Hydrodynamics is a theory of the collective properties of fluids and gases that can also be successfully applied to the description of the dynamics of quark-gluon plasma. It is an effective field theory formulated in terms of an infinite-order gradient expansion. For any collective physical mode, hydrodynamics will predict a dispersion relation that expresses this mode’s frequency in terms of an infinite series in powers of momentum. By using the theory of complex spectral curves from the mathematical field of algebraic geometry, I will describe how these dispersion relations can be understood as Puiseux series in (fractional powers of) complex momentum. The series have finite radii of convergence determined by the critical points of the associated spectral curves. For theories that admit a dual gravitational description through holography (AdS/CFT), the critical points correspond to level-crossings in the quasinormal spectrum of a dual black hole. Interestingly, holography implies that the convergence radii can be orders of magnitude larger than what may be naively expected. This fact could help explain the “unreasonable effectiveness of hydrodynamics” in describing the evolution of quark-gluon plasma. In the second part of my talk, I will discuss a recently discovered phenomenon called “pole-skipping” that relates hydrodynamics to the underlying microscopic quantum many-body chaos. This new and special property of quantum correlation functions allows for a precise analytic connection between resummed, all-order hydrodynamics and the properties of quantum chaos (the Lyapunov exponent and the butterfly velocity).