Robert Bryant, Phillip Griffiths Professor of Mathematics and Professor of Computer Science
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.
Typical Courses Taught:
- MATH 123S, GEOMETRY
- MATH 267, DIFFERENTIAL GEOMETRY
- MATH 268, Topics in DIFFERENTIAL GEOMETRY: Symplectic Geometry
- The first third of the course is devoted to
'classical' symplectic geometry: Lagrangians,
Legendre transformations, Hamiltonians, symplectic
manifolds and the Darboux-Weinstein theorem,
symmetries and conservation laws and the Arnold-
Liouville theorem, momentum mappings,
reduction, and convexity.
The second third of the course is devoted to
developing elliptic methods: pseudo-holomorphic
curves, Gromov compactness and moduli,
applications to packing and (non)-squeezing
The final third covers related topics and recent
developments, such as relations with toric
varieties, representation theory, or other topics
that depend on the interests of the class.
- Office Hours:
- Tuesdays and Thursdays, 10:000-11:30AM, and by appointment (currently, all office hours are virtual, reached by Zoom: https://duke.zoom.us/j/97527269166; note that there is a waiting room enabled)
|Ph.D.||University of North Carolina - Chapel Hill||1979|
|B.S.||North Carolina State University||1974|
- Research Interests: Nonlinear Partial Differential Equations and Differential Geometry
I'm interested in the geometry of partial differential equations (as always), but, more specifically, I have been thinking about conservation laws for PDE, Finsler geometry, calibrations, holonomy, and trying to learn more about Seiberg-Witten invariants and symplectic geometry.
- Areas of Interest:
- exterior differential systems
- Almost complex manifolds • calibrations • CR hypersurfaces • exterior differential systems • Finsler • Geometry, Differential • Global differential geometry • solitons
- Departmental Ombud
- Current Ph.D. Students
- Recent Publications
- Bryant, RL; Foulon, P; Ivanov, SV; Matveev, VS; Ziller, W, Geodesic behavior for Finsler metrics of constant positive flag curvature on S2,
Journal of Differential Geometry, vol. 117 no. 1
pp. 1-22 [doi] [abs]
- Acharya, BS; Bryant, RL; Salamon, S, A circle quotient of a G2 cone,
Differential Geometry and Its Applications, vol. 73
(December, 2020) [doi] [abs]
- Bryant, RL; Clelland, JN, Flat metrics with a prescribed derived coframing,
Symmetry, Integrability and Geometry: Methods and Applications, vol. 16
(January, 2020) [doi] [abs]
- Bryant, RL; Eastwood, MG; Gover, AR; Neusser, K, Some differential complexes within and beyond parabolic geometry,
Advanced Studies in Pure Mathematics, vol. 82 no. Differential Geometry and Tanaka Theory
pp. 13-40, Mathematical Society of Japan [abs]
- Bryant, R; Buckmire, R; Khadjavi, L; Lind, D, The origins of spectra, an organization for LGBT mathematicians,
Notices of the American Mathematical Society, vol. 66 no. 6
pp. 875-882 [doi]
- Selected Invited Lectures
- Parking Cars, Rolling Balls, and Falling Cats: The Concept of Holonomy, May 22, 2013, Simons Foundation in New York City
- Geometry Old and New: From Euclid to String Theory, November 1, 2012, Humboldt State University, Arcata, CA [title.]
- Applications of Cartan's generalization of Lie's Third Theorem, June 13, 2011, Centre de Rescherches Mathématiques, Montreal, CA [various]
- The idea of Holonomy, October 14, 2010, MAA Carriage House, Washington, DC [the-idea-of-holonomy]
- On the life and work of S.-S. Chern, August 19, 2010, ICM in Hyderabad, India [Ch]
- Recent Grant Support
- Special Holonomy In Geometry, Analysis and Physics, Simons Foundation, 2020/07-2023/06.
- Special Holonomy In Geometry, Analysis and Physics, Simons Foundation, 2016/07-2020/06.
- Conferences Organized
- Annual East Coast Geometry Festival, Member of Organizing Board, 1 January 1990
- Transforming Post-Secondary Education in Mathematics (Duke Regional Meeting), Organizer, December 05, 2015 - December 06, 2015
- Member, Organizing Committee : Bay Area Differential Geometry Seminar. 2008 - 2013, Member, Organizing Committee : Bay Area Differential Geometry Seminar, 2008 - 2013
- Co-Organizer : Chern Centennial Conference. October 24, 2011 - November 4, 2011, Co-Organizer : Chern Centennial Conference, October 24, 2011 - November 04, 2011
- AIM Workshop on Calibrations, Conference Organizer, June 26, 2006 - June 30, 2006
- Singularities in Analysis, Geometry, and Topology: A Conference Honoring the Retirements of F. Reese Harvey and John Polking, Scientific Advisor on Organizing Committee, November 11, 2005 - November 13, 2005
- International Symposium on Finsler Geometry, Member of organizing committee, August 10, 2004 - August 14, 2004
- Oberwolfach Bi-Annual Differential Geometry in the Large, co-Organizer, June 15, 2001 - May 31, 2004
- MSRI Year-Long Program in Geometry (2003-2004), co-Organizer, 2001-04