It was recently argued that topological operators (at least those associated with continuous symmetries) need regularization. However, such regularization seems to be ill-defined when the underlying QFT is coupled to gravity. If both of these claims are correct, it means that charges cannot be meaningfully measured in the presence of gravity. I will review the evidence supporting these claims as discussed in [arXiv:2411.08858]. Given the audience's high level of expertise, I hope this will spark discussion about whether this is a promising approach to understanding the fate of global symmetries in quantum gravity.

In this talk, I will discuss a conjecture of Penrose, which asserts a lower bound on the mass of a spacetime in terms of the area of a suitable horizon. Whilst Penrose presented a physical motivation for this inequality in the 1970s, the only proofs heavily rely upon PDE arguments, and in particular the use of geometric flows. I hope to show in this talk, through this concrete example (and without unpleasant technical details!), how ideas from geometric PDE theory can be helpful in obtaining results in physics.