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Robert L. Bryant studies differential geometry and geometric partial differential equations. His Duke research statement describes the use of geometric structures to understand differential equations.^[ref]

His An Introduction to Lie Groups and Symplectic Geometry by Robert L. Bryant develops symplectic reduction and its compatibility with the geometry of Kähler manifolds. In the scalar-circle example, reduction of a complex vector space at a positive level yields Complex projective space. The construction explains how imposing a constraint and identifying points along a symmetry can produce a new geometric space.^[ref]