In algebraic geometry, there is an important correspondence between commutative graded algebras and projective schemes. Noncommutative projective geometry generalizes aspects of this correspondence to the noncommutative setting, for example by using the category of graded modules over the ring as a substitute for an actual geometric space. Noncommutative projective curves are well-understood, so noncommutative surfaces and special kinds of threefolds are the major subjects of current interest. A major goal of this project is to study surfaces in the birational classes of noncommutative projective planes, especially those of Sklyanin type.
Noncommutativity is the phenomenon of two operations giving different results when performed in a different order. An important example in physics is the fact that the position and momentum operators do not commute; this is the part of the mathematical formulation of the Heisenberg Uncertainty principle. The subject of noncommutative geometry attempts to give some geometric intuition behind collections of noncommuting operations which are closed under addition and multiplication (these are called rings). Then the study of points, lines, and other such geometric concepts can help to illuminate our understanding of such rings.
