Algebraic Link Invariants and Resolution of Singularities

This thesis focuses on the local study of isolated singularities in plane algebraic curves, with the goal of unifying different mathematical approaches. We begin by presenting the algebraic perspective given by singularity resolution, through blow-up methods and Newton polygons. We then shift to a topological perspective, examining the ties to link theory and using differential geometry to define the Milnor number as a topological invariant. Finally we show how this number can be seen as a pivot to connect the two frameworks, proving that classical algebraic tools may be used to compute this topological object. Throughout the thesis we see that using different methods to study singularities leads to coherent results, which mutually reinforce each other and provide a complete understanding of singularity theory in plane algebraic curves.