Differential geometry and topology
Differential geometry is the study of geometric objects that can be described smoothly via local coordinates. These objects are called smooth manifolds, so for example, a smooth curve is a 1-dimensional manifold, and a surface is a 2-dimensional manifold. On smooth manifolds of arbitrary dimension, one can consider various additional structures that encode geometric information, e.g. in Riemannian geometry, a Riemannian metric determines distances and angles: one can then study the manifold's curvature and relate this to its symmetries or its topological properties. Alternatively, one can probe its geometry via the "modes of vibration" (i.e. the spectrum) of solutions to certain partial differential equations, generalizing the classic question of spectral geometry: "can one hear the shape of a drum?" More generally, pseudo-Riemannian manifolds are endowed with a metric in which distance can be positive or negative, a meaningful notion in Einstein's theory of gravitation that determines the distinction between space and time on a unified "spacetime" manifold. A different type of geometric structure is present on symplectic manifolds, which arose originally from physics as the natural geometric setting for classical mechanics, but have more recently given rise to a rich variety of algebraic structures that are also of interest in modern mathematical physics. The connection to physics is even more explicit in mathematical gauge theory, which uses partial differential equations originating in quantum field theory to probe the global structure of smooth manifolds.
In many of these subfields of differential geometry, the most interesting questions are topological -- meaning they concern global rather than local properties of manifolds -- and the most powerful methods come from geometric analysis, i.e. the study of partial differential equations on manifolds.
The range of topics in these areas covered by the research and teaching at the HU includes the following:
- symplectic manifolds, contact manifolds, and pseudoholomorphic curves
- Gromov-Witten invariants, symplectic field theory
- gauge theory and invariants in differential topology
- manifolds with exceptional holonomy
- spectral geometry
- geometric analysis and its applications to Riemannian geometry and topology
- algebraic and low-dimensional topology
Former professors:
Secretary
Kati Blaudzun
Raum 1.313
Tel. +49 (0)30 2093 45430
Fax +49 (0)30 2093 2727
blaudzun@math.hu-berlin.de