# Robert Bryant

- Phillip 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

##### Analysis

differential geometry, exterior differential systems, complex geometry

##### Computational Mathematics

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.

Bryant, R. I. “On Surfaces with Prescribed Shape Operator.” *Results in Mathematics*, vol. 40, no. 1–4, Oct. 2001, pp. 88–121. *Scopus*, doi:10.1007/BF03322701.
Full Text

Bryant, R. L. “Bochner-kähler metrics.” *Journal of the American Mathematical Society*, vol. 14, no. 3, July 2001, pp. 623–715. *Scopus*, doi:10.1090/S0894-0347-01-00366-6.
Full Text Open Access Copy

Bryant, R. L. *Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric
spaces*. Mar. 2001.
Open Access Copy

Bryant, R. “Recent advances in the theory of holonomy.” *Asterisque*, vol. 266, Dec. 2000, pp. 351–74.
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, Jan. 2000, pp. 405–35. *Scopus*, doi:10.1023/a:1006780703789.
Full Text 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, R. “Élie Cartan and geometric duality.” *Journées Élie Cartan 1998 Et 1999*, vol. 16, Institut Élie Cartan, 2000, pp. 5–20.
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