# Robert Bryant

- Philip 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

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, RL, Chern, SS, Gardner, RB, Goldschmidt, HL, and Griffiths, PA.* Exterior Differential Systems.* Springer, December 14, 2011.

Gu, C, Berger, M, and Bryant, RL.* Differential Geometry and Differential Equations Proceedings of a Symposium, held in Shanghai, June 21 - July 6, 1985.* Springer, November 15, 2006.

* A Sampler of Riemann-Finsler Geometry.* Ed. D Bao, RL Bryant, S-S Chern, and Z Shen. Cambridge University Press, November 1, 2004. (Edited Book)

* Integral Geometry.* Ed. R Bryant, V Guillemin, S Helgason, and RO Wells. Providence, RI: American Mathematical Society, 1987. (Edited Book)

Bryant, RL, Griffiths, PA, and Grossman, DA.* Exterior Differential Systems and Euler-Lagrange Partial Differential
Equations.*.
Open Access Copy

Bryant, R. "Holonomy and Special Geometries." *Dirac Operators: Yesterday and Today.* Ed. JP Bourguinon, T Branson, A Chamseddine, O Hijazi, and R Stanton. Somerville, MA: International Press, 2005. 71-90. (Chapter)
Open Access Copy

Bryant, R. "Classical, exceptional, and exotic holonomies: a status report." *Actes de la Table Ronde de Géométrie Différentielle.* Ed. A Besse. Paris: Société Mathématique de France, 1996. 93-165. (Chapter)

Bryant, R. "An introduction to Lie groups and symplectic geometry." *Geometry and quantum field theory (Park City, UT, 1991).* Ed. D Freed and K Uhlenbeck. Providence, RI: American Mathematical Society, 1995. 5-181. (Chapter)
Open Access Copy

Bryant, R, and Gardner, RB. "Control Structures." *Geometry in nonlinear control and differential inclusions (Warsaw, 1993).* Ed. B Jakubczyk, W Respondek, and T Rzezuchowski. Warsaw, Poland: Polish Academy of Sciences, 1995. 111-121. (Chapter)

Bryant, R. "Two exotic holonomies in dimension four, path geometries, and twistor theory." *Complex geometry and Lie theory (Sundance, UT, 1989).* Ed. J Carlson, H Clemens, and D Morrison. Providence, RI: American Mathematical Society, 1991. 33-88. (Chapter)

Bryant, R. "Surfaces in conformal geometry." *The mathematical heritage of Hermann Weyl (Durham, NC, 1987).* Ed. RO Wells. Providence, RI: American Mathematical Society, 1988. 227-240. (Chapter)

Bryant, R. "Surfaces of mean curvature one in hyperbolic space." *Théorie des variétés minimales et applications (Palaiseau, 1983–1984).* Paris: Société Mathématique de France, 1988. 321-347. (Chapter)

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)

## Pages

Bryant, R, Huang, L, and Mo, X. "On Finsler surfaces of constant flag curvature with a Killing field." *Journal of Geometry and Physics* 116 (June 2017): 345-357.
Full Text Open Access Copy

Bryant, RL. "On the convex Pfaff-Darboux Theorem of Ekeland and Nirenberg." (December 22, 2015). Open Access Copy

Bryant, RL. "On the conformal volume of 2-tori." (July 6, 2015). Open Access Copy

Bryant, RL. "S.-S. Chern's study of almost-complex structures on the six-sphere." (May 14, 2014). Open Access Copy

Bryant, RL. "Notes on exterior differential systems." (May 13, 2014). Open Access Copy

Bryant, RL, Eastwood, MG, Gover, AR, and Neusser, K. "Some differential complexes within and beyond parabolic geometry." (March 19, 2012).

Bryant, R, and Xu, F. "Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior." (January 11, 2011). Open Access Copy

Bryant, RL. "Commentary." *Bulletin of the American Mathematical Society* 46.2 (2009): 177-178.
Full Text

Bryant, RL, Manno, G, and Matveev, VS. "A solution of a problem of Sophus Lie: Normal forms of two-dimensional metrics admitting two projective vector fields." *Mathematische Annalen* 340.2 (2008): 437-463.
Full Text Open Access Copy

Bryant, RL. "Gradient Kähler Ricci solitons." *Astérisque* 321 (2008): 51-97.
Open Access Copy

## Pages

Bryant, RL. "Geometry of manifolds with special holonomy: "100 years of holonomy"." 2006.

BRYANT, R, GRIFFITHS, P, and HSU, L. "Toward a geometry of differential equations." 1995.

Bryant, R, Foulon, P, Ivanov, S, Matveev, VS, and Ziller, W. "Geodesic behavior for Finsler metrics of constant positive flag curvature on S^2 (Submitted)."

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 »