Topology is a branch of mathematics that studies properties that remain unchanged under continuous deformations. In physics, topology is used to describe phenomena that are robust against small perturbations. A well-known example is topological insulators—materials that act as insulators in their interior while conducting along their surface. These conducting states are protected by topological properties and persist even when the material is slightly modified. Over the past few decades, significant effort has been devoted to understanding and classifying different types of topological phases, which describe the various ways in which such robust properties can emerge in nature.

In recent years, interest has grown in dissipative systems, where energy losses play a central role. These systems are described using non-Hermitian Hamiltonians, which extend the conventional quantum mechanical framework.

This dissertation explores how non-Hermitian physics affects the topology and classification of topological phases. In particular, we investigate a type of topological charge known as exceptional points, which arise exclusively in non-Hermitian systems. These points are characterized by a topological charge that describes how energy bands intertwine around them. We focus specifically on how certain symmetries can stabilize exceptional points and shape their properties. Finally, we examine multifold exceptional points—a more intricate class of these singularities—and their topological characteristics.