# 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., 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.
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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.
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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.
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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.
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Bryant, R. L. “Some remarks on the geometry of austere manifolds.” *Boletim Da Sociedade Brasileira De Matemática*, vol. 21, no. 2, Sept. 1991, pp. 133–57. *Scopus*, doi:10.1007/BF01237361.
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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.
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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.
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Bryant, R. L. “Metrics with exceptional holonomy.” *Ann. of Math. (2)*, vol. 126, no. 3, 1987, pp. 525–76.

Bryant, Robert, and Phillip Griffiths. “Reduction for Constrained Variational Problems and � κ 2 2 ds.” *American Journal of Mathematics*, vol. 108, no. 3, JSTOR, June 1986, pp. 525–525. *Crossref*, doi:10.2307/2374654.
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Bryant, R. L. “Lie groups and twlstor spaces.” *Duke Mathematical Journal*, vol. 52, no. 1, Jan. 1985, pp. 223–61. *Scopus*, doi:10.1215/S0012-7094-85-05213-5.
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