Differential Geometry and Geometric Analysis
Differential manifolds provide higher dimensional generalizations of surfaces. They appear in a very natural manner in many areas of mathematics and physics. On a differential manifold or more generally on a geodesic metric space, one investigates geometric and analytic quantities and concepts, such as geodesics, the curvature of Riemannian metrics or potential theory, and also group actions on such spaces. Of particular interest are connections between such quantities and global, topological properties of the underlying manifold.
The focus topic Differential Geometry and Geometric Analysis is closely related to topology, analysis, stochastics, group theory and to physic, e.g. Einstein's general relativity. A good background in algebra is helpful.