Georgia Institute of Technology College of Sciences

I will survey recent progress toward Khovanov homology for links in general 3-manifolds based on categorification of $q$-series invariants labeled by Spin$^c$ structures. Much of the talk will focus on the $q$-series invariants themselves. In particular, I hope to explain how to compute them for a general 3-manifold and to describe some of their properties, e.g. relation to other invariants labeled by Spin or Spin$^c$ structures, such as Turaev torsion, Rokhlin invariants, and the "correction terms'' of the Heegaard Floer theory. There are many problems to work on in this relatively new research area. If time permits, I will outline some of them, and, in the context of plumbed 3-manifolds, comment on the relation to lattice cohomology proposed by Akhmechet, Johnson, and Krushkal.