Speaker : Pablo Portilla cuadrado (University of Lille)
Place : 상산관 129동 406호
Schedule : 7/11 (Mon) Survival kit on plane curve singularities I
7/12 (Tue) Survival kit on plane curve singularities II
7/18 (Mon) Characterizing the geometric monodromy group of an isolated plane curve singularity
7/19 (Tue) A quadratic form associated with pseudo-periodic homeomorphisms arising from singularity theory
(each day) 10:30 ~ 11:30, 11:45 ~ 12:45
TiTle : Survival kit on plane curve singularities I & II
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.
Title : Characterizing the geometric monodromy group of an isolated plane curve singularity
In this talk we will explain an intrinsic characterisation of the geometric monodromy group of an isolated plane curve singularity as the stabiliser in the mapping class group of the Milnor fiber of the relative isotopy class of a canonical vector field. We will also discuss two interesting consequences of this result: an easy and efficient criterion for detecting whether a simple closed curve in the Milnor fiber is a geometric vanishing cycle or not, and the non-injectivity of the natural representation of the versal unfolding of the singularity. This is a joint work with Nick Salter.
Title : A quadratic form associated with pseudo-periodic homeomorphisms arising from singularity theory
In this talk, we study the nilpotent part of a pseudo-periodic automorphism of a real oriented surface with boundary. We associate a quadratic form Q defined on the first homology group of the surface. Using some techniques from mapping class group theory, we prove that a related form is positive definite under some conditions that are always satisfied for monodromies of Milnor fibers of germs of curves on normal surface singularities. Moreover, the form Q is computable in terms of the dual resolution or semistable reduction graph. Numerical invariants associated with Q are able to distinguish plane curve singularities with different topological types but same spectral pairs. This is a joint work with L. Alanís, E. Artal, C. Bonatti, X. Gómez-Mont and M. González Villa.