# 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

Special Holonomy In Geometry, Analysis and Physics awarded by Simons Foundation (Principal Investigator). 2016 to 2020

The Geometry of Partial Differential Equations and Applications awarded by National Science Foundation (Principal Investigator). 2013 to 2015

The Geometry of Partial Differential Equations awarded by National Science Foundation (Principal Investigator). 2006 to 2008

The Differential Geometry of Partial Differential Equations awarded by National Science Foundation (Principal Investigator). 2001 to 2006

The Differential Geometry of Partial Differential Equations awarded by National Science Foundation (Principal Investigator). 1998 to 2001

(97-0653) Mathematical Sciences: The Differential Geometry of Partial Differential Equations awarded by National Science Foundation (Principal Investigator). 1995 to 1998

(95-0297) The Differential Geometry of Partial Differential Equations awarded by National Science Foundation (Principal Investigator). 1995 to 1997

(96-0876) Mathematical Sciences: The Differential Geometry of Partial Differential Equations awarded by National Science Foundation (Principal Investigator). 1995 to 1997

(92-0349) The Differential Geometry of Partial Differential Equations awarded by National Science Foundation (Principal Investigator). 1992 to 1995

(92-0042) Mathematical Sciences: Differential Geometry awarded by National Science Foundation (Principal Investigator). 1989 to 1993

## Pages

Bryant, Robert L., et al. *Exterior Differential Systems*. Springer, 2011.

Gu, Chaohao, et al. *Differential Geometry and Differential Equations Proceedings of a Symposium, held in Shanghai, June 21 - July 6, 1985*. Springer, 2006.

Bao, D., et al., editors. *A Sampler of Riemann-Finsler Geometry*. Vol. 50, Cambridge University Press, 2004.

Bryant, Robert, et al. *Exterior Differential Systems and Euler-Lagrange Partial Differential Equations*. University of Chicago Press, 2003.
Open Access Copy

Bryant, R., et al., editors. *Integral Geometry*. Vol. 63, American Mathematical Society, 1987.

Bryant, R. “Geodesically reversible Finsler 2-spheres of constant curvature.” *Inspired by S. S. Chern---A Memorial Volume in Honor of a Great Mathematician*, edited by P. A. Griffiths, vol. 11, World Scientific Publishers, 2006, pp. 95–111.
Open Access Copy

Bryant, R. “Holonomy and Special Geometries.” *Dirac Operators: Yesterday and Today*, edited by J. P. Bourguinon et al., International Press, 2005, pp. 71–90.
Open Access Copy

Bryant, R. “Pseudo-Reimannian metrics with parallel spinor fields and vanishing Ricci tensor.” *Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999)*, edited by J. P. Bourguinon et al., vol. 4, Société Mathématique de France, 2000, pp. 53–94.
Open Access Copy

Bryant, R. “On extremals with prescribed Lagrangian densities.” *Manifolds and Geometry (Pisa, 1993)*, edited by Pd Bartolomeis et al., vol. 36, Cambridge University Press, 1996, pp. 86–111.
Open Access Copy

Bryant, R. “Classical, exceptional, and exotic holonomies: a status report.” *Actes de La Table Ronde de Géométrie Différentielle*, edited by A. Besse, vol. 1, Société Mathématique de France, 1996, pp. 93–165.

Bryant, R., and R. B. Gardner. “Control Structures.” *Geometry in Nonlinear Control and Differential Inclusions (Warsaw, 1993)*, edited by B. Jakubczyk et al., vol. 12, Polish Academy of Sciences, 1995, pp. 111–21.

Bryant, R. “An introduction to Lie groups and symplectic geometry.” *Geometry and Quantum Field Theory (Park City, UT, 1991)*, edited by D. Freed and K. Uhlenbeck, vol. 1, American Mathematical Society, 1995, pp. 5–181.
Open Access Copy

Bryant, R. “Two exotic holonomies in dimension four, path geometries, and twistor theory.” *Complex Geometry and Lie Theory (Sundance, UT, 1989)*, edited by J. Carlson et al., vol. 53, American Mathematical Society, 1991, pp. 33–88.

Bryant, R. “Surfaces in conformal geometry.” *The Mathematical Heritage of Hermann Weyl (Durham, NC, 1987)*, edited by R. O. Wells, vol. 48, American Mathematical Society, 1988, pp. 227–40.

Bryant, R. “Surfaces of mean curvature one in hyperbolic space.” *Théorie Des Variétés Minimales et Applications (Palaiseau, 1983–1984)*, vol. 154–155, Société Mathématique de France, 1988, pp. 321–47.

## Pages

Bryant, R. L., et al. “Some differential complexes within and beyond parabolic geometry.” *Advanced Studies in Pure Mathematics*, vol. 82, no. Differential Geometry and Tanaka Theory, Mathematical Society of Japan, Nov. 2019, pp. 13–40.
Open Access Copy

Bryant, R., et al. “The origins of spectra, an organization for LGBT mathematicians.” *Notices of the American Mathematical Society*, vol. 66, no. 6, June 2019, pp. 875–82. *Scopus*, doi:10.1090/noti1890.
Full Text Open Access Copy

Bryant, R. L., et al. “On Finsler surfaces of constant flag curvature with a Killing field.” *Journal of Geometry and Physics*, vol. 116, June 2017, pp. 345–57. *Scopus*, doi:10.1016/j.geomphys.2017.02.012.
Full Text Open Access Copy

Bryant, R. L. *On the convex Pfaff-Darboux Theorem of Ekeland and Nirenberg*. Dec. 2015.
Open Access Copy

Bryant, R. L. *On the conformal volume of 2-tori*. July 2015.
Open Access Copy

Bryant, R. L. *S.-S. Chern's study of almost-complex structures on the six-sphere*. May 2014.
Open Access Copy

Bryant, R. L. *Notes on exterior differential systems*. May 2014.
Open Access Copy

Bryant, R., and F. Xu. *Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior*. Jan. 2011.
Open Access Copy

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.
Full Text

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

## Pages

Bryant, Robert L. “Second order families of special Lagrangian 3-folds.” *Perspectives in Riemannian Geometry*, edited by V. Apostolov et al., vol. 40, AMER MATHEMATICAL SOC, 2006, pp. 63–98.
Open Access Copy

Bryant, R. L. “Geometry of manifolds with special holonomy: "100 years of holonomy".” *150 Years of Mathematics at Washington University in St. Louis*, edited by G. R. Jensen and S. G. Krantz, vol. 395, AMER MATHEMATICAL SOC, 2006, pp. 29–38.

BRYANT, R., et al. “TOWARD A GEOMETRY OF DIFFERENTIAL-EQUATIONS.” *Geometry, Topology & Physics*, edited by S. T. Yau, vol. 4, INTERNATIONAL PRESS INC BOSTON, 1995, pp. 1–76.

Bryant, Robert L. *Complex analysis and a class of Weingarten surfaces*.
Open Access Copy

Bryant, Robert, and Jeanne Nielsen Clelland. *Flat metrics with a prescribed derived coframing*.

Bryant, Robert. *Notes on Projective, Contact, and Null Curves*.

Bryant, Robert, et al. *A circle quotient of a G_2 cone*.

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The Simons Foundation is pleased to announce the establishment of the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics, directed by Robert Bryant at Duke University.... read more »