The goal of the proposed research project is to attack open problems

in complex geometry, algebraic geometry and weak solutions of elliptic PDE's.

In particular we would like to investigate the complex singularity exponents

and their relation to geometric invariants such as the log canonical threshold,

Moser-Trudinger and Brezis-Merle type inequalities and their applications;

weak solutions of complex Hessian type equations - in particular the

existence, stability and regularity issues, the (degenerate) J-flow

and sigma_k flow; and cone type Kähler metrics with emphasis on

their behavior and asymptotics under a change of the cone angle.

We would like to investigate concrete long-standing conjectures

like the one of Guedj and Rashkovskii and also try to find an

analytical proof of the birational rigidity of smooth hypersurfaces

in projective space.