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Ishii Lab
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Research on the McKay correspondence, which links the geometry of singularities with the representation theory of groups.
AboutAbout the lab
Connecting the geometry of singularities with algebra.
Prof. Ishii studies algebraic geometry, focusing especially on a phenomenon called the McKay correspondence.
The McKay correspondence is a beautiful and mysterious relationship between the geometric description of the resolution of two-dimensional singularities (Kleinian singularities) and the representation theory of finite groups.
Generalizations of this relationship are also called by the same name.
The professor has described resolutions of singularities as moduli spaces and studied the structure of the related derived categories.
Key terms
- singularity
- A point on a shape such as a surface where it has a sharp point or crosses itself, so that it is not smooth
- McKay correspondence
- A correspondence between the geometry of resolutions of singularities and the representation theory of finite groups
- moduli space
- A single space that collects all objects of a given kind, with each object corresponding to one point
- derived category
- An algebraic framework that handles sheaves and similar objects on a shape in the form of "complexes"
ResearchResearch themes
Centering on the phenomenon known as the McKay correspondence, the lab studies the correspondence between geometry and algebra surrounding certain kinds of resolutions of singularities. It has described resolutions of singularities as moduli spaces and studied the structure of the related derived categories.
*Illustrative imageMcKay Correspondence and Crepant Resolutions
The lab has studied the correspondence between resolutions and the representation theory of groups for singularities obtained as quotients of the plane or three-dimensional space by a finite group. In dimension three, a good resolution (a crepant resolution) is not necessarily unique. Together with Craw, the lab showed that in the case of abelian groups, every crepant resolution can be realized as a moduli space.
*Illustrative imageDimer Models
This was studied mainly together with Kazushi Ueda. A dimer model determines a noncommutative ring, and when the dimer model is consistent, that noncommutative ring is connected through derived categories to a certain three-dimensional singularity. This relationship was found by string theory physicists as an extension of the McKay correspondence for abelian groups.
*Illustrative imageStructure of Derived Categories
The lab studies the structure of the derived category of coherent sheaves on an algebraic variety itself. In joint work with Uehara and Ueda, it obtained results on the group of autoequivalences and the space of Bridgeland stability conditions for the minimal resolution of a Kleinian singularity of type A.
MembersFaculty
- ProfessorAkira Ishii
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Surface of degree 7” Jean Constant / CC BY-SA 4.0 (cropped)
- McKay Correspondence and Crepant Resolutions: “Cayley graph of quaternion group on torus” PSL27 / CC BY-SA 4.0 (cropped)
- Dimer Models: “Domino” Dietmar Rabich / CC BY-SA 4.0 (cropped)
- Structure of Derived Categories: “Encyclopedia of Mathematics, Complete Set” Policron / CC BY-SA 3.0 (cropped)
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