- 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. Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces. Mar. 2001. Open Access Copy
Bryant, R. L. “Calibrated Embeddings in the Special Lagrangian and Coassociative Cases.” Annals of Global Analysis and Geometry, vol. 18, no. 3–4, Dec. 2000, pp. 405–35. Open Access Copy
Bryant, R. L. “Harmonic morphisms with fibers of dimension one.” Communications in Analysis and Geometry, vol. 8, no. 2, Jan. 2000, pp. 219–65. Scopus, doi:10.4310/CAG.2000.v8.n2.a1. Full Text 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
Bryant, R. L., and E. Sharpe. “D-branes and Spinc structures.” Physics Letters, Section B: Nuclear, Elementary Particle and High Energy Physics, vol. 450, no. 4, Jan. 1999, pp. 353–57. Scopus, doi:10.1016/S0370-2693(99)00161-6. Full Text Open Access Copy
Bryant, R. L. “Some examples of special Lagrangian Tori.” Advances in Theoretical and Mathematical Physics, vol. 3, no. 1, Jan. 1999, pp. 83–90. Scopus, doi:10.4310/ATMP.1999.v3.n1.a2. Full Text Open Access Copy
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
On families of special holonomy metrics defined by algebraic curvature conditions. Inaugural Conference of the Simons Collaboration "Special Holonomy in Geometry, Analysis, and Physics". Simons Center for Geometry and Physics. September 6, 2016
Integration techniques for exterior differential systems I & II. London Mathematical Society -- EPSRC Durham Symposium Geometric and Algebraic Aspects of Integrability. Durham University. July 27, 2016
Beyond Holonomy: Constrained Structure Equations. IHP conference "Relativity and Geometry" in memorial to André Lichnerowicz (in honor of his 100th Birthday). Institut Henri Poincaré, Paris, France. December 14, 2015