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.