Topological quantum field theories are one of the few cases of QFT that can be treated mathematically.  The goal of this talk will be to recall a mathematician's definition of TFT, and demonstrate how it recovers features of QFT that are familiar to physicists.  I will then introduce the notion of extended TFT, and explain how dualizability properties can be used to classify TFTs.

We study 5d N=1 theories engineered by (p,q) five-brane webs and their 3d N=4 magnetic quivers, focusing on the N + SU(N)^(k-1) + N gauge theory. We establish an exact parameter dictionary mapping the 5d flavour masses and instanton couplings to the FI parameters of the strong-coupling magnetic quiver. We test this dictionary against mass deformations, weak-coupling limits, and moment map VEVs, implemented via the 5d superconformal index and the 3d squashed-sphere partition function. We analyze these deformations on the magnetic-quiver side using a new framework that exactly tracks Higgsing patterns, decoupled sectors, and infrared theories. This promotes magnetic quivers to quantitative tools for studying how the 5d superconformal index responds to physical deformations
We show that large 5d moment map VEVs render the superconformal index distributional, mirroring bad theories in lower dimensions. We also develop a brane-web interpretation of these results, identifying the Hanany–Witten moves associated with the cascade of VEVs and relating inequivalent frames to distinct tessellations of the generalized toric diagram. Furthermore, we demonstrate that magnetic quivers remain powerful predictive tools even when the theory enters the bad regime.