Moduli spaces attached to singular surfaces and representation theory
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
The aim of our project is to contribute to the study of two very classical constructions in mathematics: singularities of geometric spaces on the one hand, and representation theory (the theory of symmetry) on the other.
One of the enduring patterns in mathematics is the so-called A-D-E classification: there are several, seemingly totally unrelated, questions in mathematics to which the answer involves a certain list of simple combinatorial patterns. One such question is very classical, in some sense going back to Euclid: find all possible finite 3-dimensional (rotational) symmetry groups. The answer is that such groups must be symmetry groups of one of the following polyhedra: a cone based on a regular n-gon (type A or cyclic); a prism based on a regular n-gon (type D or dihedral); or one of the five regular solids such as the tetrahedron, cube or dodecahedron (type E or exceptional). A classical construction translates these symmetry groups into groups of 2x2 (complex) matrices; we then obtain some singular spaces called simple (surface) singularities using these matrix groups.
A seemingly totally unrelated instance of the A-D-E classification is that of simple (simply laced) Lie algebras. Lie algebras are closely related to continuous groups of symmetries. The challenge then is to understand how do these continuous groups (or algebras) of symmetries relate to simple singularities.
A large part of the answer has been known for some time, and is part of what's called the McKay correspondence: given the singularity, it has a resolution, and the geometry of the resolution can be related in different ways to A-D-E patterns and continuous symmetries. Recently however, a tantalising connection has been observed in work of the PI and collaborators that suggests a relationship between the geometry of the singular space itself, and aspects of representations of Lie algebras.
The objectives of our project are to study this connection in different ways:
- Understand in concrete geometric ways a certain auxiliary space, the Hilbert scheme of points of the singularity;
- Relate the geometry of the Hilbert scheme directly to Lie algebra symmetries;
- Find new geometries attached to the singular space and study their properties;
- Extend the connection to other, higher-dimensional singular spaces.
Success in this project will further our understanding of singular geometric spaces and their hidden symmetries.
One of the enduring patterns in mathematics is the so-called A-D-E classification: there are several, seemingly totally unrelated, questions in mathematics to which the answer involves a certain list of simple combinatorial patterns. One such question is very classical, in some sense going back to Euclid: find all possible finite 3-dimensional (rotational) symmetry groups. The answer is that such groups must be symmetry groups of one of the following polyhedra: a cone based on a regular n-gon (type A or cyclic); a prism based on a regular n-gon (type D or dihedral); or one of the five regular solids such as the tetrahedron, cube or dodecahedron (type E or exceptional). A classical construction translates these symmetry groups into groups of 2x2 (complex) matrices; we then obtain some singular spaces called simple (surface) singularities using these matrix groups.
A seemingly totally unrelated instance of the A-D-E classification is that of simple (simply laced) Lie algebras. Lie algebras are closely related to continuous groups of symmetries. The challenge then is to understand how do these continuous groups (or algebras) of symmetries relate to simple singularities.
A large part of the answer has been known for some time, and is part of what's called the McKay correspondence: given the singularity, it has a resolution, and the geometry of the resolution can be related in different ways to A-D-E patterns and continuous symmetries. Recently however, a tantalising connection has been observed in work of the PI and collaborators that suggests a relationship between the geometry of the singular space itself, and aspects of representations of Lie algebras.
The objectives of our project are to study this connection in different ways:
- Understand in concrete geometric ways a certain auxiliary space, the Hilbert scheme of points of the singularity;
- Relate the geometry of the Hilbert scheme directly to Lie algebra symmetries;
- Find new geometries attached to the singular space and study their properties;
- Extend the connection to other, higher-dimensional singular spaces.
Success in this project will further our understanding of singular geometric spaces and their hidden symmetries.
Planned Impact
The immediate beneficiaries of our project will be the RAs whose research skills will be substantially enhanced by their participation in the project, leading to better employability. A wider group of academics who will benefit from our proposed interdisciplinary research project will be mathematicians working in algebraic geometry and representation theory. Further, there will be dividends also for string theorists, and researchers working in singularity theory, enumerative combinatorics, and Lie theory.
One key strand of the proposed research programme is our collaboration with two researchers from the University of Nairobi. Our impact here will consist of contributing to mathematics research development in a disadvantaged area of Africa that currently has very limited mathematics research output or contact with contemporary research.
No direct commercial exploitation is envisaged for our work in fundamental mathematics, nor do we expect any direct economic impact. However, strengthening mathematics within International Development is likely to have long-term benefits to the participating regions and countries in terms of capacity building, leading to better training of mathematical scientists and therefore a positive impact on the economy at large.
One key strand of the proposed research programme is our collaboration with two researchers from the University of Nairobi. Our impact here will consist of contributing to mathematics research development in a disadvantaged area of Africa that currently has very limited mathematics research output or contact with contemporary research.
No direct commercial exploitation is envisaged for our work in fundamental mathematics, nor do we expect any direct economic impact. However, strengthening mathematics within International Development is likely to have long-term benefits to the participating regions and countries in terms of capacity building, leading to better training of mathematical scientists and therefore a positive impact on the economy at large.
Organisations
Publications
Clemens Koppensteiner
(2020)
Traces, Schubert calculus, and Hochschild cohomology of category O
Craw A
(2021)
Punctual Hilbert schemes for Kleinian singularities as quiver varieties
in Algebraic Geometry
Craw A
(2021)
Quot Schemes for Kleinian Orbifolds
in Symmetry, Integrability and Geometry: Methods and Applications
Craw Alastair
(2019)
Punctual Hilbert schemes for Kleinian singularities as quiver varieties
in arXiv e-prints
DAVISON B
(2019)
Enumerating coloured partitions in 2 and 3 dimensions
in Mathematical Proceedings of the Cambridge Philosophical Society
Gyenge
(2019)
Young walls and equivariant Hilbert schemes of points in type D
in arXiv e-prints
Gyenge Á
(2020)
Canonical spectral coordinates for the Calogero-Moser space associated with the cyclic quiver
in Journal of Nonlinear Mathematical Physics
Hauenstein J
(2022)
On the Equations Defining Some Hilbert Schemes
in Vietnam Journal of Mathematics
Hauenstein J
(2021)
On the equations defining some Hilbert schemes
| Title | Applied Pure: an exhibition and concert performance by Kate Beaugie (artwork), Medea Bindewald (harpsichord), Balazs Szendroi (curator) |
| Description | Applied Pure was a unique collaboration between light sculptor Katharine Beaugié and international concert harpsichordist Medea Bindewald, combining the patterns made by water and light with the sound of harpsichord music in a mathematical environment. Curated by Balazs Szendroi. Concert: 18 November, 6.45pm; Exhibition: 18th November - 6th December 2019. |
| Type Of Art | Artistic/Creative Exhibition |
| Year Produced | 2019 |
| Impact | The live performance was attended by over 200 people in the Mathematical Institute, University of Oxford, many of whom commented very positively on the experience of an interaction between artists and science. |
| URL | https://www.youtube.com/watch?v=EyhGWcC0y3c |
| Description | The first set of objectives set in the grant application have been achieved during the first year of the award: the Hilbert scheme of points on a Kleinian singularity can indeed be described by a so-called quiver variety. This allows the methods developed in the extensive literature on quiver varieties to be brought to bear on problems around Hilbert schemes, which was one of the underlying motivations of the present project. On the other hand, it also led to the definition of some new spaces attached to these singularities, the topology of which is now under active investigation. All of these will enhance the interaction between the representation theory of affine Lie algebras and singularities, the core underlying idea behind the project. During the second year of the award, these investigations continued in different directions. Work of the PI and collaborators has studied the new spaces arising in earlier work further, finding formulae for their topological invariants fitting into the framework of representation theory. There was also progress in understanding the three-dimensional case (publication in an international journal). The two PDRA's on the project have jointly initiated a project on categorification, lifting a vertex algebra symmetry appearing on homologies of Hilbert schemes to the categorical level in an attractive way. During the third year of the grant, tjhe project on categorification was completed, with the release of a substantial paper which brings together deep results on category theory with ideas in representation theory and geometry. Further work was performed also on the theory of Hilbert schemes in the three-dimensional case, leading to a publication which relied on theoretical ideas as well as computer-based calculations. Further work was performed on the representation-theoretic side of the project, with a preprint in preparation getting us closer to a conjecture about the kind of Lie algebra that could act on the cohomology of our singular moduli spaces. Proving this conjecture appears quite challenging at the moment. Following the departure of a postdoctoral assistant, Clemens Koppensteiner, on the grant, another, Wicher Malten, was recruited who used his time to complete his publications on Slodowy slices, geometries closely related to certain Hilbert schemes of points on surfaces. |
| Exploitation Route | At the moment, the output is mainly on the theoretical side. |
| Sectors | Other |
| Description | Capacity building in Africa via technology-driven research in algebraic and arithmetic geometry |
| Amount | £170,892 (GBP) |
| Funding ID | EP/T001968/1 |
| Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
| Sector | Public |
| Country | United Kingdom |
| Start | 03/2020 |
| End | 03/2022 |
| Description | Marie Sklodowska-Curie Individual Fellowship |
| Amount | € 151,850 (EUR) |
| Funding ID | ModSingLDT |
| Organisation | European Commission H2020 |
| Sector | Public |
| Country | Belgium |
| Start | 01/2021 |
| End | 12/2022 |