The geometry 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. Cooperation among the Berlin mathematicians working in these fields has a long tradition. The current SFB 647 Space–Time–Matter combines many research activities including work on the following topics: the special geometries considered in string theory; mathematical relativity theory; applications of nonlinear PDEs to differential geometry, topology and algebraic geometry; and dynamical systems. These  have applications in several branches of science.

The research activities at HU in differential geometry and global analysis focus on the study of geometrically defined differential operators and equations, on their solutions and solution spaces, and on the resulting geometric classification problems. Often the analytic properties of differential operators have consequences for the geometry and topology of the spaces on which they are defined (like curvature, 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). Particular topics of research here are: symplectic geometry and topology including the quantitative and qualitative properties of Lagrangian embeddings (Mohnke), spectral properties of Dirac and Laplace operators in the presence of singularities (Brüning, Schüth), index theorems for elliptic operators (Brüning), isospectrality problems for Riemannian manifolds and orbifolds (Schüth), spectral properties of Dirac operators and field quations on manifolds with nonintegrable geometric structures (Friedrich), and Dirac operators and spinor field equations, holonomy theory and symmetries on Lorentzian manifolds or other manifolds with indefinite metrics (Baum). There is a strong cooperation with the differential geometry group at U Potsdam (Bär) working on analytic and spectral properties of geometric operators, in particular on Dirac operators in Riemannian and Lorentzian geometry.

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 (AEI) and with U Potsdam within the framework of the IMPRS Geometric Analysis, Gravitation and String Theory. 

Differential geometry research at TU (Bobenko, Pinkall, Sullivan, Suris) and FU (Polthier) is concerned with global differential geometry of surfaces, geometric optimization problems, and the theory of integrable systems, including applications to mathematical visualization. This branch of geometric research in Berlin is described in more detail in the research area Geometry, topology, and visualization.

 The group Mathematical Physics of Space, Time and Matter (Staudacher) focuses on exploring the mathematical structure of so-called gauge–string dualities, in particular the AdS/CFT correspondence, which conjectures the exact equivalence of a ten-dimensional superstring theory on an anti de Sitter space-time to a four-dimensional supersymmetric Yang–Mills quantum field theory. This entails a very rich fabric of interwoven mathematics, including such diverse topics as the differential geometry and topology of curved space-times, the representation theory of ordinary and affine non-compact super Lie algebras and Yangians, and the theory of classical and quantum integrability. The group Structure of local quantum field theory (Kreimer) focuses on mathematical aspects of local, renormalizable quantum field theories as they appear in physics. This brings in Hopf algebras and – via the Milnor–Moore theorem – free Lie algebras, as well as the study of periods extending those of the moduli spaces M_0;n of spheres with n marked points. While the physics of renormalizable field theory is the most precisely tested part of physics available, it also is a source of mathematical problems relating to algebra, algebraic geometry and number theory.