Robert Bryant

Robert Bryant
  • Philip Griffiths Professor of Mathematics
  • Professor in the Department 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
differential geometry, exterior differential systems, complex geometry
Computational Mathematics
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
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

Bryant, RL. "Calibrated embeddings in the special Lagrangian and coassociative cases." Annals of Global Analysis and Geometry 18.3-4 (2000): 405-435. Open Access Copy

Bryant, RL. "Harmonic morphisms with fibers of dimension one." Communications in Analysis and Geometry 8.2 (2000): 219-265. (Academic Article) Open Access Copy

Bryant, RL. "Recent advances in the theory of holonomy." Asterisque 266.5 (2000): 351-374. Open Access Copy

Bryant, RL. "Calibrated Embeddings in the Special Lagrangian and Coassociative Cases." Annals of Global Analysis and Geometry 18.3-4 (2000): 405-435. Open Access Copy

Sharpe, E, and Bryant, RL. "D-branes and Spin^c-structures." Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics 450.4 (1999): 353-357. Open Access Copy

Bryant, RL. "Some examples of special Lagrangian tori." Adv. Theor. Math. Phys. 3.1 (1999): 83-90. Open Access Copy

Bryant, RL. "Projectively flat Finsler 2-spheres of constant curvature." Selecta Math. (N.S.) 3.2 (1997): 161-203. Open Access Copy

Bryant, RL, and Griffiths, PA. "Characteristic cohomology of differential systems. I. General theory." Journal of the American Mathematical Society 8.3 (September 1, 1995): 507-507. Full Text

Bryant, RL, and Griffiths, PA. "Characteristic cohomology of differential systems, II: Conservation laws for a class of parabolic equations." Duke Math. Journal 78.3 (1995): 531-676.

Griffiths, PA, Hsu, L, and Bryant, RL. "Hyperbolic exterior differential systems and their conservation laws, Part I." Selecta Math. (N.S.) 1.1 (1995): 21-112.