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Robert Bryant, Phillip Griffiths Professor and Chair

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
Office Phone:  +1 919 660 2800
Email Address: send me a message
Web Page:  https://fds.duke.edu/db/aas/math/faculty/bryant

Teaching (Fall 2024):  (typical courses)

  • MATH 323S.01, GEOMETRY Synopsis
    Physics 205, MW 10:05 AM-11:20 AM
  • MATH 620.01, SMOOTH MANIFOLDS Synopsis
    Physics 205, MW 08:30 AM-09:45 AM
  • MATH 723S.01, GEOMETRY Synopsis
    Physics 205, MW 10:05 AM-11:20 AM
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. 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]
    2. 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
    3. Bryant, R; Cheeger, J; Griffiths, P; Blum, L; Burns, D; Connes, A; Donnelly, H; Ebin, D; Guillemin, V; Palais, R; Rossi, H; Simons, J; Singer, E; Singer, N; Stanton, N; Sternberg, S, Isadore M. Singer (1924–2021) In Memoriam Part 2: Personal Recollections, Notices of the American Mathematical Society, vol. 69 no. 10 (November, 2022), pp. 1-1, American Mathematical Society (AMS) [doi]
    4. Bryant, R; Bismut, J-M; Cheeger, J; Griffiths, P; Donaldson, S; Hitchin, N; Lawson, HB; Gromov, M; Marcus, A; Spielman, D; Srivastava, N; Witten, E, Isadore M. Singer (1924–2021) In Memoriam Part 1: Scientific Works, Notices of the American Mathematical Society, vol. 69 no. 09 (October, 2022), pp. 1-1, American Mathematical Society (AMS) [doi]
    5. Phong, DH; Siu, Y-T; Bryant, R; Chau, A; Falbel, E; Fefferman, C; Friedman, R; Morgan, J; Futaki, A; Griffiths, P; Kohn, JJ; Mok, N; Mori, S; Namba, M; Noguchi, J; Ohsawa, T; Sato, M; Yau, S-T, Masatake Kuranishi (1924–2021), Notices of the American Mathematical Society, vol. 69 no. 05 (May, 2022), pp. 1-1, American Mathematical Society (AMS) [doi]
    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