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Project
Non-commutative resolutions of quotient singularities for reductive groups - INCOMING [Pegasus]² Marie Sklodowska-Curie Fellowship: Appllicant Spela Spenko (FWOTM843)
A singularity is a point which is different from a general point in a space, lacking a well-behaved local structure; e.g. a corner. A cone has one distinguished point, which is a singularity, while a cooling tower is nonsingular. Cusps, nodes and self-intersection points are singular points of a curve.
Our research proposal is about resolutions of singularities. The basic idea is to replace a singular space with a nonsingular one which closely resembles it. For example, a double cone could be replaced by a cooling tower, while for a curve with one self-intersection a piece of the curve could be lifted off itself. Given a singular space, it is an interesting and important problem to understand all of its resolutions. For a curve or a surface there exists a unique minimal resolution of singularities, but in higher dimension this is no longer true.
An important emerging tool in the study of resolutions is the passage to “non-commutative spaces”. There are many instances where nice geometric resolutions fail to exist, or are difficult to understand explicitly. The novel idea, which occurred first in mathematical physics, and later independently in mathematics, is to study “non-commutative resolutions” (given by an algebraic object), which exist often in cases where their geometric counterparts do not. In this proposal, we shall seek to build on our previous work to construct and analyze non-commutative resolutions for very general classes of singularities.
Our research proposal is about resolutions of singularities. The basic idea is to replace a singular space with a nonsingular one which closely resembles it. For example, a double cone could be replaced by a cooling tower, while for a curve with one self-intersection a piece of the curve could be lifted off itself. Given a singular space, it is an interesting and important problem to understand all of its resolutions. For a curve or a surface there exists a unique minimal resolution of singularities, but in higher dimension this is no longer true.
An important emerging tool in the study of resolutions is the passage to “non-commutative spaces”. There are many instances where nice geometric resolutions fail to exist, or are difficult to understand explicitly. The novel idea, which occurred first in mathematical physics, and later independently in mathematics, is to study “non-commutative resolutions” (given by an algebraic object), which exist often in cases where their geometric counterparts do not. In this proposal, we shall seek to build on our previous work to construct and analyze non-commutative resolutions for very general classes of singularities.
Date:1 Mar 2017 → 1 Oct 2019
Keywords:singularity
Disciplines:General mathematics