Algebraic geometers like to study varieties — those subsets of projective space that can be defined by polynomial equations. And in many applications within algebraic geometry smooth points — that is, points where the implicit function theorem works or points where not all the partial derivatives vanish — behave much more nicely than singular points — those points which are not smooth, such as nodes and cusps. Because of this, many people have developed techniques to deal with singularities that can be broadly classified as the 'resolution of singularities', which roughly speaking takes an arbitrary variety X which may have singular points and defines a variety X' which does not have any singular points so that there is a nice surjection mapping X' to X. There are many ways of approaching this problem, each of which comes with various pros and cons, and in János Kollár's new book Lectures on Resolution of Singularities he gives a comprehensive and very readable overview of the topic.
The book is based on lectures Kollár gave at Princeton and at the University of Utah, and is divided into three chapters. The first chapter deals with the resolution of singularities on curves, and discusses thirteen distinct approaches to the problem (several of which have sub-approaches). These range from Newton's method of rotating rulers and Hensel's Lemma to the Albanese method and the so-called 'blowing up' of singularities. Some of these approaches involve lots of background and machinery from algebraic geometry, topology, and other branches of mathematics, but others are quite elementary. The second chapter takes several of these methods and generalizes to the case of surfaces, where one often needs to use more care (and at times more machinery) in order to make the program work. The third chapter deals with the general case, where one can actually make quite a bit of headway using (relatively) elementary methods as long as you are willing to limit yourself to working in characteristic zero.
Throughout his lectures, Kollár uses plenty of motivations and examples, and the text is very readable. Any graduate student or mathematician who wishes to learn about the subject would be well-served to use this book as a starting point.
Darren Glass is an Assistant Professor at Gettysburg College. His mathematical interests include Algebraic Geometry, Number Theory, and Cryptography. He can be reached at firstname.lastname@example.org.