Symplectic fillings of virtually overtwisted contact structures on lens spaces

Symplectic fillings of standard tight contact structures on lens spaces are understood and classified. The situation is different if one considers non-standard tight structures (i.e. those that are virtually overtwisted), for which a classification scheme is still missing. In this work we use different approaches and employ various techniques to improve our knowledge of symplectic fillings of virtually overtwisted contact structures. We study curves configurations on surfaces to solve the problem in the case of a specific family of lens spaces. Then we give general constraints on the topology of Stein fillings of any lens space by looking at algebraic properties of integer lattices and at geometric slicing of solid tori. Furthermore, we try to place these manifolds in the context of algebraic geometry, in order to determine whether Stein fillings can be realized as Milnor fibers of hypersurfce singularities, finding a series of necessary conditions for this to happen. In the concluding part of the thesis, we focus on the connections between planar contact 3-manifolds and the theory of Artin presentations.