Spectral geometry, index theory and geometric flows on singular spaces

• Applicants: Professor Dr. Bernd Eberhard Ammann, since 1/2022; Professor Dr. Boris Vertman
• Subject Area: Mathematics
• Term: since 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 338891943

Our research proposal is a continuation of our research in the three fundamental aspects of geometric analysis on singular spaces: spectral theory, index theory as well as geometric flows. As in the previous funding period of the SPP, our projects are again organized into the following mutually related sections: (I) SpectralgeometryandtheCheeger-MüllerTheorem, (II) Index theory, eta and Cheeger–Gromov rho invariants, (III) Geometric flows, Ricci, Yamabe and mean curvature flows.The first two projects are concerned with two fundamental achievements of modern geometric analysis, the Atiyah-Patodi-Singer index theorem as well as the proof of the Ray-Singer conjecture by Cheeger and Müller. The third project is devoted to the analysis of geometric flows, which has been inspired by the Hamilton’s Ricci flow and its application to three-manifold topology, including the Hamilton-Perelman proof of the Poincare conjecture. Since then all the three fields have seen tremendous progress with important applications in geometry, topology, physics and nonlinear analysis. The general theme of our research is the study of these topics in the setting of singular and non-compact manifolds.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity
• International Connection: Brazil
• Cooperation Partner: Professor Dr. Luiz Hartmann
• Ehemaliger Antragsteller: Professor Dr. Klaus Kröncke, until 12/2021