The research in the geometry of Banach spaces has for many years been concentrated on the investigation of the local structure of Banach spaces, for example properties which only depend on the structure of the finite-dimensional subspaces of the given space.
Recently an investigation of the structure of closed subspaces of von Neumann algebras has begun, thus combining Banach space theory with the theory of operator algebras.
The methods used in this research are probabilistic in nature and this has motivated a deep investigation of random variables with vector values and vector-valued analogues of the classical integral transformations, including, for example, the Fourier and the Hilbert transforms. This leads to connections with classical analysis and the theory of analytic functions.
Niels Jørgen Nielsen