Lattice cohomology and q-series invariants of 3-manifolds

I will discuss a new invariant of negative definite plumbed 3-manifolds called a weighted graded root. It unifies and extends two theories with quite different origins and structures. The first is lattice cohomology, due to Némethi, whose degree zero part is described by a certain graph and is isomorphic to Heegaard Floer homology for a large class of plumbings. The second theory is the Z-hat q-series of Gukov-Pei-Putrov-Vafa, a power series which conjecturally recovers SU(2) quantum invariants at roots of unity. I will explain lattice cohomology, Z-hat, and our unification of these theories. I will also discuss some key features of the weighted graded root: it leads to a 2-variable refinement of Z-hat, and, unlike both lattice cohomology and Z-hat, it is sensitive to spin^c  conjugation. This is joint work with Peter Johnson and Slava Krushkal.