Research Interests

Complex Analysis, Complex Geometry, Complex Differential Geometry (MSC 32).

The main interest of my research lies in the interplay of analysis and geometry on singular spaces. On complex manifolds, analytic methods have led to fundamental advances in geometry (transcendental methods), but most analytic tools are not very well developed on singular spaces. It is a promising endeavor to create appropriate analytic instruments in the presence of singularities. On the other hand, this is not possible without considerable insight in the geometric nature of the underlying space.
Particularly, I am interested in the theory of the Cauchy-Riemann-Operator and the d-bar-Neumann-Operator on singular spaces. The intention is to develop d-bar-methods - one of the most powerful tools in complex analysis - for such spaces. The focus lies on L²-methods (which are most promising), but also integral formulas and other tools from Complex Geometry and Differential Geometry play a profound role. My research is concerned with the following topics:

• Analysis on Singular (Complex) Spaces
• Complex Analytic Geometry and Complex Differential Geometry on Singular Complex Spaces
• Differential Operators, in particular d-bar and d-bar-Neumann Operators
• L2-Methods
• Integralformulas
• Canonical Sheaves
• Duality on Singular Complex Spaces
• Singularities of the Minimal Model Program
• Modifications of Complex Spaces
• Extension Phenomena in Complex Analysis
• Notions of Convexity, q-complete and q-convex Spaces
• Residue Currents in the Context of Singular Spaces
• Analysis on Positive Currents
• Singular Hermitian Metrics on Vector Bundles

Publications and Preprints