Principal Investigator: Robert L. Bryant
Bryant will apply techniques of exterior differential systems and differential invariants to problems of current interest in geometry, economics, and physics. In soliton flows, two problems will be studied: Characterizing the complete Ricci solitons in terms of extra equations that they must satisfy, such as having higher multiplicity of eigenvalues of the Ricci tensor or its (symmetrized) covariant derivatives and exploring a flow for CR-nondegenerate hypersurfaces in complex manifolds with trivialized canonical bundle. In minimal submanifolds, Bryant will work on constructing and understanding calibrated submanifolds in Riemannian manifolds of special holonomy. In almost-complex geometry, Bryant will develop a theory of almost complex 6-manifolds by studying the functionals on the space of almost complex structures on a given 6-manifold and the `quasi-integrable' class of almost complex structures, which has many of the good bundle-theoretic properties of the integrable case but is considerably more flexible. It has a good theory of Hermitian-Yang-Mills connections, at least locally, and one may hope to define, study, and apply the associated moduli spaces. Bryant will also study the minimizing properties of pseudo-holomorphic curves in almost complex manifolds. In Finsler geometry, Bryant will develop the theory of Finsler manifolds with constant flag curvature, constructing new examples, understanding the properties of their geodesic flows, and pursuing relations with twistor theory and exotic holonomy. In smaller projects, Bryant will continue to study the problem of Hessian representability of metrics and its close relations with integrable systems and will also begin developing and generalizing the convex Darboux theory introduced by Ekeland and Nirenberg for applications to econometric models. (The methods of exterior differential systems are particularly suited to this latter problem and Bryant expects some interesting interaction with econometricians along the way.)
Bryant will continue to develop geometric approaches to problems arising in analysis, physics, economics, and biology. Many of the important advances in the use of mathematics in the sciences have depended on developing a geometric understanding of those problems, i.e., an understanding that focusses on intrinsic aspects that are not tied to arbitrarily chosen coordinates. (The most famous example of this is Einstein's general theory of relativity, which he found by re-interpreting gravitation as `curvature of space', rather than as a force whose complicated expression in observer coordinates was largely irrelevant. More recently, the development of string theory and M-theory has also resulted in geometric formulations of physics.) Geometric formulations often lead to degeneracies in the mathematics that can be treated by differential systems and their invariants (nonstandard tools that Bryant and his coworkers are systematically developing), and Bryant is exploring applications of these ideas to new areas.
