SPP 2026:  Geometry at Infinity

• Subject Area: Mathematics ; Physics
• Term: since 2017
• Website: Homepage
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 313840899

This programme combines research in differential geometry, geometric topology, and global analysis. Crossing and transcending the frontiers of these disciplines it is concerned with convergence and limits in geometric-topological settings and with asymptotic properties of objects of infinite size. The overall theme can roughly be divided into the three cross-sectional topics convergence, compactifications, and rigidity.Examples of convergence arise in Gromov-Hausdorff limits and geometric evolution equations. The behaviour of geometric, topological and analytic invariants under limits is of fundamental interest. Often limit spaces are non-smooth so that it is desirable to generalize notions like curvature or spectral invariants appropriately. Limits can also be used to construct asymptotic invariants in geometry and topology such as simplicial volume or L2-invariants.Compactifications reflect asymptotic properties of geometric objects under suitable curvature conditions. Methods from topology, differential geometry, operator algebras and probability play a role in this study. Important issues are boundary value problems for Laplace or Dirac type operators, both in the Riemannian and Lorentzian setting, as well as spectral geometry and Brownian motion on non-compact manifolds.Besides continuous deformations rigidity is essential for many classification problems in geometry and topology. It appears in geometric contexts, typically in the presence of negative curvature, and in topological and even algebraic settings. Rigidity also underlies isomorphism conjectures relating analytic, geometric and homological invariants of infinite groups and more general coarse spaces.The priority programme supports individual research projects and coordinated research activities. These activities will ensure a coherence of research directions, identify promising lines of interdisciplinary research, encourage the establishment of new research cooperations, and realize gender equality measures.

• DFG Programme: Priority Programmes
• International Connection: Australia, Austria, Belgium, Brazil, China, France, Israel, Italy, Spain, Switzerland, United Kingdom, USA

Projects

• A unified approach to Euclidean Buildings and symmetric spaces of non-compact type (Applicants Kramer, Linus ;  Schwer, Petra Nora )
• Action of mapping class groups and its subgroups (Applicant Disarlo, Ph.D., Valentina )
• Alexandrov Geometry in the light of symmetry and topology (Applicant Zarei, Masoumeh )
• Analysis on spaces with fibred cusps, II (Applicant Grieser, Daniel )
• Analytic L2-invariants of non-positively curved spaces (Applicants Kammeyer, Holger ;  Sauer, Roman ;  Schick, Thomas )
• Anosov representations and Margulis spacetimes (Applicant Ghosh, Sourav )
• Asymptotic geometry of moduli spaces of curves (Applicant Bielawski, Roger )
• Asymptotic geometry of sofic groups and manifolds (Applicant Alekseev, Vadim )
• Asymptotic geometry of the Higgs bundle moduli space (Applicant Weiß, Hartmut )
• Asymptotics of singularities and deformations (Applicant Mäder-Baumdicker, Elena )
• At Infinity of Symmetric Spaces (Applicant Köhl, Ralf )
• Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces (Applicants Keller, Matthias ;  Lenz, Daniel )
• Boundaries of acylindrically hyperbolic groups and applications (Applicant Hamenstädt, Ursula )
• Boundary value problems and index theory on Riemannian and Lorentzian manifolds (Applicant Bär, Christian )
• Cohomogeneity, Curvature, Cohomology (Applicant Amann, Manuel )
• Cohomology of Symmetric Spaces as seen from Infinity (Applicants Hartnick, Tobias ;  Ott, Andreas )
• Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings (Applicants Kramer, Linus ;  Schwer, Petra Nora )
• Construction of Riemannian manifolds with scalar curvature constraints and applications to general relativity (Applicant Cabrera Pacheco, Armando )
• Coordination Funds (Applicant Hanke, Bernhard )
• Curvature flows without singularities (Applicant Schnürer, Oliver )
• Diffeomorphisms and the topology of positive scalar curvature (Applicants Ebert, Johannes ;  Schick, Thomas ;  Steimle, Wolfgang )
• Duality and the coarse assembly map (Applicants Wulff, Christopher ;  Zeidler, Rudolf )
• Existence, regularity and uniqueness results of geometric variational problems (Applicant Hirsch, Jonas )
• Gauge-theoretic methods in the geometry of G2 manifolds. (Applicant Haydys, Andriy )
• Geometric Chern characters for p-adic equivariant K-theory and K-homology (Applicant Schick, Thomas )
• Geometric invariants of discrete and locally compact groups (Applicants Bux, Kai-Uwe ;  Witzel, Stefan )
• Geometric operators on singular domains (Applicants Ammann, Bernd Eberhard ;  Große, Nadine )
• Geometrically defined asymptotic coordinates in General Relativity (Applicants Cederbaum, Carla ;  Metzger, Jan )
• Geometry of surface homeomorphism groups (Applicants Bowden, Jonathan ;  Hensel, Sebastian )
• Gerbes in Renormalization and Quantization of Infinite-Dimensional Moduli Spaces (Applicant Upmeier, Markus )
• Hitchin components for orbifolds (Applicants Alessandrini, Ph.D., Daniele ;  Lee, Gye-Seon )
• Index theory on Lorentzian manifolds (Applicant Bär, Christian )
• Invariants and Boundaries of Spaces (Applicant Ott, Andreas )
• Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces (Applicants Keller, Matthias ;  Lenz, Daniel ;  Schmidt, Marcel )
• Large Genus Limit of Energy Minimizing Compact Minimal Surfaces in the 3-Sphere (Applicant Bielawski, Roger )
• Limits of invariants of translation surfaces (Applicant Randecker, Anja )
• Macroscopic invariants of manifolds (Applicant Engel, Alexander )
• Minimal Lagrangian connections and related structures (Applicant Mettler, Thomas )
• Minimal surfaces in metric spaces (Applicants Lytchak, Alexander ;  Stadler, Stephan )
• Minimizer of the Willmore energy with prescribed rectangular conformal class (Applicant Heller, Lynn )
• New hyper-Kähler spaces from the self-duality equations (Applicant Weiß, Hartmut )
• Nonlinear evolution equations on singular manifolds (Applicant Schrohe, Elmar )
• Parabolics and Invariants (Applicants Bux, Kai-Uwe ;  Kielak, Dawid ;  Witzel, Stefan )
• Probabilistic and Spectral Properties of Weighted Riemannian Manifolds with Kato-bounded Bakry-Emery-Ricci Curvature (Applicants Güneysu, Batu ;  von Renesse, Max-Konstantin )
• Profinite and RFRS groups (Applicant Gardam, Giles )
• Profinite perspectives on L2-cohomology (Applicants Kammeyer, Holger ;  Kionke, Steffen ;  Sauer, Roman ;  Schick, Thomas )
• Property (T) (Applicant Sauer, Roman )
• Resonances for non-compact locally symmetric spaces (Applicant Delarue, Benjamin )
• Ricci flows for non-smooth spaces, monotonic quantities, and rigidity (Applicants Erbar, Matthias ;  Sturm, Karl-Theodor )
• Rigidity, deformations and limits of maximal representations (Applicants Pozzetti, Maria Beatrice ;  Wienhard, Anna )
• Rigidity, stability and deformations in nearly parallel G2 geometry (Applicant Semmelmann, Uwe )
• Secondary invariants for foliations (Applicants Azzali, Sara ;  Goette, Sebastian )
• Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces (Applicant Finster, Felix )
• Singular Riemannian foliations and collapse (Applicant Corro Tapia, Diego )
• Singularities of the Lagrangian mean curvature flow (Applicant Smoczyk, Knut )
• Solutions to Ricci flow whose scalar curvature is bounded in L^p (II) (Applicant Simon, Ph.D., Miles )
• Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds (Applicants Hanke, Bernhard ;  Tuschmann, Wilderich )
• Spectral Analysis of Sub-Riemannian Structures (Applicant Bauer, Wolfram )
• Spectral geometry, index theory and geometric flows on singular spaces (Applicants Ammann, Bernd Eberhard ;  Vertman, Boris )
• Spectral theory with non-unitary twists (Applicant Pohl, Anke )
• Spin obstructions to metrics of positive scalar curvature on nonspin manifolds (Applicant Cecchini, Simone )
• Stability and instability of Einstein manifolds with prescribed asymptotic geometry (Applicant Kröncke, Klaus )
• The geometry of locally symmetric spaces via natural maps (Applicant Hamenstädt, Ursula )
• Topological and equivariant rigidity in the presence of lower curvature bounds (Applicants Galaz-García, Fernando ;  Kerin, Martin )
• Uniqueness in Mean Curvature Flow (Applicant Schnürer, Oliver )
• Wall-crossing and hyperkähler geometry of moduli spaces (Applicant Meneses, Ph.D., Claudio )
• Willmore functional and Lagrangian surfaces (Applicants Kuwert, Ernst ;  Wang, Guofang )

• Spokesperson: Professor Dr. Bernhard Hanke