Abstract: I will introduce the homotopy-theoretic classification of fully extended reflection-positive invertible field theories and their application to classifying symmetry-protected topological phases (SPTs) in physics. In ongoing work with Arun Debray, Natalia Pacheco-Tallaj, and Luuk Stehouwer, we use twisted index maps to relate invertible field theories and K-theory, modeling a physics process comparing free and interacting SPTs. In our example of interest, we witness the "Morita variance" of invertible field theories, in which Bott periodicity in K-theory is mapped to a non-periodic "Bott spiral" of invertible field theories.

Shifted quantum affine algebras are quantum groups parameterized by a coweight of the underlying Lie algebra. Hernandez introduced a category O of representations of these algebras, and Geiss–Hernandez–Leclerc proved that the Grothendieck ring of this category O admits a cluster algebra structure. In this talk, after introducing the necessary background, I will explain how to construct a quantization of this cluster algebra, leading to a definition of the quantum Grothendieck ring for category O. If time permits, I will also discuss some applications and directions for future research.