Topology, braiding, and disorder in planar Josephson junctions*

In this talk I will describe how the Josephson effect may be employed to realize one dimensional topological superconductivity, with the Josephson phase difference being a "user-friendly" knob to drive the system to become topological. I will describe the basic idea, the experimental observations, the relation to topological superconductivity based on quantum wires, a surprising effect of disorder, and a scheme for braiding Majorana zero modes in Josephson junctions. I will also describe how this system manifests a first order phase transition into a topological phase.