Due to a celebrated theorem of Bondal and Orlov, a projective variety with an ample canonical or anticanonical divisor is uniquely determined by its derived category. However, the same category can also arise in other contexts, such as the derived category of modules over a noncommutative algebra or the derived Fukaya category of the mirror symplectic manifold. This opens the possibility of interpreting deformations of algebraic varieties in terms of representation theory or symplectic geometry. I will survey one of the best-studied test cases of this line of thought: deformations of algebraic surfaces with cyclic quotient singularities, Kalck–Karmazyn algebras, and Kawamata vector bundles and Lagrangians. The original results were developed in collaborations with Giancarlo Urzúa (AG side) and Yankı Lekili (SG side).