Each fusion higher category has a "framed S-matrix" which encodes the commutator of operators of complementary dimension. I will explain how to construct and interpret this pairing, and I will emphasize that it may fail to exist if you drop semisimplicity requirements. I will then outline a proof that the framed S-matrix detects (non)degeneracy of the fusion higher category. This is joint work in progress with David Reutter.

This is a report on joint work in progress with L. Katzarkov, M. Kontsevich, and P. Pandit. The Homological Mirror Symmetry conjecture is stated as an equivalence of triangulated categories, one coming from algebraic geometry and the other from symplectic topology. An enhancement of the conjecture also identifies stability conditions (in the sense of Bridgeland) on these categories. We adopt the point of view that triangulated (DG/A-infinity) categories are non-commutative spaces of an algebraic nature. A stability condition can then be thought of as the analog of a Kähler class or polarization. Many, often still conjectural, constructions of stability conditions hint at a richer structure which we think of as analogous to a Kähler metric. For instance, a type of Donaldson and Uhlenbeck-Yau theorem is expected to hold. I will discuss these examples and common features among them, leading to a tentative definition.