The Cobordism Conjecture postulates that the cobordism classes in a consistent theory of quantum gravity should be trivial. Non-vanishing cobordism classes can be made consistent either by gauging or breaking the symmetry, possibly predicting new stringy defects.
In the first part of this talk, I will discuss how cobordism charges can be related to K-theory charges, precisely reproducing the structure of known tadpole cancellation conditions. This postulated relation is tested under dimensional reduction, reproducing the expected patterns.
In the second part of the talk, I will describe the dynamical cobordism induced by the backreaction of a 9-dimensional non-supersymmetric, positive-tension domain wall in string theory. I will provide an explicit description of the accompanying 8-dimensional defect, predicted by the Cobordism Conjecture, in terms of a new non-isotropic solution of the dilaton-gravity equations of motion.