There are multiple interactions between noncommutative algebra and representation theory on the one hand and classical algebraic geometry on the other, and the aim of this book is to expand upon this interplay. One of the most obvious areas of interaction is in noncommutative algebraic geometry, where the ideas and techniques of algebraic geometry are used to study noncommutative algebra. An introduction to this material is given in Chapter I. Many of the algebras that appear naturally in that, and other, areas of mathematics are deformations of commutative algebras, and so in Chapter II we provide a comprehensive introduction to that theory. One of the most interesting classes of algebras to have appeared recently in representation theory, and discussed in Chapter III, is that of symplectic reflection algebras. Finally, one of the strengths of these topics is that they have applications back in the commutative universe. Illustrations of this appear throughout the book, but one particularly important instance is that of noncommutative (crepant) resolutions of singularities. This forms the subject of Chapter IV.

These notes have been written up as an introduction to these topics, suitable for advanced graduate students or early postdocs. In keeping with the lectures upon which the book is based, we have included a large number of exercises, for which we have given partial solutions at the end of book. Some of these exercises involve computer computations, and for these we have either included the code or indicated web sources for that code.

We now turn to the individual topics in this book. Throughout the introduction k will denote an algebraically closed base field and all algebras will be k-algebras.

This subject seeks to use the results and intuition from algebraic geometry to understand noncommutative algebras. There are many different versions of noncommutative algebraic geometry, but the one that concerns us is noncommutative projective algebraic geometry, as introduced by Artin, Tate, and Van den Bergh [9, 10].