A famous conjecture due to Yau states that a Kahler manifold admits a constant scalar curvature Kahler metric if and only if it is ""stable"" in the sense of Geometric Invariant Theory. This conjecture has been expanded and clarified by Tian and Donaldson and through their work and the work of others one direction of this conjecture has essentially be proved: the existence of such a metric implies stability. The converse is largely open and should be considered as a central problem in Kahler geometry. The PI will study stability of manifolds and orbifolds from the viewpoint of algebraic geometry and as an obstruction to the existence of constant scalar curvature metrics.
A fundamental concept in mathematics is that of a metric which defines a notion of distance and thus the ""shape"" of a geometric object. In some cases it is possible to find a good metric with particular properties. For instance on a ball there exists the round metric, making the ball into a sphere (rather than an ellipsoid). The analogous metrics in higher dimension are called Kahler-Einstein metrics and extremal metrics. These metrics have applications in geometry and mathematical physics, and it is important to understand when they exist. A remarkable conjecture connects the existence of such metrics to a problem in algebraic geometry concerning stability. The PI will study this notion of stability, and extend the conjecture to a more general setting.
