ROBERT L. BRYANT

B.S. North Carolina State University

Ph.D. The University of North Carolina at Chapel Hill

Juanita M. Kreps Professor of Mathematics

Areas of Expertise: Geometric Partial Differential Equations, Differential Geometry.

Research Summary:

Professor Bryant's research interests are in the areas of differential geometry and the geometry of partial differential equations.

Differential geometry is concerned with the study of differential equations and problems in the calculus of variations which have `invariance' or `symmetry' properties. For example, the study of `minimal' or `soap-film' surfaces in space is simpler (and more interesting) than minimizers of the `generic' first-order variational problem for surfaces because they are invariant under the group of rigid motions in space. More advanced examples are the Einstein equations or the Yang-Mills equations, which also describe extrema of variational problems which have a great deal of symmetry.

The work of Lie, Cartan, and Noether led them to develop a theory of differential equations which tries to keep invariance properties in the forefront of the analysis. Lie's development of the theory of Lie groups and differential invariants; Cartan's theories of exterior differential systems, the moving frame, and the method of equivalence; and Noether's far-reaching insight into the relationship between symmetries and conservation laws have been milestones in this theory.

Professor Bryant has been involved in the modern development of exterior differential systems and the applications of its techniques to a wide variety of differential equations that arise in geometric contexts. These range from applications to what might be called classical differential geometry, exemplified by his work on the holonomy problem in Riemannian geometry, to his more recent and current work on conservation laws for partial differential equations and their relationship with their differential invariants in the sense of Lie and Cartan.

Other interests include global analysis, integral geometry, Finsler geometry, twistor theory, and algebraic geometry.

Recent Publications:

1. (with S.-S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths), Exterior Differential Systems, Springer-Verlag, New York, 1991.
2. Rigidity of intergral curves or rank 2 distributions, Inventiones Mathematicae 114 (1993), 435-461.
3. An introduction to Lie groups and symplectic geometry, IAS/Park City Mathematics Series 1 (1995), 7-181.
4. (with P. A. Griffiths) Characteristic cohomology of differential systems I, Journal of the AMS 8 (1995), 507-596.
5. (with P. A. Griffiths), Characteristic cohomology of differential systems II, Duke Math. Journal 78 (1995), 531-676.
6. Projectively flat Finsler 2-spheres of constant curvature, Selecta Mathematica (New Series) 33 (1997), pp. 161-203.
7. Harmonic morphisms with fibers of dimension one, to appear in Communications in Analysis and Geometry.

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