Research Field

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.

Berlin Research Groups

The research activities at HU in differential geometry and global analysis (Mohnke, Schüth, Baum, Walpuski, Wendl) 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, spectral properties of Dirac and Laplace operators in the presence of singularities, and Dirac operators and spinor field equations, holonomy theory and symmetries on Lorentzian manifolds or other manifolds with indefinite metrics.

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. Because it emphasizes techniques of discrete differential geometry, including applications to mathematical visualization, this research is described in more detail as part of Area 5.

The HU hosts several mathematical physics research groups (Borot, Kreimer, Staudacher), who join forces in the Kolleg Mathematik Physik Berlin. They deal with a broad spectrum of interwoven mathematics, such as differential and algebraic geometry (relating to RTA 2), low-dimensional topology (surfaces, 3-manifolds, knot theory, ...), combinatorics and enumerative geometry, algebraic structures (infinite-dimensional Lie groups, quantum groups, Hopf algebras, vertex operator algebras, operads, higher structures, etc.),  representation theory, functional analysis and asymptotic analysis (relating to RTA 7). They can be used to advance the understanding of quantum field theories (supersymmetric gauge theories, conformal field theory, 2d quantum gravity, etc.), and string theory, but can also take inspiration from theoretical physics to discover new mathematics and establish relations between apparently distant problems.

Basic Courses



The two Basic Courses give an introduction to the most important concepts of differential geometry; these are fundamental for Riemannian and symplectic geometry as well as for geometric analysis and mathematical physics. The first course provides an introduction to the basic notions of geometry and analysis on manifolds, while the second one imparts fundamental knowledge in global Riemannian geometry.

Outline of the contents:

Analysis and geometry on manifolds
◦ Differentiable manifolds, tangent bundles, tensor fields
◦ Differential forms, Stokes’ theorem, de Rham cohomology
◦ Riemannian metrics
◦ Connections, parallel transport, curvature
◦ Geodesics, exponential map, Hopf–Rinow theorem

Riemannian geometry
◦ Jacobi fields, variation of length and energy
◦ Relations between curvature and topology
(theorems of Gauß–Bonnet, Hadamard–Cartan, Bonnet–Myers, Synge, etc.)
◦ Geometry of Riemannian immersions and submersions
◦ Lie groups, geometry of homogeneous and symmetric spaces
◦ Geometric evolution equations and differential equations from geometric analysis
◦ Symplectic and complex geometry (Riemann–Roch theorem, Hamiltonian dynamics)

Spectrum of Advanced Courses

Geometric Analysis, Geometric Measure Theory, Quantum Field Theory, The Renormalization Group, Gauge Theories, Group Theory in Physics, Integrable Systems, Scattering Theory, Spectral Geometry, Pseudo-Riemannian Geometry, Symplectic Geometry and Topology, Advanced Surface Theory, Riemann Surfaces.