In this talk, I explore the set of line defects supported by 3D N=4 theories, and their significance in the context of generalised symmetries. I begin by discussing mirror symmetry of line defects using the example of Sp(k) SQCD and its two mirror theories. I then introduce rational Q-systems, a powerful technique borrowed from spin-chains/integrability, for evaluating twisted indices and studying line operator correlation functions. Finally, I highlight the role of line defects in realising mirror symmetry in the presence of non-trivial higher symmetries

In this talk, I will discuss the generalised and basic fuzzy 4-sphere in the context of the IKKT matrix model. These spaces arise as SO(5)-equivariant projections of quantised SO(6) coadjoint orbits and exhibit full SO(5) covariance. I will sketch how (basic and generalised) 4-sphere arise as solutions in a Yang-Mills matrix model, such that the fluctuations on the 4-sphere lead to a higher-spin gauge theory.

In this talk, I will discuss how the two geometric notions "fan" and "monoid" can be very fruitful for the understanding of the monopole formula for 3d N=4 gauge theories. After a brief reminder of the monopole formula, I will introduce the matter fan and reorganise the monopole formula accordingly. I then discuss the resulting benefits such as: (1) Explicit expressions for the Hilbert series built from well-studied constituents. (2) Proof that the order of the pole at t=1 and t → ∞ equals the complex or quaternionic dimension of the Coulomb branch. (3) Identification of a sufficient set of chiral ring generators.