Welcome!

This book aims to provide a short introduction to the topology of algebraic varieties, with a special emphasis placed on singular varieties and their homotopy groups. It is divided into six chapters. In chapter 1, we discuss the notion of homotopy depth (introduced by A. Grothendieck) which plays an important role in the study of singular varieties. Chapters 2 and 3 are devoted to Lefschetz-Zariski hyperplane section theorems and Zariski-van Kampen theorems, in both cases, again, with the main focus on singular varieties.  In Chapters 4 and 5, we focus on fundamental groups of plane curve complements. Finally, in Chapter 6, we investigate Zariski-van Kampen type theorems for higher homotopy groups of hypersurface complements.

This book can be purchased at: https://www.impan.pl/pl/wydawnictwa/impan-lecture-notes/114956/homotopy-theory-on-singular-complex-algebraic-varieties

This book is an introduction to equisingularity theory through a specific problem, namely the famous Zariski multiplicity conjecture. This conjecture says that the multiplicity of a reduced complex hypersurface singularity is an embedded topological invariant. Motivated by this problem, the book aims to present a number of major advances in equisingularity theory over the last five decades. A special emphasis is placed on recent developments on non-isolated singularities.

This book can be purchased at: https://www.impan.pl/pl/wydawnictwa/impan-lecture-notes/92014/topics-in-equisingularity-theory