Risi Kondor: you can take the fourier transform over many more things than you thought! for any function f : G → C (where G is, well, I think it has to be like a locally compact group, but definitely finite groups work) you can get out its fourier transform as a function \hat f : \hat G → C where \hat G is the irreps ("irreducible representations") of G in the complex numbers. In the case of finite cyclic groups and the real numbers, the irreps correspond to just frequencies, which is why the frequency domain is the frequency domain, but for general groups, they're much trickier. This is old news, apparently. Kondor figured out a way of turning graphs into fourier-transformable signals and efficiently transforming them so as to give a group invariant that is very close to faithful in practice and performs well as a preprocessing step in graph classification tasks.
Erik Demaine: gave positive answers to a whole mess of linkage and folding problems. It's really quite amazing what can be folded. Very nice enthusiastic talk — it was a bit short on technical detail, but I get the feeling this wasn't supposed to be a technical talk, since it was for the receipt of some prize or another, for which I guess I should expect a feel-good high-level "here is all the research that I think is awesome!!!" deal.
Kenny easwaran: Fallis argues that probabilistic proofs are no worse than long axiomatic proofs, because, for instance, the latter also are wrong with some nonzero probability. Kenny pointed out that we can observe that axiomatic proofs are more "transferable" because they can be independently checked without relying on testimony of other mathematicians, and without the checker having to run her own experiment which is of uncertain relationship to the original. I think the idea of modularity of axiomatic proofs — that we can map failure of checking to a particular component of the proof — brought up by a question from the audience is also very interesting.