Classification of Noncommutative Projective Surfaces

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The objective of this project is to make progress on parts of theclassification of noncommutative projective surfaces. This is animportant problem, because many interesting new phenomena occur forsurfaces that are absent for curves; thus understanding the surfacecase better will advance our understanding of noncommutativegeometry in general. First, the PI plans to complete theclassification of birationally commutative surfaces---which arenoncommutative surfaces with a commutative function field oftranscendence degree 2---in terms of the process of naivenoncommutative blowing up. Next, one wants to study noncommutativesurfaces containing a commutative curve of points---we will start byexamining a class of subalgebras of elliptic regular algebras ofdimension 3, which may be related to M. Van den Bergh's differentnotion of noncommutative blowing up. The PI plans to study fat pointmodules, about which there are many foundational questions.Finally, one should study the generalization of naive blowing up tohigher dimensional noncommutative varieties.Noncommutative projective geometry is a relatively new subject,which attempts to use some of the same intuition and techniquesunderlying commutative algebraic geometry, but in the noncommutativesetting. Modern physics tells us the importance of consideringoperations that may not commute with each other, which means itmatters which order you perform them. For example, oneinterpretation of the celebrated Heisenberg Uncertainty Principle isthat position and momentum do not commute. There are many otherconnections between noncommutative algebra and physics. Thisproject hopes to gain a better abstract understanding of certainkinds of structures in algebra (noncommutative graded rings of lowdimension) which include among them some famous examples, such asthe Sklyanin algebras, which are related to the study of thephysical world.