Speaker:	Charlie Beil (UC Santa Barbara)
Title:	The geometry of noncommutative singularity resolutions: shrinking exceptional loci to zero size
Date (JST):	Mon, May 17, 2010, 14:00 - 17:00
Place:	Seminar Room A
Abstract:	Part I. In the first talk I will introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining appropriate conventional resolutions using noncommutative coordinate rings. This conjecture provides a possible new generalization of the classical McKay correspondence. We then explain how symplectic reduction may be used within these rings to obtain new, non-conventional resolutions that are hidden if only commutative functions are considered. We discuss how these non-conventional resolutions result from shrinking exceptional loci to stack-like (non-Azumaya) points. Part II. In the second talk we verify the conjecture for the conifold and different surface quotient singularities (including the A_n, D_n, and E_6 singularities), and describe what happens to their minimal resolution when exceptional loci are shrunk to zero size. A large class of algebras for which the conifold is a special case, called square superpotential algebras, will then be considered. We show that these algebras are noncommutatve crepant resolutions with 3 dimensional normal Gorenstein centers, and describe what happens when the exceptional locus of a blow-up of their center is shrunk to zero size.
Remarks:	we will have a break at 15:00-15:30.

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