Workshop on non-commutative geometry and the McKay correspondence

WORKSHOP ON NON-COMMUTATIVE GEOMETRY AND THE McKAY CORRESPONDENCE

14-16 March 2011
Graduate School of Mathematics
Nagoya University

SCHEDULE

14th March (Monday)

   10:00-11:00 Alvaro Nolla de Celis (Nagoya), "Equivariant Hilbert schemes and crepant resolutions of polyhedral singularities".
   11:30-12:30 Oskar Kedzierski (Warsaw), "Resolutions of the 3-dimensional cyclic terminal singularity".
     Lunch
   14:00-15:00 Akira Ishii (Hiroshima), "A remark on invertible polynomials and exceptional collections".
     Tea break
   16:00-17:00 Yuhi Sekiya (Nagoya), "Mutations of CY-algebras and moduli spaces".

15th March (Tuesday)

10:00-11:00 Takehiko Yasuda (Kagoshima), "Noncommutative resolution of pure subrings and Frobenius morphisms".
11:30-12:30 Osamu Iyama (Nagoya), "Algebraic McKay correspondence and Cluster tilting".
Lunch
14:00-15:00 Izuru Mori (Shizuoka), "McKay correspondence in noncommutative algebraic geometry".
Tea break
16:00-17:00 Hokuto Uehara (Tokyo Metropolitan), "Generators by Frobenius push-forward on smooth toric surfaces".

16th March (Wednesday)

10:00-11:00 Kentaro Nagao (Nagoya), "Motivic Donaldson-Thomas invariants and wall-crossing".
11:15-12:15 Joseph Grant (Nagoya), "Periodic algebras, derived autoequivalences, and preprojective algebras".

All the talks be at Room I-309.
How to arrive

Titles and Abstracts

Alvaro Nolla de Celis (Nagoya)
Title: "Equivariant Hilbert schemes and crepant resolutions of polyhedral singularities".
Abstract: I will explain the construction of the equivariant Hilbert scheme G/N-Hilb(N-Hilb) as a moduli of representations of the McKay quiver Q. To present a family of examples of this Hilbert schemes I will treat the case of finite subgroups of SO(3) and describe explicitely every crepant resolution of the polyhedral singularity C^3/G, which are in 1-to-1 correspondence with mutations of Q. The talk pretend to cover join works with Y. Ito and Y. Sekiya.
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Oskar Kedzierski (Warsaw)
Title: Resolutions of the 3-dimensional cyclic terminal singularity".
Abstract: The talk will be devoted to the geometry of the two resolutions: the G-Hilbert scheme and the Danilov resolution. Both will be interpreted as moduli spaces of McKay quiver representations and the full chamber of corresponding stability conditions will be computed.
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Akira Ishii (Hiroshima)
Title: "A remark on invertible polynomials and exceptional collections".
Abstract: I will talk about a conjecture on the derived categories of Deligne-Mumford stacks associated with invertible polynomials, where the two-dimensional case is related to the special McKay correspondence.
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Yuhi Sekiya (Nagoya)
Title: "Mutations of CY-algebras and moduli spaces".
Abstract: I will talk about a relation between moduli spaces of modules over two different 3-CY algebras which are related by a mutation. Moreover I explain how to compute endomorphism rings of tilting modules by using mutations of quivers with potentials in special case. Finally I give a correspondence between crepant resolutions and non-commutative crepant resolutions of polyhedral singularities which are quotient singularities by finite subgroups of SO(3).
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Takehiko Yasuda (Kagoshima)
Title: "Noncommutative resolution of pure subrings and Frobenius morphisms".
Abstract: We will see that a certain pure subring of a regular local ring or a ring of finite representation type admits a noncommutative resolution. We will also discuss its relation to (commutative or noncommutative) Frobenius morphisms.
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Osamu Iyama (Nagoya)
Title: "Algebraic McKay correspondence and Cluster tilting".
Abstract: Algebraic McKay correspondence was given by Auslander as a categorical equivalence between Cohen-Macaulay modules over Kleinian singularities and projective modules over skew group algebras. Its higher dimensional analogue is given by "cluster tilting", which is a kind of noncommutative resolutions. As an application, we give a positive solution to a noncommutative analogue of Bondal-Orlov conjecture in dimension 3 by tilting theory.
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Izuru Mori (Shizuoka)
Title: "McKay correspondence in noncommutative algebraic geometry".
Abstract: This is a very preliminary report on work in progress. We will see that there may be McKay type correspondence in noncommutative algebraic geometry.
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Hokuto Uehara (Tokyo Metropolitan)
Title: "Generators by Frobenius push-forward on smooth toric surfaces".
Abstract: We show that for any smooth projective toric surface, the Frobenius push-forward of the structure sheaf is a classical generator of the derived category of coherent sheaves. My talk is based on a joint work with Ryo Ohkawa.
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Kentaro Nagao (Nagoya)
Title: "Motivic Donaldson-Thomas invariants and wall-crossing".
Abstract: The motivic Donaldson-Thomas invariant, introduced by Kontsevich-Soibelman and Behrend-Bryan-Szendroi, is defined using the motivic Milnor fiber of the holomorphic Chern-Simons functional. In this talk I will show a wall-crossing formula for the motivic Donaldson-Thomas invariants.
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Joseph Grant (Nagoya)
Title: "Periodic algebras, derived autoequivalences, and preprojective algebras"
Abstract: I will explain how we can construct derived autoequivalences of finite dimensional algebras based on periodic algebras. This construction is related to the spherical twists of Seidel and Thomas. I will also give an example of how the construction can be applied to some (infinite-dimensional) preprojective algebras.
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