EPSRC Reference: 
EP/R045038/1 
Title: 
Moduli spaces attached to singular surfaces and representation theory 
Principal Investigator: 
Szendroi, Professor B 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematical Institute 
Organisation: 
University of Oxford 
Scheme: 
Standard Research 
Starts: 
01 January 2019 
Ends: 
31 December 2021 
Value (£): 
518,956

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
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 socalled ADE 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 3dimensional (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 ngon (type A or cyclic); a prism based on a regular ngon (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 ADE 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 ADE 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, higherdimensional singular spaces.
Success in this project will further our understanding of singular geometric spaces and their hidden symmetries.

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Organisation Website: 
http://www.ox.ac.uk 