# 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

##### Analysis

##### Computational Mathematics

##### Geometry: Differential & Algebraic

##### Mathematical Physics

##### PDE & Dynamical Systems

##### Topology

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

(90-0718) Mathematical Sciences: Differential Geometry awarded by National Science Foundation (Principal Investigator). 1989 to 1992

(89-0300) 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).* Ed. M Luksic, C Martin, and W Shadwick. Providence, RI: American Mathematical Society, 1987. 65-76. (Chapter)

Bryant, R. "A survey of Riemannian metrics with special holonomy groups." *Proceedings of the International Congress of Mathematicians. Vol. 1, 2. (Berkeley, Calif., 1986).* Ed. A Gleason. Providence, RI: American Mathematical Society, 1987. 505-514. (Chapter)

Bryant, R. "Minimal Lagrangian submanifolds of Kähler-Einstein manifolds." *Differential geometry and differential equations (Shanghai, 1985).* Ed. C Gu, M Berger, and RL Bryant. Berlin, Germany: Springer-Verlag, 1987. 1-12. (Chapter)

Bryant, R. "Metrics with holonomy G2 or Spin(7)." *Workshop Bonn 1984 (Bonn, 1984).* Ed. F Hirzebruch, J Schwermer, and S Suter. Berlin, Germany: Springer, 1985. 269-277. (Chapter)

Bryant, R, and Griffiths, PA. "Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle." *Arithmetic and geometry, Vol. II.* Ed. M Artin and J Tate. Boston, MA: Birkhäuser Boston, 1983. 77-102. (Chapter)

Bryant, R, Chern, SS, and Griffiths, PA. "Exterior Differential Systems." *Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations (Beijing, 1980).* Ed. SS Chern and WT Wu. Beijing, PRC; New York, NY: Science Press; Gordon & Breach Science Publishers, 1982. 219-338. (Chapter)

Bryant, RL. "Some remarks on G_2-structures." Open Access Copy

## Pages

Bryant, RL. "On the geometry of almost complex 6-manifolds." *The Asian Journal of Mathematics* 10.3 (September 2006): 561-606.

BRYANT, RL. "Conformal geometry and 3-plane fields on 6-manifolds." *RIMS Kokyuroku* 1502 (Developments of Cartan Geometry and Related Mathematical Problems) (July 2006): 1-15.
Open Access Copy

Bryant, R, and Freed, D. "Shiing-Shen Chern - Obituary." *PHYSICS TODAY* 59.1 (January 2006): 70-72.
Full Text

Bryant, RL. "SO(n)-invariant special Lagrangian submanifolds of C^{n+1} with fixed loci." *Chinese Annals of Mathematics, Series B* 27.1 (January 2006): 95-112.
Open Access Copy

Bryant, RL. "Real hypersurfaces in unimodular complex surfaces." (July 27, 2004). Open Access Copy

Bryant, R, Edelsbrunner, H, Koehl, P, and Levitt, M. "The area derivative of a space-filling diagram." *Discrete and Computanional Geometry* 32.3 (2004): 293-308.
Full Text

Bryant, RL. "Some remarks on Finsler manifolds with constant flag curvature." *Houston Journal of Mathematics* 28.2 (January 1, 2002): 221-262. (Academic Article)
Open Access Copy

Bryant, RL. "Bochner-kähler metrics." *Journal of the American Mathematical Society* 14.3 (July 1, 2001): 623-715.
Full Text Open Access Copy

Bryant, RL. "Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces." (March 5, 2001). Open Access Copy

Bryant, R. "Élie Cartan and geometric duality." *Journées Élie Cartan 1998 et 1999* 16 (2000): 5-20.
Open Access Copy