Robert Bryant
Phillip Griffiths Professor of Mathematics
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.
Current Research Interests
Office Hours
Current Appointments & Affiliations
 Phillip Griffiths Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2013
 Professor in the Department of Mathematics, Mathematics, Trinity College of Arts & Sciences 1987
Contact Information
 120 Science Drive, Durham, NC 27708
 Box 90320, Durham, NC 277080320
 bryant@math.duke.edu (919) 6602817
 #Personal Website
 Interview with Notices of AMS
 Background

Education, Training, & Certifications
 Ph.D., University of North Carolina  Chapel Hill 1979
 B.S., North Carolina State University 1974

Duke Appointment History
 Professor in the Department of Computer Science, Computer Science, Trinity College of Arts & Sciences 2014  2016

Leadership & Clinical Positions at Duke
 Ombudsperson, July 2019

Academic Positions Outside Duke
 Emeritus Professor of Mathematics, University of California at Berkeley. 2013  2053
 Director, Mathematical Sciences Research Institute. 2007  2013
 Professor of Mathematics, University of California at Berkeley. 2007  2013
 Noah Harding Professor of Mathematics, Rice University. 1986  1988
 Professor of Mathematics, Rice University. 1982  1986
 Recognition

In the News

JUN 1, 2019 The Notices of the American Mathematical Society 
OCT 1, 2018 The Notices of the American Mathematical Society 
JUN 1, 2016 Notices of the AMS 
MAR 1, 2015 The Notices of the American Mathematical Society 
DEC 10, 2013


Awards & Honors
 Expertise

Subject Headings
 Research

Selected Grants
 Special Holonomy In Geometry, Analysis and Physics awarded by Simons Foundation 2020  2023
 Special Holonomy In Geometry, Analysis and Physics awarded by Simons Foundation 2016  2020
 The Geometry of Partial Differential Equations and Applications awarded by National Science Foundation 2013  2015
 The Geometry of Partial Differential Equations awarded by National Science Foundation 2006  2008
 The Differential Geometry of Partial Differential Equations awarded by National Science Foundation 2001  2006
 The Differential Geometry of Partial Differential Equations awarded by National Science Foundation 1998  2001
 Mathematical Sciences: The Differential Geometry of Partial Differential Equations awarded by National Science Foundation 1995  1998
 Mathematical Sciences: The Differential Geometry of Partial Differential Equations awarded by National Science Foundation 1995  1997
 The Differential Geometry of Partial Differential Equations awarded by National Science Foundation 1995  1997
 The Differential Geometry of Partial Differential Equations awarded by National Science Foundation 1992  1995
 Mathematical Sciences: Differential Geometry awarded by National Science Foundation 1989  1993
 Differential Geometry awarded by National Science Foundation 1989  1990

External Relationships
 Institut des Hautes Etudes Scientifiques
 MathX
 Simons Foundation
 Publications & Artistic Works

Selected Publications

Books

Bryant, Robert L., S. S. Chern, Robert B. Gardner, Hubert L. Goldschmidt, and P. A. Griffiths. Exterior Differential Systems. Springer, 2011.

Gu, Chaohao, Marcel Berger, and Robert L. Bryant. Differential Geometry and Differential Equations Proceedings of a Symposium, held in Shanghai, June 21  July 6, 1985. Springer, 2006.

Bao, D., R. L. Bryant, S. . S. Chern, and Z. Shen, eds. A Sampler of RiemannFinsler Geometry. Vol. 50. Cambridge University Press, 2004.

Bao, D., R. L. Bryant, S. . S. Chern, and Z. Shen, eds. A Sampler of RiemannFinsler Geometry. Vol. 50. Cambridge University Press, 2004.

Bryant, Robert, Phillip Griffiths, and Daniel Grossman. Exterior Differential Systems and EulerLagrange Partial Differential Equations. University of Chicago Press, 2003.Open Access Copy

Bryant, R., V. Guillemin, S. Helgason, and R. O. Wells, eds. Integral Geometry. Vol. 63. Providence, RI: American Mathematical Society, 1987.

Bryant, R., V. Guillemin, S. Helgason, and R. O. Wells, eds. Integral Geometry. Vol. 63. Providence, RI: American Mathematical Society, 1987.


Academic Articles

Bryant, R. L., P. Foulon, S. V. Ivanov, V. S. Matveev, and W. Ziller. “Geodesic behavior for Finsler metrics of constant positive flag curvature on S^{2}.” Journal of Differential Geometry 117, no. 1 (January 1, 2021): 1–22. https://doi.org/10.4310/JDG/1609902015.Full Text Open Access Copy

Acharya, B. S., R. L. Bryant, and S. Salamon. “A circle quotient of a G2 cone.” Differential Geometry and Its Application 73 (December 1, 2020). https://doi.org/10.1016/j.difgeo.2020.101681.Full Text Open Access Copy

Bryant, R. L., and J. N. Clelland. “Flat metrics with a prescribed derived coframing.” Symmetry, Integrability and Geometry: Methods and Applications (Sigma) 16 (January 1, 2020). https://doi.org/10.3842/SIGMA.2020.004.Full Text Open Access Copy

Bryant, R. L., M. G. Eastwood, A. R. Gover, and K. Neusser. “Some differential complexes within and beyond parabolic geometry.” Advanced Studies in Pure Mathematics 82, no. Differential Geometry and Tanaka Theory (November 22, 2019): 13–40.Open Access Copy Link to Item

Bryant, R., R. Buckmire, L. Khadjavi, and D. Lind. “The origins of spectra, an organization for LGBT mathematicians.” Notices of the American Mathematical Society 66, no. 6 (June 1, 2019): 875–82. https://doi.org/10.1090/noti1890.Full Text Open Access Copy

Bryant, R. L., L. Huang, and X. Mo. “On Finsler surfaces of constant flag curvature with a Killing field.” Journal of Geometry and Physics 116 (June 1, 2017): 345–57. https://doi.org/10.1016/j.geomphys.2017.02.012.Full Text Open Access Copy

Bryant, R. L. “On the convex PfaffDarboux Theorem of Ekeland and Nirenberg,” December 22, 2015.Open Access Copy Link to Item

Bryant, R. L. “On the conformal volume of 2tori,” July 6, 2015.Open Access Copy Link to Item

Bryant, R. L. “S.S. Chern's study of almostcomplex structures on the sixsphere,” May 14, 2014.Open Access Copy Link to Item

Bryant, R. L. “Notes on exterior differential systems,” May 13, 2014.Open Access Copy Link to Item

Bryant, R., and F. Xu. “Laplacian Flow for Closed $G_2$Structures: Short Time Behavior,” January 11, 2011.Open Access Copy Link to Item

Bryant, R. L. “Commentary.” Bulletin of the American Mathematical Society 46, no. 2 (April 1, 2009): 177–78. https://doi.org/10.1090/S0273097909012488.Full Text

Bryant, R. L. “Gradient Kähler Ricci solitons.” Astérisque, no. 321 (2008): 51–97.Open Access Copy Link to Item

Bryant, R. L., G. Manno, and V. S. Matveev. “A solution of a problem of Sophus Lie: Normal forms of twodimensional metrics admitting two projective vector fields.” Mathematische Annalen 340, no. 2 (2008): 437–63. https://doi.org/10.1007/s0020800701583.Full Text Open Access Copy

BRYANT, R. L. “Conformal geometry and 3plane fields on 6manifolds.” Rims Kokyuroku 1502 (Developments of Cartan Geometry and Related Mathematical Problems) (July 2006): 1–15.Open Access Copy Link to Item

Bryant, R. L. “SO(n)Invariant special Lagrangian submanifolds of ℂ ^{n+1} with fixed loci.” Chinese Annals of Mathematics. Series B 27, no. 1 (January 1, 2006): 95–112. https://doi.org/10.1007/s1140100503685.Full Text Open Access Copy

Bryant, R. L. “On the geometry of almost complex 6manifolds.” Asian Journal of Mathematics 10, no. 3 (January 1, 2006): 561–605. https://doi.org/10.4310/AJM.2006.v10.n3.a4.Full Text

Bryant, Robert, and Dan Freed. “ShiingShen Chern.” Physics Today 59, no. 1 (January 2006): 70–72. https://doi.org/10.1063/1.2180187.Full Text

Bryant, R. L. “Real hypersurfaces in unimodular complex surfaces,” July 27, 2004.Open Access Copy Link to Item

Bryant, R., H. Edelsbrunner, P. Koehl, and M. Levitt. “The area derivative of a spacefilling diagram.” Discrete and Computational Geometry 32, no. 3 (January 1, 2004): 293–308. https://doi.org/10.1007/s0045400410991.Full Text

Bryant, R. L. “Some remarks on Finsler manifolds with constant flag curvature.” Houston Journal of Mathematics 28, no. 2 (January 1, 2002): 221–62.Open Access Copy Link to Item

Bryant, R. I. “On Surfaces with Prescribed Shape Operator.” Results in Mathematics 40, no. 1–4 (October 1, 2001): 88–121. https://doi.org/10.1007/BF03322701.Full Text

Bryant, R. L. “Bochnerkähler metrics.” Journal of the American Mathematical Society 14, no. 3 (July 1, 2001): 623–715. https://doi.org/10.1090/S0894034701003666.Full Text Open Access Copy

Bryant, R. L. “Rigidity and quasirigidity of extremal cycles in Hermitian symmetric spaces,” March 5, 2001.Open Access Copy Link to Item

Bryant, R. “Recent advances in the theory of holonomy.” Asterisque 266 (December 1, 2000): 351–74.Open Access Copy

Bryant, R. L. “Harmonic morphisms with fibers of dimension one.” Communications in Analysis and Geometry 8, no. 2 (January 1, 2000): 219–65. https://doi.org/10.4310/CAG.2000.v8.n2.a1.Full Text Open Access Copy

Bryant, R. L. “Calibrated Embeddings in the Special Lagrangian and Coassociative Cases.” Annals of Global Analysis and Geometry 18, no. 3–4 (January 1, 2000): 405–35. https://doi.org/10.1023/a:1006780703789.Full Text Open Access Copy

Bryant, R. “Élie Cartan and geometric duality.” Journées Élie Cartan 1998 Et 1999 16 (2000): 5–20.Open Access Copy
 Bryant, RL. "Calibrated embeddings in the special Lagrangian and coassociative cases." Annals of Global Analysis and Geometry 18, no. 34 (2000): 405435. Open Access Copy

Bryant, R. L. “Some examples of special Lagrangian Tori.” Advances in Theoretical and Mathematical Physics 3, no. 1 (January 1, 1999): 83–90. https://doi.org/10.4310/ATMP.1999.v3.n1.a2.Full Text Open Access Copy

Bryant, R. L., and E. Sharpe. “Dbranes and Spin^{c} structures.” Physics Letters, Section B: Nuclear, Elementary Particle and High Energy Physics 450, no. 4 (January 1, 1999): 353–57. https://doi.org/10.1016/S03702693(99)001616.Full Text Open Access Copy

Bryant, R. L. “Projectively flat finsler 2spheres of constant curvature.” Selecta Mathematica, New Series 3, no. 2 (January 1, 1997): 161–203. https://doi.org/10.1007/s000290050009.Full Text Open Access Copy

Bryant, R., P. Griffiths, and L. Hsu. “Hyperbolic exterior differential systems and their conservation laws, part II.” Selecta Mathematica, New Series 1, no. 2 (September 1, 1995): 265–323. https://doi.org/10.1007/BF01671567.Full Text

Bryant, Robert L., and Phillip A. Griffiths. “Characteristic cohomology of differential systems. I. General theory.” Journal of the American Mathematical Society 8, no. 3 (September 1, 1995): 507–507. https://doi.org/10.1090/s0894034719951311820x.Full Text

Bryant, R., P. Griffiths, and L. Hsu. “Hyperbolic exterior differential systems and their conservation laws, part I.” Selecta Mathematica, New Series 1, no. 1 (March 1, 1995): 21–112. https://doi.org/10.1007/BF01614073.Full Text

Bryant, R. L., and P. A. Griffiths. “Characteristic cohomology of differential systems II: Conservation laws for a class of parabolic equations.” Duke Mathematical Journal 78, no. 3 (January 1, 1995): 531–676. https://doi.org/10.1215/S0012709495078247.Full Text

Bryant, R. L., and L. Hsu. “Rigidity of integral curves of rank 2 distributions.” Inventiones Mathematicae 114, no. 1 (December 1, 1993): 435–61. https://doi.org/10.1007/BF01232676.Full Text

Bryant, R. L. “Some remarks on the geometry of austere manifolds.” Boletim Da Sociedade Brasileira De Matemática 21, no. 2 (September 1, 1991): 133–57. https://doi.org/10.1007/BF01237361.Full Text

Bryant, R. L., and S. M. Salamon. “On the construction of some complete metrics with exceptional holonomy.” Duke Mathematical Journal 58, no. 3 (January 1, 1989): 829–50. https://doi.org/10.1215/S0012709489058390.Full Text

Bryant, R., and R. Harvey. “Submanifolds in hyperKähler geometry.” Journal of the American Mathematical Society 2, no. 1 (January 1, 1989): 1–31. https://doi.org/10.1090/S08940347198909531698.Full Text

Bryant, R. L. “Metrics with exceptional holonomy.” Ann. of Math. (2) 126, no. 3 (1987): 525–76.

Bryant, Robert, and Phillip Griffiths. “Reduction for Constrained Variational Problems and � κ 2 2 ds.” American Journal of Mathematics 108, no. 3 (June 1986): 525–525. https://doi.org/10.2307/2374654.Full Text

Bryant, R. L. “Lie groups and twlstor spaces.” Duke Mathematical Journal 52, no. 1 (January 1, 1985): 223–61. https://doi.org/10.1215/S0012709485052135.Full Text

Bryant, R. L. “Minimal surfaces of constant curvature in s^{n}.” Transactions of the American Mathematical Society 290, no. 1 (January 1, 1985): 259–71. https://doi.org/10.1090/S00029947198507879648.Full Text

Bryant, R. L. “A duality theorem for willmore surfaces.” Journal of Differential Geometry 20, no. 1 (January 1, 1984): 23–53. https://doi.org/10.4310/jdg/1214438991.Full Text

Berger, E., R. Bryant, and P. Griffiths. “The Gauss equations and rigidity of isometric embeddings.” Duke Mathematical Journal 50, no. 3 (January 1, 1983): 803–92. https://doi.org/10.1215/S0012709483050391.Full Text

Bryant, R. L., P. A. Griffiths, and D. Yang. “Characteristics and existence of isometric embeddings.” Duke Mathematical Journal 50, no. 4 (January 1, 1983): 893–994. https://doi.org/10.1215/S0012709483050408.Full Text

Bryant, R. L. “Conformal and minimal immersions of compact surfaces into the 4sphere.” Journal of Differential Geometry 17, no. 3 (January 1, 1982): 455–73. https://doi.org/10.4310/jdg/1214437137.Full Text

Bryant, R. L. “Submanifolds and special structures on the octonians.” Journal of Differential Geometry 17, no. 2 (January 1, 1982): 185–232. https://doi.org/10.4310/jdg/1214436919.Full Text

Bryant, R. L. “Holomorphic curves in lorentzian crmanifolds.” Transactions of the American Mathematical Society 272, no. 1 (January 1, 1982): 203–21. https://doi.org/10.1090/S00029947198206564864.Full Text

Berger, E., and P. Griffiths. “Some isometric embedding and rigidity results for Riemannian manifolds.” Proc. Nat. Acad. Sci. U.S.A. 78, no. 8 (1981): 4657–60.

Acharya, Bobby Samir, Robert L. Bryant, and Simon Salamon. “A circle quotient of a $G_2$ cone,” n.d.Open Access Copy Link to Item

Bryant, R., P. Foulon, S. Ivanov, V. S. Matveev, and W. Ziller. “Geodesic behavior for Finsler metrics of constant positive flag curvature on S^2 (Accepted).” Journal of Differential Geometry, n.d.Open Access Copy Link to Item

Bryant, Robert L. “Leviflat Minimal Hypersurfaces in Twodimensional Complex Space Forms.” Adv. Stud. Pure Math., 37, Math. Soc. Japan, Tokyo, 2002, 1 44, n.d.Open Access Copy Link to Item

Bryant, Robert L. “Recent Advances in the Theory of Holonomy.” Seminaire Bourbaki 1998, no. 99 (n.d.): 351–74.Open Access Copy Link to Item

Bryant, Robert L. “Notes on Projective, Contact, and Null Curves,” n.d.Open Access Copy Link to Item

Bryant, Robert L. “Notes on spinors in low dimension,” n.d.Open Access Copy Link to Item

Bryant, Robert L. “On surfaces with prescribed shape operator.” Results Math. 40 (2001), No. 1 4, 88 121, n.d.Open Access Copy Link to Item

Bryant, Robert L., Maciej Dunajski, and Michael Eastwood. “Metrisability of twodimensional projective structures,” n.d.Open Access Copy Link to Item


Book Sections

Bryant, R. “Geodesically reversible Finsler 2spheres of constant curvature.” In Inspired by S. S. ChernA Memorial Volume in Honor of a Great Mathematician, edited by P. A. Griffiths, 11:95–111. Hackensack, NJ: World Scientific Publishers, 2006.Open Access Copy

Bryant, R. “Holonomy and Special Geometries.” In Dirac Operators: Yesterday and Today, edited by J. P. Bourguinon, T. Branson, A. Chamseddine, O. Hijazi, and R. Stanton, 71–90. Somerville, MA: International Press, 2005.Open Access Copy

Bryant, R. “PseudoReimannian metrics with parallel spinor fields and vanishing Ricci tensor.” In Global Analysis and Harmonic Analysis (MarseilleLuminy, 1999), edited by J. P. Bourguinon, T. Branson, and O. Hijazi, 4:53–94. Paris: Société Mathématique de France, 2000.Open Access Copy

Bryant, R. “On extremals with prescribed Lagrangian densities.” In Manifolds and Geometry (Pisa, 1993), edited by Pd Bartolomeis, F. Tricerri, and E. Vesentini, 36:86–111. Cambridge, UK: Cambridge University Press, 1996.Open Access Copy

Bryant, R. “Classical, exceptional, and exotic holonomies: a status report.” In Actes de La Table Ronde de Géométrie Différentielle, edited by A. Besse, 1:93–165. Paris: Société Mathématique de France, 1996.

Bryant, R. “An introduction to Lie groups and symplectic geometry.” In Geometry and Quantum Field Theory (Park City, UT, 1991), edited by D. Freed and K. Uhlenbeck, 1:5–181. Providence, RI: American Mathematical Society, 1995.Open Access Copy

Bryant, R., and R. B. Gardner. “Control Structures.” In Geometry in Nonlinear Control and Differential Inclusions (Warsaw, 1993), edited by B. Jakubczyk, W. Respondek, and T. Rzezuchowski, 12:111–21. Warsaw, Poland: Polish Academy of Sciences, 1995.

Bryant, R. “Two exotic holonomies in dimension four, path geometries, and twistor theory.” In Complex Geometry and Lie Theory (Sundance, UT, 1989), edited by J. Carlson, H. Clemens, and D. Morrison, 53:33–88. Providence, RI: American Mathematical Society, 1991.

Bryant, R. “Surfaces in conformal geometry.” In The Mathematical Heritage of Hermann Weyl (Durham, NC, 1987), edited by R. O. Wells, 48:227–40. Providence, RI: American Mathematical Society, 1988.

Bryant, R. “Surfaces of mean curvature one in hyperbolic space.” In Théorie Des Variétés Minimales et Applications (Palaiseau, 1983–1984), 154–155:321–47. Paris: Société Mathématique de France, 1988.

Bryant, R. “On notions of equivalence of variational problems with one independent variable.” In Differential Geometry: The Interface between Pure and Applied Mathematics (San Antonio, Tex., 1986), edited by M. Luksic, C. Martin, and W. Shadwick, 68:65–76. Providence, RI: American Mathematical Society, 1987.

Bryant, R. “A survey of Riemannian metrics with special holonomy groups.” In Proceedings of the International Congress of Mathematicians. Vol. 1, 2. (Berkeley, Calif., 1986), edited by A. Gleason, 505–14. Providence, RI: American Mathematical Society, 1987.

Bryant, R. “Minimal Lagrangian submanifolds of KählerEinstein manifolds.” In Differential Geometry and Differential Equations (Shanghai, 1985), edited by C. Gu, M. Berger, and R. L. Bryant, 1255:1–12. Berlin, Germany: SpringerVerlag, 1987.

Bryant, R. “Minimal Lagrangian submanifolds of KählerEinstein manifolds.” In Differential Geometry and Differential Equations (Shanghai, 1985), edited by C. Gu, M. Berger, and R. L. Bryant, 1255:1–12. Berlin, Germany: SpringerVerlag, 1987.

Bryant, R. “Metrics with holonomy G2 or Spin(7).” In Workshop Bonn 1984 (Bonn, 1984), edited by F. Hirzebruch, J. Schwermer, and S. Suter, 1111:269–77. Berlin, Germany: Springer, 1985.

Bryant, R., and P. A. Griffiths. “Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle.” In Arithmetic and Geometry, Vol. II, edited by M. Artin and J. Tate, 36:77–102. Boston, MA: Birkhäuser Boston, 1983.

Bryant, R., S. S. Chern, and P. A. Griffiths. “Exterior Differential Systems.” In Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations (Beijing, 1980), edited by S. S. Chern and W. T. Wu, 1:219–338. Beijing, PRC; New York, NY: Science Press; Gordon & Breach Science Publishers, 1982.

Bryant, Robert L. “Nonembedding and nonextension results in special holonomy,” n.d.Open Access Copy Link to Item

Bryant, Robert L. “Some remarks on G_2structures,” n.d.Open Access Copy Link to Item


Other Articles

Bryant, Robert L. “Complex analysis and a class of Weingarten surfaces,” n.d.Open Access Copy Link to Item

Bryant, Robert, Bobby Acharya, and Simon Salamon. “A circle quotient of a G_2 cone (Submitted),” n.d.Link to Item

Bryant, Robert, and Jeanne Nielsen Clelland. “Flat metrics with a prescribed derived coframing (Submitted),” n.d.Link to Item

Bryant, Robert. “Notes on Projective, Contact, and Null Curves,” n.d.Link to Item


Conference Papers

Bryant, R. L. “Geometry of manifolds with special holonomy: "100 years of holonomy".” In 150 Years of Mathematics at Washington University in St. Louis, edited by G. R. Jensen and S. G. Krantz, 395:29–38. AMER MATHEMATICAL SOC, 2006.Link to Item

Bryant, Robert L. “Second order families of special Lagrangian 3folds.” In Perspectives in Riemannian Geometry, edited by V. Apostolov, A. Dancer, N. Hitchin, and M. Wang, 40:63–98. AMER MATHEMATICAL SOC, 2006.Open Access Copy Link to Item

BRYANT, R., P. GRIFFITHS, and L. HSU. “TOWARD A GEOMETRY OF DIFFERENTIALEQUATIONS.” In Geometry, Topology & Physics, edited by S. T. Yau, 4:1–76. INTERNATIONAL PRESS INC BOSTON, 1995.Link to Item


 Teaching & Mentoring

Recent Courses

Advising & Mentoring

Anuk Dayaprema, Duke undergraduate
Eric Lin, Duke undergraduate
Yuanling Wang, Duke undergraduate


Teaching Activities

Math 323S: Euclidean and NonEuclidean Geometry
Math 421: Differential Geometry

Math 323S: Euclidean and NonEuclidean Geometry
 Scholarly, Clinical, & Service Activities

Presentations & Appearances
 The geometry of pseudoholomorphic curves in the 6sphere. Special Holonomy in Geometry, Analysis, and Physics: Progress and Open Problems. Duke University. September 18, 2020 2020
 An Overview of the Progress and Goals of the Special Holonomy Simons Collaboration. Fourth Annual Meeting of the Simons Collaboration on Special Holonomy. Simons Foundation. September 9, 2020 2020
 On solitons for the closed G2Laplacian flow. Special Session on the Geometry of Differential Equations. Joint Mathematics Meetings. January 17, 2020 2020
 An Overview of the Progress and Goals of the Special Holonomy Simons Collaboration. Third Annual Meeting of the Simons Collaboration on Special Holonomy. Simons Foundation. September 12, 2019 2019
 On solitons for the G_2 –Laplacian flow . LG&TBQ Conference. University of Michigan. June 11, 2019 2019
 Limits, Bubbles, and Singularities: An introduction to the fundamental ideas of Karen Uhlenbeck. The 2019 Abel Prize Ceremony Lectures. Norwegian Academy of Science and Letters. May 22, 2019  May 22, 2019 2019
 2019 Porcelli Lectures by Robert Bryant. Porcelli Lectures. Louisiana State University. March 28, 2019  March 29, 2019 2019
 Algebraically Special Metrics with Special Holonomy. Conference on Differential Geometry in the Large. MATRIX. February 11, 2019  February 11, 2019 2019
 Solitons for the closed G_2Laplacian Flow. Conference on Differential Geometry in the Large. MATRIX. February 8, 2019 2019
 The Best Possible Shapes of Surfaces. Masaryk University Seminar Series in Mathematics, Physics, and Computer Science. Masaryk University, Brno, CZ. December 12, 2018 2018
 An introduction to Lie groups, symmetries, and symplectic geometry. Gulbenkian Summer School in Differential Geometry. Gulbenkian Foundation, LISBON. July 23, 2018  July 27, 2018 2018
 Finsler metrics with constant positive flag curvature and holonomy. New Methods in Finsler Geometry. Clay Mathematics Institute. May 22, 2018 2018
 On selfdual curves. Griffiths Conference 2018. University of Miami. March 2, 2018 2018
 From Mechanics, to Algebra, to Geometry: The Notion of Holonomy. McKnight Zame Lecture in Mathematics. University of Miami. March 1, 2018 2018
 The concept of Holonomyits history and recent developments. AMS Retiring Presidential Address. American Mathematical Society. January 13, 2018 2018
 Algebraically Constrained Special Holonomy Metrics and Secondorder Associative 3folds. Riemannian Geometry Past, Present, and Future: an Homage to Marcel Berger. Institute des Hautes Études Scientifiques. December 6, 2017 2017
 Algebraically special associative submanifolds and special holonomy metrics. Special Holonomy: Progress and Open Problems 2017. Simons Center for Geometry and Physics. September 10, 2017 2017
 Symmetry and Gstructures. Topological and Geometric Methods in QFT. University of Montana in Bozeman Mathematics. July 31, 2017 2017
 Recent progress on manifolds with special holonomy. PanAfrican Congress of Mathematicians 2017. Mohammed V University, Rabat. July 7, 2017 2017
 Curvaturehomogeneous metrics in dimension 3.. Colorado University Mathematics Colloquium. Colorado UniversityBoulder Mathematics. May 2, 2017 2017
 On the geometry of geodesics on Finsler surfaces. Swarthmore Dresden Lectures (2016–17). Swarthmore College Mathematics Department. April 25, 2017 2017
 The idea of holonomy. Swarthmore Dresden Lectures (2016–17). Swarthmore College Mathematics Department. April 24, 2017 2017
 Engel structures, rigid curves, and compatible complex and symplectic structcures. AIM Workshop on Engel Structures. American Institute of Mathematics. April 20, 2017 2017
 Finsler Surfaces of Constant Curvature. First Annual Meeting of the Indian Mathematics Consortium. The Indian Mathematics Consortium. December 14, 2016 2016
 On families of special holonomy metrics defined by algebraic curvature conditions. Inaugural Conference of the Simons Collaboration "Special Holonomy in Geometry, Analysis, and Physics". Simons Center for Geometry and Physics. September 6, 2016 2016
 Integration techniques for exterior differential systems I & II. London Mathematical Society  EPSRC Durham Symposium Geometric and Algebraic Aspects of Integrability. Durham University. July 27, 2016 2016
 On curvaturehomogeneous metrics in dimension 3. "Differential Geometry in the Large (Conference in honor of Wolfgang Meyer's 80th birthday). Università degli studi Firenze. July 11, 2016 2016
 Curvaturehomogeneous metrics in dimension 3. Stony Brook University Colloquium Series. Stony Brook University Mathematics Department. April 28, 2016 2016
 The geometry and dynamics of uniformly periodic recursion relations. Vanderbilt University Mathematics Colloquium Series. Vanderbilt University Mathematics Department. April 21, 2016 2016
 The geometry and dynamics of periodic equiareal sequences. Tulane Colloquia in Mathematics. Tulane University Mathematics Department. February 25, 2016 2016
 Beyond Holonomy: Constrained Structure Equations. IHP conference "Relativity and Geometry" in memorial to André Lichnerowicz (in honor of his 100th Birthday). Institut Henri Poincaré, Paris, France. December 14, 2015 2015
 On curvaturehomogeneous 3manifolds. BaltimoreWashington Metro Area Differential Geometry Seminar. Howard University. October 24, 2015 2015
 The idea of holonomy. Public Lecture celebrating the 200th anniversary of George Boole. University College Cork, Cork, Ireland. October 12, 2015 2015
 The geometry of periodic equiPoisson sequences. Colloquium. University of California, Irvine, Mathematics Department. May 29, 2015 2015
 The geometry of equiareal sequences. Geometry Topology Seminar. North Carolina State University Mathematics Department. January 29, 2015 2015
 Gstructures with prescribed geometry. The 35th Winter School in GEOMETRY AND PHYSICS, Srní, Czech Republic . January 18, 2015  January 20, 2015 2015
 Some geometric constructions of holonomy plane fields and their analysis. IHP Workshop "Equivalence, invariants, and symmetries of vector distributions and related structures: from Cartan to Tanaka and beyond". Institut Henri Poincaré, Paris France. December 10, 2014 2014
 Some exceptional geometry of rolling surfaces. AMS Southeastern Regional Meeting, Special Session "Exceptional Groups in Physics, Algebra, and Geometry". UNC Greensboro. November 9, 2014 2014
 Convex billiards and nonholonomic systems. FiftyFirst Annual DeLong Lectures. University of Colorado at Boulder . October 14, 2014 2014
 The idea of holonomy. FiftyFirst Annual DeLong Lectures. University of Colorado at Boulder. October 13, 2014 2014
 Riemannian manifolds with special holonomy and special curvature. Simons Center program "G2 manifolds". Simons Center for Geometry and Physics . September 30, 2014 2014
 On G_2 and complex and almostcomplex structures on 6manifolds. July 14, 2014 2014
 On G_2 and complex and almostcomplex structures on 6manifolds. University College London. July 14, 2014 2014
 The Geometry of Special Holonomy Metrics. July 7, 2014  July 11, 2014 2014
 The geometry of periodic equiareal sequences. June 5, 2014 2014
 Existence and NonExistence of Canonical Connections. June 3, 2014 2014
 Differential Systems and Holonomy. May 1, 2014 2014
 Rolling surfaces and exceptional geometry. April 11, 2014 2014
 The geometry of rolling surfaces and nonholonomic mechanics. March 21, 2014 2014
 Exterior Differential Systems and Lie Theory, IV: Examples and Applications. December 12, 2013 2013
 Exterior Differential Systems and Lie Theory, II: The basics of CartanKähler Theory. December 11, 2013 2013
 Exterior Differential Systems and Lie Theory, III: The structure theory and existence. December 10, 2013 2013
 Exterior Differential Systems and Lie Theory, I: Introduction. December 9, 2013 2013
 Finsler geometry and manifolds of special holonomy. November 20, 2013 2013
 Prescribed curvature in Finsler geometry and PDE. November 20, 2013 2013
 Finsler spaces of constant flag curvature. November 18, 2013 2013
 RiemannFinsler geometry and the Method of Equivalence. November 15, 2013 2013
 The geometry of equiPoisson sequences. November 8, 2013 2013
 The exceptional geometry of binary sextics. November 5, 2013 2013
 Recent developments in special holonomy manifolds. November 1, 2013 2013
 The geometry of rolling surfaces and nonholonomic mechanics. November 1, 2013 2013
 Finsler manifolds of constant flag curvature. October 31, 2013 2013
 The origins of Finsler geometry in the calculus of variations. October 31, 2013 2013
 The geometry of calibrations. October 24, 2013 2013
 The geometry of equiPoisson sequences. October 17, 2013 2013
 Convex Billiards and nonholonomic systems. October 9, 2013 2013
 Recent advances in the geometry of PDE. October 3, 2013 2013
 The Affine Bonnet Problem. October 1, 2013 2013
 On the geometry of rolling surfaces. September 17, 2013 2013
 A visit to the Finsler world. September 13, 2013 2013
 The affine Bonnet Problem. September 10, 2013 2013
 Convex Billiards and nonholonomic systems. September 6, 2013 2013
 Introduction to Holonomy, II. May 29, 2013 2013
 Introduction to Holonomy, I. May 28, 2013 2013
 Parking Cars, Rolling Balls, and Falling Cats: The Concept of Holonomy. May 22, 2013 2013
 Control theory, rolling surfaces, and the exceptional group G_2. April 8, 2013 2013
 Convex billiards and nonholonomic systems. November 1, 2012 2012
 Geometry Old and New: From Euclid to String Theory. November 1, 2012 2012
 On the local classification of ambiKähler structures in dimension 4. October 27, 2012 2012
 Geometry Old and New: From Euclid to String Theory. September 29, 2012 2012
 On normal forms for Pfaffian systems and their application. September 12, 2012 2012
 Some geometry of characteristic cohomology. January 23, 2012 2012
 Applications of Cartan's generalization of Lie's Third Theorem. June 13, 2011 2011
 On the affine Bonnet problem. May 17, 2011 2011
 On the affine Bonnet problem. January 24, 2011 2011
 The Affine Bonnet Problem. December 16, 2010 2010
 Cartan's Generalization of Lie's Third Fundamental Theorem. December 15, 2010 2010
 A visit to the Finsler world. December 13, 2010 2010
 Invariants of projective structures on surfaces and some applications. November 19, 2010 2010
 The Affine Bonnet Problem. November 14, 2010 2010
 The Affine Bonnet Problem. October 22, 2010 2010
 The Affine Bonnet Problem. October 17, 2010 2010
 The idea of Holonomy. October 14, 2010 2010
 On the life and work of S.S. Chern. August 19, 2010 2010
 A visit to the Finsler world. June 2, 2010 2010
 On the geometry of projective surfaces. May 25, 2010 2010
 Auf Wiedersehen Surfaces, Revisited. March 10, 2010 2010
 The nearest way: from lines to geometry. May 28, 2009 2009
 The infinite groupoids of Lie and Cartan. May 23, 2009 2009
 Involutive Rational Normal Structures. May 12, 2009 2009
 Auf Wiedersehen Surfaces, Revisited. April 3, 2009 2009
 On Cartan's generalization of Lie's Third Theorem. December 17, 2008 2008
 On natural equations for metrics and potentials in dimension 3. November 24, 2008 2008
 Cartan's Generalization of Lie's Third Theorem. November 13, 2008 2008
 Riemannian Submersions as PDE. October 27, 2008 2008
 Involutive Rational Normal Structures. October 17, 2008 2008
 Geometry on moduli spaces of rational curves, III. September 29, 2008 2008
 Geometry on moduli spaces of rational curves, II. September 15, 2008 2008
 Geometry on moduli spaces of rational curves, I. September 8, 2008 2008
 Riemannian Submersions as PDE. September 1, 2008 2008
 The idea of holonomy. August 7, 2008 2008
 Involutive rational normal structures. August 4, 2008 2008
 The geometry of Riemannian submersions and foliations. June 6, 2008 2008
 Geometric flows and special holonomy. June 5, 2008 2008
 Riemannian Submersions as PDE. May 5, 2008 2008
 Riemannian Submersions as PDE. April 25, 2008 2008
 The Hessian representation problem. April 24, 2008 2008
 Geometric flows and special holonomy. April 23, 2008 2008
 The unreasonable effectiveness of geometric methods. April 22, 2008 2008
 Riemannian Submersions as PDE. February 2, 2008 2008

Service to the Profession
 Member. Search Advisory Committee for the Assistant Director of the MPS Directorate. National Science Foundation. February 2020  June 2020 2020
 Member. US National Committee for Mathematics. National Academy of Sciences. January 29, 2019  December 31, 2021 2019  2021
 Section Chair of Section 11 (Mathematics). National Academy of Sciences. May 2018  May 2021 2018  2021
 Member. Scientific Advisory Board. Berlin Mathematical School. 2018 2018
 Immediate Past President. American Mathematical Society. February 2017  February 2018 2017  2018
 Chair. Search Committee for new Director of the Division of Mathematical Sciences. National Science Foundation. January 2017  June 2017 2017
 Trustee. Board of Trustees. The EDGE Foundation. January 2017 2017
 Chair. Scientific Council. Fondation Mathématique Jacques Hadamard. March 2016  March 2020 2016  2020
 Member. Advisory Board. Project Euclid. 2016 2016
 President of the American Mathematical Society. American Mathematical Society. February 1, 2015  January 31, 2017 2015  2017
 Member. Mathematics and Physical Sciences Advisory Committee. National Science Foundation. October 2014  March 2021 2014  2021
 Member, Class I Membership Committee. National Academy of Sciences. November 15, 2013  November 15, 2014 2013  2014
 Member : Scientific Committee of the IHÉS. June 1, 2013 2013
 Member : StatsNSF. November 1, 2012  July 2013 2012  2013
 Chair of Membership Committee. Class I, Section 1 (Mathematics). American Academy of Arts and Sciences. July 2012  June 2018 2012  2018
 Chair : 2014 Chern Medal Committee of the International Mathematical Union. June 1, 2012  August 15, 2014 2012  2014
 CoOrganizer : Chern Centennial Conference. October 24, 2011  November 4, 2011 2011
 Chair : Section I:1 (Mathematics and Statistics) panel for 2014 AAAS elections. June 1, 2011 2011
 Exterior Member : Academic Council of the Institute Henri Poincaré. July 1, 2010 2010
 Member : Board of Trustees of the Banff International Research Station. July 1, 2009  July 31, 2013 2009  2013
 Deputy Chair, Chair, then Program Chair : Executive Committee of the Golden Section of the MAA. March 1, 2008  February 28, 2011 2008  2011
 Member : International Committee for the National Mathematics Centre of Nigeria. January 1, 2008 2008
 Member, Organizing Committee : Bay Area Differential Geometry Seminar. 2008  2013 2008  2013
 Editor. Communications in Analysis and Geometry. 2003 2003

Service to Duke
 Duke University. President's Council on Black Affairs. July 2019  June 2022 2019  2022
 Ombud. Duke Department of Mathematics. July 2019 2019
 Academic Council of Arts and Sciences. August 2018  May 2020 2018  2020
 Advisory Committee on Distinguished Professorships in Arts and Sciences . July 2016  June 2018 2016  2018
 Provost Advisory Committee on Appointments, Promotion, and Tenure. July 2016  December 2019 2016  2019
 Chamber Arts Society of Durham. July 1, 2013 2013
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