I will start with a basic introduction to planar algebras. I will then discuss recent joint work with Scott Morrison (arXiv:1208.3637) and recent joint work with Stephen Bigelow (arXiv:1208:1564). With Morrison, we construct a new exotic subfactor planar algebra using Bigelow's jellyfish algorithm. With Bigelow, we determine exactly when a planar algebra has a presentation by generators and jellyfish relations.

The Z/2-graded dg-category MF of matrix factorizations is a categorical singularity invariant appearing on the algebro-geometric side of mirror symmetry. Though it lives in commutative algebra, and the first definitions were very explicit, it turns out to admit a few descriptions very much in the spirit of homotopy theory. After recalling a bit about sheaf theories in derived algebraic geometry, we'll sketch at least one of these definitions and show how it can be used to give conceptual proofs of some fundamental properties of MF.