Abstract: Noncommutative projective algebraic geometry uses the techniques and intuition behind classical algebraic geometry to understand noncommutative graded algebras and their associated categories. Over the last two decades this has been surprisingly effective, both in classifying large and important classes of algebras and, as significantly, in introducing a wide range of fundamental techniques that have in turn had a wide range of applications. The aim of these lectures will be to provide an introduction into this subject. In particular I will try to keep the technicalities to a minimum, especially in the first lecture, so that it will be intelligible to a wide audience.

Let me give a little more detail. Formally, noncommutative projective varieties do not exist, so in order to obtain an analogy with the commutative case we work with their categories of sheaves (or modules). Thus the category $qgr(R)$ of finitely generated graded modules modulo those of finite length over a graded ring $R=k\oplus R_1\oplus R_2\oplus\cdots$ of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has been remarkably efffective. Indeed, the problem of classifying noncommutative curves (aka noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, the attempts at such classification have led to a rich body of nontrivial examples and techniques. Describing these results will form the heart of these lectures.