Plane curve singularities

A plane curve ( in the sense of the seminar) is a complex curve in C^2 defined by some equation f(z_1, z_2)=0.

The topic of this class is to study what happens near a singular point of a plane curve. Thus we will mainly study curves just in a neighborhood of the origin in the complex plane.  Here by complex plane we mean  C^2. The study of singular points of algebraic curves (where f is a polynomial)  in the complex plane is a meeting point for many different areas of mathematics like geometry, algebra, topology and function theory.  The first systematic study of plane curve singularities is due to Newton. During the nineteenth and early twenteeth century algebraic geometers  developed methods which allowed them to deal with singular curves. One of their notable achievement was the resolution of singularities.  In the late 20th century several interesting results were obtained in the area of topology by looking at the neighborhood of such a singularity.

Our goal for this semester is to understand the basics of the plane curve singularities and study some of its applications in topology.

This class is suitable for advanced bachelors and masters students.

Prior knowledge of complex analysis and some basic topology will be desired.