Classification of Complex Analytic Map-Germs

Classification of maps is an important part of Singularity Theory. One can see that the pioneers of the subject Whitney and Thom both gave important classifications that stimulated significant amounts of research and were applied in many different contexts. In fact, Thom's classification of the seven elementary catastrophes took on a life of its own and penetrated into the public consciousness during the 1970s. In this book we develop computational method suitable for performing the classification theory. A computer package called CAST is developed. This is written in the Singular program and performs calculations such as complete transversals, finite determinacy and triviality. We discuss the package in detail and give examples of calculations performed in this book. Also, we classify map-germs $(\C^{p},0)\to (\C^{q} ,0)$ under $_V\KK$-equivalence: the restriction of $\KK $-equivalences to those preserving a particular subset of the singularity's domain. We consider the case where $V$ is the image of the minimal crosscap of multiplicity $d\geq 2$. In the final part, we give an application to classification problems. We classify corank $1$ $\AA_{e}$-codimension $2$ map-germs.

M. S. J. Al-Bahadeli obtained his PhD in Mathematics at University of Leeds. He is currently a lecturer in Mathematics at University of Baghdad. His research interests are primarily in the areas of Singularity Theory; classification of complex analytic map-germs, computational methods in the classification problems and liftable vector fields.