Introduction

We introduce and review the essential kinematical and dynamical differences between the two most commonly encountered examples of indistinguishable quantum particles. Namely, bosons and fermions. We discuss, in detail, paradigmatic examples of dynamical and thermodynamical instabilities arising bosonic systems that lack any fermionic counterpart. In the non-interacting setting, the emergence of these instabilities can be, in part, associated to an intrinsic non-Hermiticity present in the Heisenberg equations of motion. As such, we motivate the study of this effective non-Hermiticity from the perspective of realizing novel non-Hermitian physics known to exist in nonequilbrium and open quantum systems. A second key distinction between non-interacting fermionic and bosonic physics is the apparent lack of topology as a classifying agent for bosonic thermodynamic phases. We explain the reason for this and motivate the move into an explicitly open Markovian setting in order to obtain bosonic signatures of non-trivial topology reminiscent of their fermionic counterparts. We then summarize all of the main results presented in the thesis.