# Robert Bryant

• Phillip Griffiths Professor of Mathematics
External address: 103 Physics Bldg, West Campus, Durham, NC 27708
Internal office address: Box 90320, Durham, NC 27708-0320
Phone: (919) 660-2817
Office Hours:

Tuesdays and Thursdays, 10:30-12:00PM, and by appointment

### Research Areas and Keywords

##### Algebra & Combinatorics
integrability, symplectic geometry
##### Analysis
differential geometry, exterior differential systems, complex geometry
integrability
##### Geometry: Differential & Algebraic
differential geometry, holonomy, exterior differential systems, integrability, curvature, Lie groups, symplectic geometry, complex geometry, homology
##### Mathematical Physics
holonomy, exterior differential systems, symplectic geometry
##### PDE & Dynamical Systems
differential geometry, holonomy, exterior differential systems, integrability, symplectic geometry
##### Topology
curvature, Lie groups, homology

My research concerns problems in the geometric theory of partial differential equations.  More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy.

Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems.  Because of their built-in invariance properties, these systems often have special features that make them difficult to treat by the standard tools of analysis, and so my approach uses ideas and techniques from the theory of exterior differential systems, a collection of tools for analyzing such PDE systems that treats them in a coordinate-free way, focusing instead on their properties that are invariant under diffeomorphism or other transformations.

I’m particularly interested in geometric structures constrained by natural conditions, such as Riemannian manifolds whose curvature tensor satisfies some identity or that supports some additional geometric structure, such as a parallel differential form or other geometric structures that satisfy some partial integrability conditions and in constructing examples of such geometric structures, such as Finsler metrics with constant flag curvature.

I am also the Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics, and a considerable focus of my research and that of my students is directed towards problems in this area.

##### Education & Training
• Ph.D., University of North Carolina at Chapel Hill 1979

• B.A., North Carolina State University 1974

Griffiths, PA, Hsu, L, and Bryant, RL. "Hyperbolic exterior differential systems and their conservation laws, Part II." Selecta Math. (N.S.) 1.2 (1995): 265-323.

Bryant, RL, and Hsu, L. "Rigidity of integral curves of rank 2 distributions." Inventiones Mathematicae 114.1 (1993): 435-461. Full Text

Bryant, RL. "Some remarks on the geometry of austere manifolds." Bol. Soc. Brasil. Mat. (N.S.) 21.2 (1991): 133-157.

Salamon, S, and Bryant, RL. "On the construction of some complete metrics with exceptional holonomy." Duke Math. J. 58.3 (1989): 829-850.

Harvey, FR, and Bryant, RL. "Submanifolds in hyper-Kähler geometry." J. Amer. Math. Soc. 2.1 (1989): 1-31.

Bryant, RL. "Metrics with exceptional holonomy." Ann. of Math. (2) 126.3 (1987): 525-576.

Griffiths, PA, and Bryant, RL. "Reduction for constrained variational problems and $\int{1\over 2}k\sp 2\,ds$." Amer. J. Math. 108.3 (1986): 525-570.

Bryant, RL. "Lie groups and twistor spaces." Duke Math. J. 52.1 (1985): 223-261.

Bryant, RL. "Minimal surfaces of constant curvature in S^n." Trans. Amer. Math. Soc. 290.1 (1985): 259-271.

Bryant, RL. "A duality theorem for Willmore surfaces." J. Differential Geom. 20.1 (1984): 23-53.