Yesterday we had another cake meeting where people "shared their ignorance".
Mine concerned: what is the relationship between the chiral (parity) anomaly in quantum field theory, edge states, and topological insulators?
I learnt something in the week from this PRL [more string theorists writing about Fermi liquids!], which at least taught me what the relevant "anomaly" equation is for the "violation" of charge conservation.
Ben Powell pointed out that there is a relevant blog post Monopoles passing through flatland by John Preskill. His beautiful post certainly provides the background information I need, in a particularly lucid form. Having read it once I now need to digest it properly. Otherwise, I will still be "ignorant.".