Hermitian and non-Hermitian topological phenomena in active matter

I will talk about aspects of Hermitian and non-Hermitian topology which can be relevant to certain types of active matter systems. Specifically, we first introduce a simple model of active particles that can exhibit the topologically protected sound mode in a steady-state regime. This robust sound mode is a classical analog of the edge mode originally discussed by Haldane in the context of the quantum anomalous Hall effect. We then discuss a possibility to use active systems to realize yet another exotic edge mode, namely, exceptional edge mode. We show that it is protected by the topological structure associated with exceptional points in complex-valued band structures and is thus immune to disorder or impurities. Time permitting, we also discuss how the exceptional edge mode could emerge in a system of coupled nonlinear oscillators and realize a phenomenon that we call topological synchronization.