Cobordism Utopia

On general grounds one expects that global symmetries are absent in quantum gravity. We discuss some aspects of this issue, focusing on the recently proposed Swampland Cobordism Conjecture, and related conjectures connected with completeness of the spectrum of states charged under symmetries. In particular, the U-dualities of M-theory provide an excellent arena both for testing aspects of these conjectures, as well as predicting the existence of new dynamical objects. We also comment on how this approach connects to related top down and holographic approaches to constructing and studying gauging and breaking symmetries in quantum gravity. Based on joint work to appear with Braeger, Debray, Dierigl, and Montero.