Rupert Klein Freie Universität Berlin Mathematics Due to the curse of dimensionality, we will not be able to numerically solve non-stationary many-body Schroedinger problems for large particle numbers and for general initial data in the foreseeable future. The situation seems equally severe for the simpler problem of approximating eigenfunctions of the non-relativistic electronic Schroedinger equation with clamped nuclei. It is known, on the other hand, from numerical analysis that very smooth functions can be approximated efficiently with few degrees of freedom, given the "right" ansatz functions for their representation. Harry Yserentant and co-workers recently analyzed the smoothness of many-electron eigenfunctions for large particle numbers [1,2,3]. By revealing an interesting competition between increasing dimensionality and increasing smoothness, they came up with results that seem outright spectacular to a newcomer in the field like myself. In this presentation I will summarize these results, the principal line of analysis, and open issues. [1] H. Yserentant, Lecture Notes in Mathematics, Vol. 2000 (2010) [2] H. Yserentant, ESAIM: M2AN, vol. 45, 803-824 (2011) [3] H.-C. Kreusler, H. Yserentant, Numer. Math., vol. 121, 781-802 (2012)