Symmetry and Geometric Structures

This is a project in differential geometry, the study of shape using calculus and differential equations. The most familiar geometry is based on length but there are many other differential geometries besides, including conformal (based on angles) and projective (based on lines). Especially interesting are geometries with a high degree of symmetry, a critical attribute that pervades both mathematics and physics. Many features of the most symmetrical cases seem to persist in the arbitrarily curved setting. Motivated by a newly discovered combination of projective and symplectic geometry, this project will investigate how geometry and partial differential equations arise from symmetry.