Conformal Differential Geometry

Differential geometry is the study of shape using calculus and differential equations. Conformal geometry allows a distortion of distance provided that angles are preserved: the Mercator projection of the Earth has this feature. Especially interesting are geometries with a high degree of symmetry, a critical feature that pervades both mathematics and physics. Symmetries will be used to study integral transforms and in constructing invariants. Many features of the most symmetrical cases seem to persist in the arbitrarily curved setting. The project will investigate how this happens and its consequences, especially in regard to geometrically defined systems of partial differential equations.