- Phillip Griffiths Professor of Mathematics
- Professor in the Department of Mathematics
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
Geometry: Differential & Algebraic
differential geometry, holonomy, exterior differential systems, integrability, curvature, Lie groups, symplectic geometry, complex geometry, homology
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.
Bryant, R. L. “Projectively flat finsler 2-spheres of constant curvature.” Selecta Mathematica, New Series, vol. 3, no. 2, Jan. 1997, pp. 161–203. Scopus, doi:10.1007/s000290050009. Full Text Open Access Copy
Bryant, Robert L., and Phillip A. Griffiths. “Characteristic cohomology of differential systems. I. General theory.” Journal of the American Mathematical Society, vol. 8, no. 3, American Mathematical Society (AMS), Sept. 1995, pp. 507–507. Crossref, doi:10.1090/s0894-0347-1995-1311820-x. Full Text
Bryant, R., et al. “Hyperbolic exterior differential systems and their conservation laws, part II.” Selecta Mathematica, New Series, vol. 1, no. 2, Sept. 1995, pp. 265–323. Scopus, doi:10.1007/BF01671567. Full Text
Bryant, R., et al. “Hyperbolic exterior differential systems and their conservation laws, part I.” Selecta Mathematica, New Series, vol. 1, no. 1, Mar. 1995, pp. 21–112. Scopus, doi:10.1007/BF01614073. Full Text
Bryant, R. L., and P. A. Griffiths. “Characteristic cohomology of differential systems II: Conservation laws for a class of parabolic equations.” Duke Mathematical Journal, vol. 78, no. 3, Jan. 1995, pp. 531–676. Scopus, doi:10.1215/S0012-7094-95-07824-7. Full Text
Bryant, R. L., and L. Hsu. “Rigidity of integral curves of rank 2 distributions.” Inventiones Mathematicae, vol. 114, no. 1, Dec. 1993, pp. 435–61. Scopus, doi:10.1007/BF01232676. Full Text
Bryant, R. L., and S. M. Salamon. “On the construction of some complete metrics with exceptional holonomy.” Duke Mathematical Journal, vol. 58, no. 3, Jan. 1989, pp. 829–50. Scopus, doi:10.1215/S0012-7094-89-05839-0. Full Text
Bryant, R., and R. Harvey. “Submanifolds in hyper-Kähler geometry.” Journal of the American Mathematical Society, vol. 2, no. 1, Jan. 1989, pp. 1–31. Scopus, doi:10.1090/S0894-0347-1989-0953169-8. Full Text
Bryant, R. L. “Metrics with exceptional holonomy.” Ann. of Math. (2), vol. 126, no. 3, 1987, pp. 525–76.
Some geometric constructions of holonomy plane fields and their analysis. IHP Workshop "Equivalence, invariants, and symmetries of vector distributions and related structures: from Cartan to Tanaka and beyond". Institut Henri Poincaré, Paris France....