This book is based on a series of lectures on noncommutative geometry given in 1999 in the framework of an EU project. The main theme of the book is the theory of Cayley-smooth orders in central simple algebra over a function field of a variety. The book has several parts.

The first part (chapters 1 - 4) reviews a lot of material needed in the main parts of the book. In particular, the first two chapters are devoted to Cayley-Hamilton algebras and to a description of orders and their centres from the point of view of invariant theory. The next two chapters deal with étale topology, Asumaya algebras, Luna slices and quivers and their representations. Chapters 5 and 6 form the main core of the book. They are devoted to a study of Cayley-smooth orders, their étale locale structure and a classification of the associated central singularity. Chapter 6 deals with a study of different strata in the Hesselink stratification of the null-cone of quiver representations. The last two chapters contain a study of Quillen-smooth algebras and their representations. The book is very well-ordered and is written in a nice and readable style. It contains a huge amount of interesting material related to many important topics in modern mathematics and mathematical physics. It could be greatly appreciated by talented students and young mathematicians as a well-written introduction to important fields of mathematics.

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