Moduli spaces attached to singular surfaces and representation theory

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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.