This thesis is devoted to studying quantum mechanical systems with gravity duals. It is interesting to study holographic correspondence for quantum mechanical systems since we have much more theoretical control over them compared to quantum field theories. At the same time, gravity duals to quantum mechanical systems are quite rich as they can include black holes and wormholes.
Chapter 2 is based on work  with J. Maldacena and studies aspects of gauge symmetry in Banks-Fishler-Shenker-Susskind(BFSS) model. In the original formulation it includes gauged SU (N) symmetry. However, we argued that non-singlet states are separated by a finite gap from the ground state. Therefore, gauging SU (N) symmetry is not important at low energies.
Chapter 3 is based on paper  with A. Almheiri and B. Swingle. It is dedicated to studying thermalization dynamics of systems with gravity duals. We argued that average null energy condition(ANEC) in the bulk leads to a universal bound on the total amount of energy exchange between two quantum systems. We study this bound perturbatively and in Sachdev-Ye-Kitaev(SYK) model at arbitrary coupling. As a byproduct, we studied the non-equilibrium dynamics of SYK, both analytically and numerically.
Chapter 4 is based on paper  with J. Maldacena. We study wormhole formation in SYK model in real time. We start from a high temperature state, let it cool by coupling to a cold bath and numerically solve for the large N dynamics. Our main result is that the system forms a wormhole by going through a region with negative specific heat, taking time that is independent of N .
Chapter 5 is based on paper  with I. Klebanov, F. Popov and G. Tarnopolsky. This paper is dedicated to studying various spectral properties of large N melonic tensor models. They have the same large N limit as SYK model, but unlike SYK they do not include disorder average. We find the exact expression for the number of singlet states and derive various bounds on energies.