The use of differential geometry to understand topological properties precedes the formal establishment of Algebraic Topology as a mathematical discipline and has been a central theme in mathematics since the early 20th Century. The goals of this project fall in this tradition, in that we propose to study a variety of geometric objects and their topological properties using tools from Differential Geometry and Analysis. The main areas of study are the following: Foliations of singular spaces. The main objective is to generalize the theory for regular spaces by using tools of non-commutative geometry. Holonomy and Lie algebroids. A central question is the extension of the notion of parallel transport to higher dimensions, beyond the case of curves. Topology and geometry of moduli spaces. Among the central objects of study are character varieties for surface groups. Through the holonomy representation and non-abelian Hodge theory these spaces are viewed as moduli spaces of gauge theoretic objects, known as Higgs bundles.