We work on the following mathematical subjects:
- Modell Theory and Geometry: O-minimal Structures and their connections to Analysis
- Real Algebraic Geometry and Real Analytic Geometry
- Real Algebra and Commmutative Algebra
Our research is characterized by an interesting interaction between Mathematical Logic, Real Geometry, Analysis and Algebra. We achieve results for all these subjects.
Model Theory and Geometry: O-minimal Structures and their connections to Analysis
O-minimal Structures have been developed by an interplay of Logic and Geometry of algebraic style and extend the latter by central concepts from Analysis. They generalize the category of semialgebraic and of semi- resp. subanalytic sets and functions.
O-minimal Structures are defined by the same finiteness conditions and share their good geometric behaviour ("tame geometry").
We want to make the theory of o-minimal structures and hence the reasoning of logic and tame geometry available for important subjects from Analysis. For this we work on Dynamical Systems, Complex Analysis and Potential Theory.
In addition to the more geometric aspects of o-minimal structures we also deal with their model theory.
Real Algebraic Geometry und Real Analytic Geometry
We investigate (globally) subanalytic sets and functions with respect to their analytic-geometric properties. We extend Real Algebrac Geometry over arbitrary real closed fields by the concept of integration.
Real Algebra and Commutative Algebra
In Real Algebra we are studying special classes of ordered rings as graded or homogeneous rings. We see possible applications for blowing up processes.
In Commutative Algebra we are engaged in valuation theory, especially with Manis valuations and Prüfer extensions. This theory allows to investigate important classes of non-noetherian rings which appear in Real Algebra.