I work on algebraic topics with a geometric flavour. In this context, my main focus has been to understand the space of representations of finitely generated groups into Lie groups. In the classical case where the source is finite, this essentially reduces to the correspondence between linear representations and characters due to Frobenius. On the other hand, when the source is infinite, the analogue of character theory is the parametrization of representations by geometric varieties. Although these so-called representation varieties are interesting in their own right, they happen to arise concretely in various settings such as mathematical physics, symplectic geometry and gauge theory. There, one often encounters basic questions about their topology; for instance, one might need to know their number of connected components. My long-standing goal is to understand the topology of these spaces by exploiting their algebro-geometric structure.