Noncommutative curves and noncommutative surfaces

In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categori es. Roughly speaking and by analogy with the commutative situation, the categor y of graded modules modulo torsion over a noncommutative graded ring of qua dratic, respectively cubic, growth should be thought of as the noncommutati ve analogue of a projective curve, respectively surface. This intuition has led to a remarkable number of nontrivial insights and results in noncommut ative algebra. Indeed, the problem of classifying noncommutative curves (an d noncommutative graded rings of quadratic growth) can be regarded as settl ed. Despite the fact that no classification of noncommutative surfaces is i n sight, a rich body of nontrivial examples and techniques, including blowi ng up and down, has been developed.