Noncommutative surfaces and Calabi-Yau algebras

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Noncommutative algebraic geometry generalizes certain aspects of the correspondence between commutative rings and schemes to the setting of noncommutative rings. In particular, noncommutative projective geometry uses a localization of the category of graded modules over a noncommutative graded ring as a substitute for a category of coherent sheaves. A major goal of this project is to advance the classification of noncommutative projective surfaces. We continue to study surfaces in the birational classes of noncommutative projective planes, especially those of Sklyanin type. Calabi-Yau algebras are an important class of noncommutative algebras with good homological properties, which are related to commutative Calabi-Yau varieties. For example, in certain cases they can be seen as providing noncommutative resolutions of singular varieties. In this project, we also propose to study certain classes of graded Calabi-Yau algebras which are closely connected to noncommutative projective geometry.The set of all functions from a geometric space to the real numbers has a natural addition and multiplication. A set with an addition and multiplication is called a ring, and the example above is called the coordinate ring of the space. Its algebraic properties often reflect and give information about the underlying geometry of the space. The goal of noncommutative geometry, which is the subject of this project, is to generalize to rings which do not directly arise as coordinate rings of spaces, because the product of two elements in these rings depends on the order you multiply them. Though there is no underlying space, the algebra of these rings can have deep structure that allows one to still study them in a geometric way. As an example, certain noncommutative rings can be thought of as giving a noncommutative resolution of a singular commutative space (the space is not smooth but has sharp points, and the resolution smooths it out). Our main project concerns the classification of noncommutative surfaces (2-dimensional spaces) and the structure of Calabi-Yau algebras, which are noncommutative rings motivated originally by string theory in theoretical physics.