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Robert Bryant, Philip Griffiths Professor of Mathematics and Professor of Computer Science

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:  103 Physics
Office Phone:  (919) 660-2800, (919) 660-2817
Email Address: send me a message
Web Pages:  https://fds.duke.edu/db/aas/math/faculty/bryant
http://math.duke.edu/people/robert-bryant

Teaching (Spring 2018):  (typical courses)

  • MATH 602.01, COMMUTATIVE ALGEBRA Synopsis
    Physics 227, MW 08:30 AM-09:45 AM
Teaching (Fall 2018):

  • MATH 603.01, REPRESENTATION THEORY Synopsis
    Physics 227, TuTh 03:05 PM-04:20 PM
Office Hours:

Tuesdays and Thursdays, 10:30-12:00PM, and by appointment
Education:

Ph.D.University of North Carolina at Chapel Hill1979
BSNorth Carolina State University at Raleigh1974
B.A.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:

Currently serving as President of the American Mathematical Society (2015-2017)
Curriculum Vitae
Current Ph.D. Students   (Former Students)

Recent Publications   (More Publications)   (search)

  1. Bryant, R; Huang, L; Mo, X, On Finsler surfaces of constant flag curvature with a Killing field, Journal of Geometry and Physics, vol. 116 (June, 2017), pp. 345-357 [doi]
  2. Bryant, RL, On the convex Pfaff-Darboux Theorem of Ekeland and Nirenberg (Preprint, December 22, 2015) [arXiv:1512.07100]  [abs]
  3. Bryant, RL, On the conformal volume of 2-tori (Preprint, July 6, 2015) [arXiv:1507.01485]  [abs]
  4. Bryant, RL, S.-S. Chern's study of almost-complex structures on the six-sphere (Preprint, May, 2014) [arXiv:1405.3405]  [abs]
  5. Bryant, RL, Notes on exterior differential systems (Preprint, May, 2014) [arXiv:1405.3116]  [abs] [author's comments]
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, 2016/07-2020/06.      
Conferences Organized

 

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

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320