MATHEMATICS RESEARCH AREAS

Differential geometry is the study of geometric structures on manifolds (generalised surfaces). Doing calculus on a manifold allows one to study the curvature (bending) and topology (rubber sheet geometry) of the manifold. As with calculus its applications are widespread and include the field theories of physics, information loss in parametric statistics, and algorithms for computer graphics.

JOHN BURNS is mainly interested in manifolds that are highly symmetric, in the sense that they have a large (Lie) group of distance preserving transformations (Homogeneous spaces). This symmetry can be captured algebraically by a Lie algebra, and the algebra used to study the geometry (e.g. shortest paths) and topology (e.g. the number of holes ) of the manifold. Sometimes computer programmes ( representation theory) can be used to make the necessary algebraic calculation. Recent work has explored applications to complex geometry. Because of the connections with topology some of John's work is joint with GRAHAM ELLIS.

IRISH GEOMETRY DAY