Robert Bryant, Phillip Griffiths Professor and Chair
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 builtin 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 coordinatefree 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.  Contact Info:
Typical Courses Taught:
 MATH 123S, GEOMETRY
Synopsis

 MATH 267, DIFFERENTIAL GEOMETRY

 MATH 268, Topics in DIFFERENTIAL GEOMETRY: Symplectic Geometry
Synopsis

 The first third of the course is devoted to
'classical' symplectic geometry: Lagrangians,
Legendre transformations, Hamiltonians, symplectic
manifolds and the DarbouxWeinstein 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: pseudoholomorphic
curves, Gromov compactness and moduli,
applications to packing and (non)squeezing
theorems, etc.
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:
 Wednesdays, 10:00AM12noon ET, and by appointment (please write to chair@math.duke.edu for appointments) Currently, all office hours are virtual, reached by Zoom: https://duke.zoom.us/j/95511599741; note that there is a waiting room enabled.
 Education:
Ph.D.  University of North Carolina  Chapel Hill  1979 
B.S.  North Carolina State University  1974 
 Specialties:

Geometry
Mathematical Physics
 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 SeibergWitten invariants and symplectic geometry.
 Areas of Interest:
exterior differential systems differential geometry algebraic geometry Finsler geometry
 Keywords:
Almost complex manifolds • calibrations • CR hypersurfaces • exterior differential systems • Finsler • Geometry, Differential • Global differential geometry • solitons
 Duties:
 Departmental Ombud
 Curriculum Vitae
 Current Ph.D. Students
(Former Students)
 Recent Publications
(More Publications)
(search)
 Bryant, RL, S.S. Chern's study of almostcomplex structures on the sixsphere, edited by Bryant, R; Cheng, SY; Griffiths, P; Ma, X; Ni, L; Wallach, N,
International Journal of Mathematics, vol. 32 no. 12
(November, 2021), World Scientific Publishing [arXiv:1405.3405] [abs]
 Bryant, R; Foulon, P; Ivanov, S; Matveev, VS; Ziller, W, Geodesic behavior for Finsler metrics of constant positive flag curvature on S^2,
Journal of Differential Geometry, vol. 117 no. 1
(January, 2021),
pp. 122 [doi] [abs]
 Bryant, RL, Notes on spinors in low dimension
(November, 2020) [abs]
 Bryant, R; Clelland, JN, Flat metrics with a prescribed derived coframing, 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
(November, 2019),
pp. 1340, Mathematical Society of Japan [abs]
 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 [theideaofholonomy]
 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, 2020/072023/06.
 Special Holonomy In Geometry, Analysis and Physics, Simons Foundation, 347349, 2016/072020/06.
 Conferences Organized
 Annual East Coast Geometry Festival, Member of Organizing Board, 1 January 1990
 Transforming PostSecondary 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
 CoOrganizer : Chern Centennial Conference. October 24, 2011  November 4, 2011, CoOrganizer : 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 BiAnnual Differential Geometry in the Large, coOrganizer, June 15, 2001  May 31, 2004
 MSRI YearLong Program in Geometry (20032004), coOrganizer, 200104
