The compactifications of moduli spaces of Burniat surfaces with $2leq k^{2}leq5$

A Burniat surface $X$ is a particular surface of general type with $p_{g}=q=0$, $K_{X}^{2}=2,3,4,5mbox{ or }6$. Alexeev and Pardini constructed an explicit compactification of the moduli space of Burniat surfaces with $K_{X}^{2}=6$.In this thesis, we describe compactifications of moduli spaces of Burniat surfaces with $2leq K_{X}^{2}leq5$ obtained by adding KSBA surfaces, i.e. slc surfaces $X$ with ample canonical class $K_{X}$. We do it in two ways: by describing all one-parameter degenerations, and by using the theory of matroid tilings by matroid polytopes.