TRR 71:  Geometric Partial Differential Equations

• Subject Area: Mathematics
• Term: from 2009 to 2013
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 64447639

The Collaborative Research Centre/ Transregio studies analytic problems, which arise in a geometric context, either in differential geometry or in applications involving geometric modelling. Motivations within geometry include questions from geometric measure theory, variational problems for the area and the Willmore functional, evolution equations for submanifolds and Riemannian metrics. Open problems posed by Gromov, Nitsche, Willmore and Yau are driving forces. Applied motivations come from continuum mechanics, in particular fluid dynamics and fluid structure interactions, liquid crystals, image segmentation and singular limits in mathematical physics. Most projects have both geometric and applied sources, leading to cooperations and cross-links. In all subjects the geometry creates specific challenges and opportunities.
The Collaborative Research Centre/ Transregio provides a unique research environment to study geometric partial differential equations from complementary viewpoints, cooperating between analysis, numerical analysis and computer simulations. Which are the key analytic notions and quantities describing the geometric or physical phenomena? How can geometric and analytic aspects be reflected and utilised in the construction of algorithms? It is the issue in each of the projects to identify and understand the central aspects, to develop the appropriate concepts of regularity and singularity. The model character of the problems ensures the broad relevance of results.
With its two locations Freiburg and Tübingen, enhanced by one member from the University of Zürich, the Collaborative Research Centre/ Transregio creates an important centre in the fields of geometry and partial differential equations. The high degree of scientific cooperation and communication among the principal investigators adds to the attractiveness of the programme for PhD students and postdocs.

• DFG Programme: CRC/Transregios
• International Connection: Switzerland

Completed projects

• A02 - Higher rectifiability (Project Head Schätzle, Reiner )
• A03 - Topology of partitioning surfaces (Project Heads Kuwert, Ernst ;  de Lellis, Camillo ;  Schätzle, Reiner )
• A04 - Min-Max-constructions of minimal surfaces (Project Heads Kuwert, Ernst ;  de Lellis, Camillo ;  Schätzle, Reiner )
• A05 - Minimizing Normal Currents and the Stable Norm (Project Head Bangert, Victor )
• A06 - Estimates for Scalar Curvature, Dirac Eigenvalues, and related geometric magnitudes (Project Heads Goette, Sebastian ;  Pedit, Franz )
• A07 - Yamabe type problems and their applications (Project Head Wang, Guofang )
• B01 - Willmore surfaces and second variation (Project Heads Kuwert, Ernst ;  Pedit, Franz ;  Schätzle, Reiner )
• B02 - Willmore minimizers in fixed conformal classes (Project Heads Kuwert, Ernst ;  Pedit, Franz ;  Schätzle, Reiner )
• B03 - Minimizers of the Willmore functional with prescribed area and volume (Project Heads Kuwert, Ernst ;  Schätzle, Reiner )
• B04 - Discretization of geometric functionals (Project Heads Dziuk, Gerhard ;  Pozzi, Ph.D., Paola ;  Prohl, Andreas )
• B05 - Experiments and Visualization of Willmore surfaces (Project Head Pedit, Franz )
• B07 - Conformal metrics of constant Q-curvature on 4-dimensional manifolds (Project Heads Ahmedou, Mohameden ;  Schätzle, Reiner )
• B11 - Surfaces with small Willmore variation (Project Head Schätzle, Reiner )
• C01 - Dynamics on and of surfaces (Project Heads Dedner, Andreas ;  Dziuk, Gerhard ;  Kröner, Dietmar ;  Lubich, Christian )
• C02 - Fluid structure interaction (Project Heads Diening, Lars ;  Kröner, Dietmar ;  Ruzicka, Michael )
• C04 - The Euler equations as a differential inclusion (Project Head de Lellis, Camillo )
• C06 - Dynamics of free discontinuity problems from image processing (Project Heads Prohl, Andreas ;  Schätzle, Reiner )
• D01 - Effective Schrödinger dynamics on submanifolds (Project Heads Lubich, Christian ;  Teufel, Stefan )
• D03 - Adiabatic perturbation theory for magnetic Bloch bands (Project Heads Pedit, Franz ;  Teufel, Stefan )
• D04 - Ricci flow of singular metric spaces (Project Heads Dziuk, Gerhard ;  Simon, Ph.D., Miles )
• S - Administrative project (Project Head Kuwert, Ernst )

• Applicant Institution: Albert-Ludwigs-Universität Freiburg
• Co-Applicant Institution: Eberhard Karls Universität Tübingen
• Participating University: Universität Zürich
• Spokesperson: Professor Dr. Ernst Kuwert