The research groups at the three Berlin universities cover a wide range of current research topics in the fields of differential geometry, geometric analysis, and mathematical physics. The cooperation among Berlin mathematicians working in these fields has a long tradition. Recently, Berlin mathematicians and physicists combined their research activities in the SFB 647 Space, Time, Matter. Specific topics of that cooperation include the special geometries considered in string theory, mathematical relativity theory, applications of nonlinear PDEs to differential geometry, topology and algebraic geometry, and dynamical systems, with applications in several branches of science. Mathematicians working in the field cooperate in national and international activities, for example within the DFG Priority Research Program Global Differential Geometry.

The research activities at HU focus on the study of geometrically defined differential operators and equations, their solutions and solution spaces. Often their properties have consequences for the geometry and topology of the spaces on hich they are defined (like holonomy, dimension, volume, injectivity radius) or, vice versa, the geometrical data have implications for the structure of the differential operators involved (like spectrum and bordism class of the solution space). Special subjects are spectral properties of Dirac and Laplace operators in the resence of singularities (Brüning, Schüth), symplectic geometry and topology (Mohnke), field equations which are defined on nonintegrable geometric structures (Friedrich) or on manifolds with indefinite metrics (Baum) and the related geometric classification problems.

Differential Geometry and Geometric Analysis is one of the main topics within the International RTG Arithmetic and Geometry, a joint initiative with ETH and the University of Zurich.
At FU, there are groups working in geometric analysis (Ecker, Huisken) and in nonlinear dynamics (Fiedler) with a joint research seminar. The research focuses on geometric evolution equations, geometric variational problems, mathematical relativity theory and nonlinear theory of dynamical systems. Particular topics include singularity formation and the longtime behavior of solutions of nonlinear evolution equations. In geometric analysis there is strong cooperation with the MPI for Gravitational Physics and Potsdam University in the framework of the IMPRS Geometric Analysis, Gravitation and String Theory. The dynamical systems group (Fiedler) is involved in the DFG Priority Research Program Analysis and Numerics of Conservation Laws.

Geometry research at TU is concerned with global differential geometry of surfaces, geometric optimization problems and theory of integrable systems, including applications to mathematical visualization. Particular topics include conformal surface theory (Pinkall), geometric knot theory (Sullivan), and investigation of special classes of surfaces (Bobenko) and the theory of isoparametric submanifolds (Scherfner). A combination of mathematical theory and numerical experiments in these fields may be seen as a Berlin speciality.