Dynamical topology change, compactification and waves in a stringy early universe

Exact string solutions are presented, where moduli fields are varying with time. They provide examples where a dynamical change of the topology of space is occuring. Some other solutions give cosmological examples where some dimensions are compactified dynamically or simulate pre-big bag type scenarios. Some lessons are drawn concerning the region of validity of effective theories and how they can be glued together, using stringy information in the region where the geometry and topology are not well defined from the low energy point of view. Other time dependent solutions are presented where a hierarchy of scales is absent. Such solutions have dynamics which is qualitatively different and resemble plane gravitational waves. Talk presented by E. Kiritsis in the 2\`eme Journ\'ee Cosmologie, Observatoire de Paris, 2-4 June 1994.