The geometry of partial differential equations

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AbstractAward: DMS-0604195Principal Investigator: Robert L. BryantBryant will apply techniques of exterior differential systems anddifferential invariants to problems of current interest ingeometry, economics, and physics. In soliton flows, two problemswill be studied: Characterizing the complete Ricci solitons interms of extra equations that they must satisfy, such as havinghigher multiplicity of eigenvalues of the Ricci tensor or its(symmetrized) covariant derivatives and exploring a flow forCR-nondegenerate hypersurfaces in complex manifolds withtrivialized canonical bundle. In minimal submanifolds, Bryantwill work on constructing and understanding calibratedsubmanifolds in Riemannian manifolds of special holonomy. Inalmost-complex geometry, Bryant will develop a theory of almostcomplex 6-manifolds by studying the functionals on the space ofalmost complex structures on a given 6-manifold and the`quasi-integrable' class of almost complex structures, which hasmany of the good bundle-theoretic properties of the integrablecase but is considerably more flexible. It has a good theory ofHermitian-Yang-Mills connections, at least locally, and one mayhope to define, study, and apply the associated moduli spaces.Bryant will also study the minimizing properties ofpseudo-holomorphic curves in almost complex manifolds. InFinsler geometry, Bryant will develop the theory of Finslermanifolds with constant flag curvature, constructing newexamples, 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 ofHessian representability of metrics and its close relations withintegrable systems and will also begin developing andgeneralizing the convex Darboux theory introduced by Ekeland andNirenberg for applications to econometric models. (The methodsof exterior differential systems are particularly suited to thislatter problem and Bryant expects some interesting interactionwith econometricians along the way.)Bryant will continue to develop geometric approaches to problemsarising in analysis, physics, economics, and biology. Many ofthe important advances in the use of mathematics in the scienceshave depended on developing a geometric understanding of thoseproblems, i.e., an understanding that focusses on intrinsicaspects that are not tied to arbitrarily chosen coordinates.(The most famous example of this is Einstein's general theory ofrelativity, which he found by re-interpreting gravitation as`curvature of space', rather than as a force whose complicatedexpression in observer coordinates was largely irrelevant. Morerecently, the development of string theory and M-theory has alsoresulted in geometric formulations of physics.) Geometricformulations often lead to degeneracies in the mathematics thatcan be treated by differential systems and their invariants(nonstandard tools that Bryant and his coworkers aresystematically developing), and Bryant is exploring applicationsof these ideas to new areas.