Title: Curvature blow up at big bang singularities

Abstract: Singularities have been accepted as a natural feature in general relativity since the appearance of the singularity theorems of Hawking and Penrose. But these theorems do not say much concerning the nature of singularities. Do the gravitational fields become unbounded? Can the spacetime be extended through the singularity? Recently, many results demonstrating the stability of spatially homogeneous solutions with big bang singularities have appeared. As a consequence, there is an open set of initial data yielding big bang singularities with curvature blow up. However, the purpose of the talk is to illustrate that it is possible to go beyond stability results. In fact, I will present a new result (joint work with Hans Oude Groeniger and Oliver Petersen) in which we identify a general condition on initial data ensuring big bang formation. The solutions need, in this case, not be close to symmetric background solutions. Moreover, the result reproduces previous results in the Einstein-scalar field and Einstein-vacuum settings. Finally, the result is in the Einstein-non-linear scalar field setting, and therefore yields future and past global non-linear stability of large classes of spatially locally homogeneous solutions.