Topological twists of 3d N=4 gauge theories naturally give rise to non-semisimple 3d TQFT's. In mathematics, prototypical examples of the latter were constructed in the 90's (by Lyubashenko and others) from representation categories of small quantum groups at roots of unity; they were recently generalized in work of Costantino-Geer-Patureau Mirand and collaborators. I will introduce a family of physical 3d quantum field theories that (conjecturally) reproduce these classic non-semisimple TQFT's. The physical theories combine Chern-Simons-like and 3d N=4-like sectors. They are also related to Feigin-Tipunin vertex algebras, much the same way that Chern-Simons theory is related to WZW vertex algebras. (Based on work with T. Creutzig, N. Garner, and N. Geer.); part of the London TQFT Journal Club; it will be possible to follow this talk online (please register at https://london-tqft.vercel.app)

Quantized complex curves play a central role in both topological string theory and Chern-Simons theory with complexified gauge group. In both cases these quantum curves yield operators that annihilate partition functions. However, in both cases, the actual quantization of these curves has only been understood indirectly (via matrix models on one hand, via recursion relations for the Jones polynomial on the other). I will discuss an intrinsic, geometric quantization scheme that should produce such quantum Riemann surfaces directly. N.B. Such quantum curves also show up in conformal field theory, as the operators that annihilate correlators with degenerate insertions.