A General Description
Geometric methods and geometric language provide a common ground of current mathematical research in arithmetic and analysis. On the one hand a lot of progress has been made by transferring ideas between arithmetic and global analysis, and on the other hand global analysis is deeply connected with geometric partial differential equations.
More specifically to the research in Regensburg, concepts as Chern characters or regulators from K-theory to various cohomology groups have enriched our understanding of central results as the Riemann-Roch theorem or the index theorem. Our view on zeta functions occurring in arithmetic and in dynamical systems has profited from structural coincidences in different areas. A characteristic feature of current research in arithmetic and global analysis is the use of the language and methods of homotopy theory, which is employed in arithmetic to study algebraic cycles. Typical aspects of homotopy theory in modern global analysis are twisted K-theory and orientation theory.
On the other hand, the basic figures of global analysis are geometric partial differential equations, most notably the Dirac operator and curvature driven flow equations (Ricci flow, Einstein equations, Willmore flow). The differential geometric and analytic aspects of the description of solutions at singularities or at infinity, which can even be used to prove fundamental theorems in topology (Poincar\'e conjecture), reflects the common interest of the research groups in Regensburg.
The Mathematics Department in Regensburg offers a unique research environment for the Graduiertenkolleg Curvature, Cycles and Cohomology centered around the above aspects. It complements Johannes-Kepler-Forschungszentrum für Mathematik on the educational side. The focus and of the Graduiertenkolleg is development and teaching of common geometric, analytic, and topological aspects of the interacting fields, the transport of ideas, principles and methods, and their application to central problems in the participating fields themselves. By its joint education programme the Graduiertenkolleg will enable its PhD students in addition to their education in their special field to progress on the basis of a broad background knowledge of different - but from a broader perspective deeply related - fields. As an important aspect it will encourage them to initiate joint projects and interactions. The guest program the Graduiertenkolleg and its support for exchange and collaboration will expose the students to the international research community. It will give them the opportunity to get into contact with leading experts and also to present their own results.