Department of Mathematics
 Search | Help | Login | printable version

Math @ Duke



Math HOME

.......................

Faculty

.......................

Robert Bryant, Phillip Griffiths Distinguished Professor

Robert Bryant

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.

Contact Info:
Office Location:  120 Science Drive, Durham, NC 27708
Email Address: send me a message
Web Page:  https://fds.duke.edu/db/aas/math/faculty/bryant

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 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 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, 10:00AM-12noon ET, and by appointment (please write to chair@math.duke.edu for appointments).
Education:

Ph.D.University of North Carolina, Chapel Hill1979
B.S.North Carolina State University1974
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 Seiberg-Witten 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)

    1. Buckmire, R; Beeton, B; Bryant, R; Gouvêa, FQ; Phillips, AV; Sullivan, D; Wolf, M, Michael Spivak: A Memorial, Notices of the American Mathematical Society, vol. 71 no. 6 (June, 2024), pp. 786-795 [doi]
    2. Bryant, RL, Hessianizability of surface metrics (May, Preprint, 2024)
    3. Bryant, R; Florit, L; Ziller, W, Curvature homogeneous hypersurfaces in space forms (April, Preprint, 2024)
    4. Bryant, R, The generality of closed G_2 solitons, edited by Cheng, S-Y; Lima-Filho, P; Yau, SS-T; Yau, S-T, Pure and Applied Mathematics Quarterly, vol. 19 no. 6 (January, 2024), pp. 2827-2840, International Press  [abs]
    5. Bryant, R; Cheeger, J; Lima-Filho, P; Rosenberg, J; White, B, The mathematical work of H. Blaine Lawson, Jr., Pure and Applied Mathematics Quarterly, vol. 19 no. 6 (January, 2024), pp. 2627-2662, International Press
    Selected Invited Lectures

    1. Parking Cars, Rolling Balls, and Falling Cats: The Concept of Holonomy, May 22, 2013, Simons Foundation in New York City    
    2. Geometry Old and New: From Euclid to String Theory, November 1, 2012, Humboldt State University, Arcata, CA [title.]    
    3. Applications of Cartan's generalization of Lie's Third Theorem, June 13, 2011, Centre de Rescherches Mathématiques, Montreal, CA [various]    
    4. The idea of Holonomy, October 14, 2010, MAA Carriage House, Washington, DC [the-idea-of-holonomy]    
    5. 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, 2020/07-2024/06.      
    Conferences Organized

     

    dept@math.duke.edu
    ph: 919.660.2800
    fax: 919.660.2821

    		Mathematics Department
    Duke University, Box 90320
    Durham, NC 27708-0320
    Mathematics Home | Arts & Sciences | Duke University