# 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.

### Selected Grants

Mathematical Sciences: Differential Geometry awarded by National Science Foundation (Principal Investigator). 1989 to 1993

Differential Geometry awarded by National Science Foundation (Principal Investigator). 1989 to 1990

## Pages

Bryant, R. “On notions of equivalence of variational problems with one independent variable.” *Differential Geometry: The Interface between Pure and Applied Mathematics (San Antonio, Tex., 1986)*, edited by M. Luksic et al., vol. 68, American Mathematical Society, 1987, pp. 65–76.

Bryant, R. “A survey of Riemannian metrics with special holonomy groups.” *Proceedings of the International Congress of Mathematicians. Vol. 1, 2. (Berkeley, Calif., 1986)*, edited by A. Gleason, American Mathematical Society, 1987, pp. 505–14.

Bryant, R. “Minimal Lagrangian submanifolds of Kähler-Einstein manifolds.” *Differential Geometry and Differential Equations (Shanghai, 1985)*, edited by C. Gu et al., vol. 1255, Springer-Verlag, 1987, pp. 1–12.

Bryant, R. “Metrics with holonomy G2 or Spin(7).” *Workshop Bonn 1984 (Bonn, 1984)*, edited by F. Hirzebruch et al., vol. 1111, Springer, 1985, pp. 269–77.

Bryant, R., and P. A. Griffiths. “Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle.” *Arithmetic and Geometry, Vol. II*, edited by M. Artin and J. Tate, vol. 36, Birkhäuser Boston, 1983, pp. 77–102.

Bryant, R., et al. “Exterior Differential Systems.” *Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations (Beijing, 1980)*, edited by S. S. Chern and W. T. Wu, vol. 1, Science Press; Gordon & Breach Science Publishers, 1982, pp. 219–338.

Bryant, Robert L. *Nonembedding and nonextension results in special holonomy*.
Open Access Copy

Bryant, Robert L. *Some remarks on G_2-structures*.
Open Access Copy

## Pages

Bryant, R. L. “Commentary.” *Bulletin of the American Mathematical Society*, vol. 46, no. 2, Apr. 2009, pp. 177–78. *Scopus*, doi:10.1090/S0273-0979-09-01248-8.
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Bryant, R. L., et al. “A solution of a problem of Sophus Lie: Normal forms of two-dimensional metrics admitting two projective vector fields.” *Mathematische Annalen*, vol. 340, no. 2, 2008, pp. 437–63. *Scival*, doi:10.1007/s00208-007-0158-3.
Full Text Open Access Copy

Bryant, R. L. “Gradient Kähler Ricci solitons.” *Astérisque*, no. 321, Soc. Math. France, 2008, pp. 51–97.
Open Access Copy

BRYANT, R. L. “Conformal geometry and 3-plane fields on 6-manifolds.” *Rims Kokyuroku*, vol. 1502 (Developments of Cartan Geometry and Related Mathematical Problems), Kyoto University, July 2006, pp. 1–15.
Open Access Copy

Bryant, R. L. “SO(n)-Invariant special Lagrangian submanifolds of ℂ n+1 with fixed loci.” *Chinese Annals of Mathematics. Series B*, vol. 27, no. 1, Jan. 2006, pp. 95–112. *Scopus*, doi:10.1007/s11401-005-0368-5.
Full Text Open Access Copy

Bryant, R. L. “On the geometry of almost complex 6-manifolds.” *Asian Journal of Mathematics*, vol. 10, no. 3, Jan. 2006, pp. 561–605. *Scopus*, doi:10.4310/AJM.2006.v10.n3.a4.
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Bryant, Robert, and Dan Freed. “Shiing-Shen Chern.” *Physics Today*, vol. 59, no. 1, AIP Publishing, Jan. 2006, pp. 70–72. *Crossref*, doi:10.1063/1.2180187.
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Bryant, R. L. *Real hypersurfaces in unimodular complex surfaces*. July 2004.
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Bryant, R., et al. “The area derivative of a space-filling diagram.” *Discrete and Computational Geometry*, vol. 32, no. 3, Jan. 2004, pp. 293–308. *Scopus*, doi:10.1007/s00454-004-1099-1.
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Bryant, R. L. “Some remarks on Finsler manifolds with constant flag curvature.” *Houston Journal of Mathematics*, vol. 28, no. 2, UNIV HOUSTON, Jan. 2002, pp. 221–62.
Open Access Copy