CURRICULUM VITAE

Biographical Data

Name:  Robert L. Bryant  Born:  August 30, 1953
Citizenship:  United States  Birthplace:  Kipling, North Carolina
Education:  B.S., North Carolina State University, 1974

Ph.D., University of North Carolina, 1979 (Ph.D. Advisor: Robert B. Gardner)

Areas of Research:  Geometric Partial Differential Equations and Differential Geometry
Positions Held:  Duke University

Juanita M. Kreps Professor of Mathematics, 1988–present Arts and Sciences Professor of Mathematics, 1987–88

Rice University Noah Harding Professor of Mathematics, 1986–88 Professor, 1982–86 Associate Professor, 1981–82 Assistant Professor, 1979–81

Visiting Positions Visitor, IMPA (Rio de Janiero, Brazil), July–August 1996 Visitor, Nankai Inst. of Mathematics (Tianjin, PRC), May 1995 André Aisenstadt Chair, Centre de Recherches Math., Montréal, Fall 1994 Member, MSRI (Berkeley, CA), January–June 1994 Member, IAS (Princeton, NJ), September–December 1993 Visitor, University of Adelaide, July-August 1993 Member, IMPA (Rio de Janiero, Brazil), July–August 1990 W. R. Reynolds Professor, UNC–Chapel Hill, Fall 1987 Member, IMPA (Rio de Janiero, Brazil), May 1986 Member, IHES (Bures-sur-Yvette, France), Nov 1985 Member, Max Planck Institute (Bonn, Germany), October 1985 Professor, Duke University, Fall 1984 Member, MSRI (Berkeley, CA), April–May 1983 Associate Professor, Harvard, Fall 1982 Member, IAS (Princeton, NJ), 1979–80

Professional Service: AMS-SIAM Committee on Chinese Graduate Students, 1984–89

Editor, Journal of Differential Geometry, 1988–93 Member of the Organizing Board, East Coast Geometry Festival, 1990–present Organizing Board Member, AMS 1990 Summer Conference on Differential Geometry AMS Centennial Fellowships Committee, 1990–92 Organizing Board Member, 1993-94 MSRI Special Year in Geometry Undergraduate Director, Regional Geometry Institute, 1991–1993 Editor, Transactions and Memoirs of the AMS, 1992–1997 Director of the Undergraduate Program, Park City/IAS Mathematics Institute, 1993–2000 Editorial Board, Duke Mathematical Journal, 1997–present Editor, Differential Geometry and its Applications, 1999–present Member at Large, Council of the AMS, 1998–present AMS Committee on Publications, 1998–present (chair, 2000–present) MSRI Board of Trustees, 1999–present (chair, 2001–present) Member of the Executive Council of the AMS, 2000–present MAA Souteastern Section Lecturer, 2001–2002

Awards:  NSF Postdoctoral Research Fellowship, 1979–80

Alfred P. Sloan Fellowship, 1982–84

Presidential Young Investigator Award, 1984–89 Trinity College Distinguished Teaching Award, 1992 MAA Southeastern Region Distinguished Teaching Award, 1993

Publications

Books:

[1] (with S.-S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths) Exterior Differential Systems, Springer-Verlag, New York, 1991. MR 92h:58007

[2] (with Chao Hao Gu and Marcel Berger (eds.)), Differential geometry and differential equations, Proceedings of the sixth symposium held at Fudan University, Shanghai, June 21–July 6, 1985., MR 88b:53002

[3] (with Victor Guillemin, Sigurdur Helgason, and R. O. Wells, Jr. (eds.)), Integral geometry, Contemporary Mathematics 63. AMS, Providence, R.I., 1987., MR 87j:53003

[4] Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces, to appear in the series Annals of Mathematics Studies, Princeton University Press.

Expository Articles:

[1] (with S.-S. Chern and P. A. Griffiths) Exterior differential systems, Proceedings of the 1980 Peking Symposium on Partial Differential Equations, 1 (1982), 219–338, Beijing Press.

[2] A survey of riemannian metrics with special holonomy groups, Proceedings of the 1986 International Congress of Mathematicians, American Mathematical Society 1 (1987), 505–514. MR 89f:53068

[3] An Introduction to Lie groups and symplectic geometry, in Geometry and Quantum Field Theory (Park City, UT, 1991), 5–181, IAS/Park City Math. Ser. 1, Amer. Math. Soc., Providence, RI, 1995. MR 96i:58002

[4] Élie Cartan and geometric duality, Journées Élie Cartan 1998 et 1999, Institut Élie Cartan 16 (2000), 5–20.

Research Articles:

[1] (with E. Berger and P. A. Griffiths) Some isometric embedding and rigidity results for Riemannian manifolds, Proc. Natl. Acad. Sci. USA 78 (1981), 4657–4660. MR 82h:53074

[2] Holomorphic curves in Lorentzian CR-manifolds, Trans. Amer. Math. Soc. 272 (1982), 203–221. MR 83i:32029

[3] Submanifolds and special structures on the octonians, Journal of Differential Geometry 17 (1982), 184– 232. MR 84h:53091 [4] Conformal and minimal immersions of compact surfaces into the 4-sphere, Journal of Differential Ge-

ometry 17 (1982), 455–473. MR 84a:53062

[5] (with P. A. Griffiths) Some observations on the infinitesimal period relations for regular three-folds with trivial canonical bundles, in Arithmetic and Geometry II, Progress in Mathematics 36 (1983), 77–102. MR 86a:32044

[6] (with E. Berger and P. A. Griffiths), The Gauss equations and rigidity of isometric embeddings, Duke Math. Journal 50 (1983), 803–892. MR 85k:53056

[7] (with P. A. Griffiths and D. Yang) Characteristics and existence of isometric embeddings, Duke Math. Journal 50 (1983), 893–994. MR 85d:53027

[8] A duality theorem for Willmore surfaces, Journal of Differential Geometry 20 (1984), 23–53. MR 86j:58029

[9] Lie groups and twistor spaces, Duke Math. Journal 52 (1985), 223–261. MR 87d:58047

[10] Metrics with holonomy G_{2} or Spin(7), Lecture Notes in Mathematics 1111 (1984), Springer-Verlag, 269–277. MR 87a:53082

[11] Minimal surfaces of constant curvature in S^{n}, Trans. Amer. Math. Soc. 290 (1985), 259–271. MR 87c:53110

[12] Minimal Langrangian submanifolds of Kähler-Einstein manifolds, Lecture Notes in Mathematics 1255 (1987), 1–12. MR 88j:53061

[13] (with P. A. Griffiths) Reduction for constrained variational problems and ∫ \frac{1}{2}κ^{2}ds, American Journal of Mathematics 108 (1986), 525–570. MR 88a:58044

[14] On notions of equivalence of variational problems with one independent variable, Contemporary Mathematics 68 (1987), 65–76. MR 89f:58037

[15] Metrics with exceptional holonomy, Annals of Mathematics 126 (1987), 525–576. MR 89b:53084

[16] Surfaces of mean curvature one in hyperbolic space, Asterisque 154–155 (1987), 321–347. CMP 955 072 [17] Surfaces in conformal geometry, Proceedings of Symposia in Pure Mathematics 48 (1988), 227–240.

MR 89m:53102

[18] (with R. Harvey) Submanifolds in hyper-Kähler geometry, Journal of the Amer. Math. Soc. 2 (1989), 1–31. MR 89m:53090

[19] (with S. Salamon) On the construction of some complete metrics with exceptional holonomy, Duke Math. Journal 58 (1989), 829–850. MR 90i:53055

[20] Some remarks on the geometry of austere submanifolds, Boletim da Sociedade Brasileira de Matematica (Nova Série) 21 (1991), 133–157. MR 92k:53112

[21] Two exotic holonomies in dimension four, path geometries, and twistor theory, in Complex Geometry and Lie Theory, Proceedings of Symposia in Pure Mathematics 53 (1991), 33–88. MR 93e:53030

[22] (with L. Hsu) Rigidity of integral curves of rank 2 distributions, Inventiones Mathematicae 114 (1993), 435–461. MR 94j:58003 [23] (with P. A. Griffiths) Characteristic cohomology of differential systems, I: General theory, Journal of

the AMS 8 (1995), 507–596. MR 96c:58183

[24] (with P. A. Griffiths) Characteristic cohomology of differential systems, II: Conservation laws for a class of parabolic equations, Duke Mathematical Journal 78 (1995), 531–676. MR 96d:58158

[25] (with P. A. Griffiths and Lucas Hsu) Hyperbolic exterior differential systems and their conservation laws, Part I, Mathematica Selecta, 1 (1995), 21–112. CMP 1 327 228

[26] (with P. A. Griffiths and Lucas Hsu) Hyperbolic exterior differential systems and their conservation laws, Part II, Mathematica Selecta, 1 (1995). CMP 1 354 599

[27] (with P. A. Griffiths and Lucas Hsu) Toward a geometry of differential equations, in Geometry, Topology & Physics, 1–76, Conf. Proc. Lecture Notes Geom. Topology, VI, Internat. Press, Cambridge, MA, 1995. CMP 1 358 612

[28] (with R. Gardner) Control Structures, Geometry in Nonlinear Control and Differential Inclusions, Banach Center Publications 32 (1995), Institute of Mathematics, Polskeij Akademii Nauk, Warszawa, Poland. MR 96h:93024

[29] On extremals with prescribed Lagrangian densities, in Manifolds and Geometry, Symposia Mathematica XXXVI (1996), Cambridge University Press, pp. 86–111. MR 99a:58043

[30] Classical, Exceptional, and Exotic Holonomies: A Status Report, in Actes de la Table Ronde de Géométrie Differentielle en l’Honneur de Marcel Berger, Collection SMF, Séminaires & Congrès 1 (1996), Soc. Math. de France, pp. 93–166. MR 98c:53037

[31] Finsler structures on the 2-sphere satisfying K = 1, in Finsler Geometry, Contemporary Mathematics 196 (1996), AMS Publications, pp. 27–42. MR 97e:53128

[32] Projectively flat Finsler 2-spheres of constant curvature, Selecta Mathematica, (New Series) 3 (1997), 161–203. dg-ga/9611010, MR 98i:53101

[33] Harmonic morphisms with fibers of dimension one, Communications in Analysis and Geometry 8 (2000), 219-265. dg-ga/9701002

[34] (with Eric. R. Sharpe) D-Branes and Spin^{c} Structures, Physics Letters B 450 (1999), 353–357. hepth/9812084, MR 2000c:53054

[35] Some examples of special Lagrangian tori, Adv. Theor. Math. Phys., vol. 1, (1999), 83–90. math.DG/9902076, MR 2000f:32033

[36] Recent Advances in the Theory of Holonomy, Exposés de la Seminar Bourbaki, no. 861 (1999), Astérisque 266 (2000), 351-374. math.DG/9910059

[37] Calibrated embeddings in the special Lagrangian and coassociative cases, Annals of Global Analysis and Geometry 18 (2000), 405-435. math.DG/9912246

[38] Levi-flat minimal hypersurfaces in two-dimensional complex space forms, in Lie Groups, Geometric Structures and Differential Equations, Advanced Studies in Pure Mathematics, accepted 2000, math.DG/9909159 [39] Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor, in Global Anal-

ysis and Harmonic Analysis, Séminaires & Congrès 4 (2000), Soc. Math. de France, pp. 53–93. math.DG/0004073

[40] Bochner-Kähler metrics, Journal of the AMS, 14 (2001), 623–715. math.DG/0003099

Submitted or In Preparation:

[1] Second order families of special Lagrangian 3-folds, preprint 2000, math.DG/0007128

[2] Calibrated embeddings in the special Lagrangian and coassociative cases, submitted, [math.DG/9912246]

[3] (with P. A Griffiths and Lucas Hsu) Poincaré-Cartan forms and the geometry of first-order functionals, in preparation.

[4] On the geometry of webs, in preparation.

[5] On metrics in 3-space for which the planes are minimal, in preparation.

[6] On the geometry of some degenerate partial differential equations in three dimensions, in preparation.

[7] Lectures on the geometry of differential equations, to appear in the CRM Monograph Series as the 1994 Andre Aisenstadt Lectures volume, in preparation.

[8] Finsler structures on the n-nphere with constant flag curvature, in preparation.

[9] (with P. A Griffiths) Calibrating minimal submanifolds, in preparation.

[10] Minimizing cycles in compact Lie groups, in preparation.

[11] The geometry of almost-complex 6-manifolds, in preparation.

Current to 25 May 2001