The study of plane curve singularities is one of the most classical parts of singularity theory going back to Newton in the XVII century.
When one studies complex polynomials in two variables, singularities appear in a very natural way. Although many times this topic is treated from an algebraic point of view, one quickly sees that it has many ties with low dimensional topology topics such as knot theory and mapping class groups.

In this mini-course we will make a gentle introduction to singularity theory through the world of plane curves. We will focus on the topological aspect of singularities and we will mainly learn techniques through rich examples. By the end of the course we will be able to compute many invariants of a plane curve singularity and we will understand the topology around a singular point of an algebraic plane curve. In particular we will learn how to find parametrizations of each irreducible component of a plane curve singularity. We will see how these parametrizations can result very useful in computing the embedded topology of each branch and how each branch interacts with the rest. We will learn to find smooth models (resolve) of plane curve singularities by repeatedly blowing up the ambient space and, from the final picture, we will understand the topology of the Milnor fibration and its geometric monodromy.
We will end the course by introducing the versal unfolding of a plane curve singularity and posing some questions that naturally emanate from it.