Title: The Topology of Complex Hypersurface Singularities
Speaker: Yilong Zhang (Ohio State University)
Abstract: Let $f(z_1,...,z_n)$ be a polynomial function in $n$ complex variables, so the zero locus of $f$ defines a hypersurface $X$ of $\mathbb C^n$. Assume that $p$ is an isolated singularity of $X$. We want to understand the topology of $X$ around the singularity $p$. Let $S_r\subset \mathbb C^n$ be a $2n-1$-sphere with radius $r$ centered at $p$, whose intersection with $X$ is a smooth. Milnor showed that for sufficiently small $r$, the map $z\mapsto f(z)/|f(z)|$ from $S_r\setminus X$ to the unit circle is a smooth fiber bundle, and the fiber $F$ is commonly called the Milnor fiber, which has homotopy type a bouquet of $n-1$-spheres. The purpose of this talk is to give an introduction to the Milnor fiber, and I will explain how it is related to stable reduction of algebraic curves.
