Hydrodynamic excitations corresponding to sound and shear modes in fluids are characterized by gapless dispersion relations. In the hydrodynamic gradient expansion, their frequencies are represented by power series in spatial momenta. In this talk we will discuss the convergence properties of the hydrodynamic series by studying the associated spectral curve in the space of complexified frequency and complexified spatial momentum. For the N=4 supersymmetric Yang-Mills plasma at infinite 't Hooft coupling, we will use the holographic methods to demonstrate that the derivative expansions have finite non-zero radii of convergence. Obstruction to the convergence of hydrodynamic series arises from level-crossings in the quasinormal spectrum at complex momenta. We will discuss how finiteness of 't Hooft coupling affects the radius of convergence. We will show that the purely perturbative calculation in terms of inverse 't Hooft coupling gives the increasing radius of convergence when the coupling is decreasing. Applying the non-perturbative resummation techniques will make radius of convergence piecewise continuous function that decreases after the initial increase. Finally, we will provide arguments in favour of the non-perturbative approach and show that the presence of nonperturbative modes in the quasinormal spectrum can be indirectly inferred from the analysis of perturbative critical points.

I will describe recent work with Zhibeodov on the boundary interpretation of orbits around an AdS black hole. When the orbits are far away from the black hole, these orbits describe heavy-light double-twist operators on the boundary. I will discuss how the dimensions of these operators can be computed exactly in terms of quasinormal modes in the bulk, using techniques from a paper to appear soon with Grassi, Iossa, Lichtig, and Zhiboedov. Then I will explain how these results are related to the concept of quantum scars, which are eigenstates that do not obey ETH. 

3d N=4 gauge theories can be studied on a circle times a closed Riemann surface. Their partition functions on this geometry, known as twisted indices, were computed some time ago using supersymmetric localisation on the Coulomb branch. An alternative perspective is to consider the theory as a supersymmetric quantum mechanics on S^1. In this talk I will review this point of view, which unveils interesting connection to topics in geometry such as wall-crossing and symplectic duality of quasi-maps.