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, 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.
At FU, there are groups working in geometric analysis (Ecker) 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.
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.
Several groups with an interest in the mathematics and physics of quantum fields have recently joined forces in the Kolleg Mathematik Physik. The group of Professor Staudacher (Mathematical Physics of Space, Time and Matter) 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 of Professor Kreimer (Structure of local quantum field theory) 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.
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
◦ 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.