I will begin by defining a canonical family of perturbations of the Dirac equation. These perturbations are complex anti-linear, thus ground states only form a real vector space. A special case of this theory is known as the Jackiw–Rossi theory, which models surface excitations on the surface of a topological insulator placed in proximity to an s-wave superconductor. While the physics of the theory is relatively well-understood, the mathematical side has not been studied, even on surfaces, not to mention its generalizations to higher dimensional and on nontrivial manifolds. I call these equations the *generalized Jackiw–Rossi equations*.

After the definitions and connections to physics, I will present my current work on the generalized Jackiw–Rossi equations. My main result is a concentration phenomenon which proves the physical expectation that such Majorana fermions concentrate around the vortices of the superconducting order parameter. Moreover, I provide approximate solutions that are exponentially sharp in the large coupling limit.

If time permits, then I will show how these Majorana fermions define a bundle of projective spaces over the *simple* part of vortex moduli spaces. The holonomies of such bundles give rise to projective representations of (surface) braid groups, and thus, speculatively, can be of interest to quantum information theorists.