Combinatorical algebraic topology: dualities for manifolds with boundary

During the last years techniques from Algebraic Topology have been applied to a variety of fields ranging from data analysis to the construction of numerical methods for PDE’s . In this thesis we will focus on this second kind of application. In particular we will deal with spaces (manifolds) which admits a "triangularization" and then we will use the combinatoric of the data to deal with the problems. In fact we will show how to create a model that solves a Poisson problem in dimension 2: we will translate the smooth geometric techniques of Differential Geometry in discrete form via Combinatorical Algebraic Topology