The Differential Geometry of Partial Differential Equations

Duke University, Durham NC

Investigators

Abstract

Abstract for DMS - 0103884 (Bryant, Duke) Robert Bryant plans to apply the theory of differential systems, the method of equivalence, and methods from the calculus of variations to study a collection of problems in differential geometry and mathematical physics. In the first problem, motivated by mathematical physics, Bryant intends to study the geometry of connections compatible with either a Riemannian or pseudo-Riemannian metric that admit parallel spinor fields and that differ from the Levi-Civita connection by a closed 3-form. In the second problem, Bryant wants to continue his investigations into the nature of singular special Lagrangian subvarieties and special Lagrangian foliations of Calabi-Yau manifolds. In the third problem, which concerns the study of the space of almost complex structures on 6-manifolds, Bryant proposes to investigate several natural functionals on the space of such almost complex structures and the geometry of the extrema of these functionals. Fourth, Bryant plans to continue his study of the space of homologically volume minimizing cycles in compact Lie groups, with the goal of finding a complete classification of the volume minimizing cycles in each homology class in a compact, simple, simply connected Lie group. Finally, Bryant plans to continue his investigations into Finsler geometry, particularly the problem of classifying the spaces of constant flag curvature (the natural generalization to Finsler geometry of constant sectional curvature in the Riemannian case). Optimization is a central problem in mathematics, in which one tries to select the 'best' configuration in a space of possible configurations of a model for a physical system. An example is the problem of navigating on a body of water in which one must take water currents into account in planning the 'best' path from origin to destination, where 'best' is taken to mean 'shortest time of traverse'. A path that is optimal for a short period (a 'geodesic') might not remain optimal if pursued far enough. This is known as instability. (For example, in a river where the current is faster in midstream it turns out that downstream geodesics are stable, but that upstream geodesics are not.) The geometric quantity that measures this notion of stability is known as 'curvature', since it was first identified in studies of the curvature of the Earth. Bryant's work studies curvature and 'over-determined' systems of differential equations, and is relevant to optimization problems in motion planning, control theory, robotics, and string theory models in high energy physics. Some of the specific problems he works on are aimed at applications to mathematical physics (e.g., connections with parallel spinor fields) or control theory (e.g., Finsler geometry, which is the subject that studies problems such as the navigation problem mentioned above), while others are aimed at more foundational questions about the nature of minimizers (e.g., volume minimizing cycles in Lie groups) or the limits and/or possibilities inherent in the current methods and techniques for minimization problems (e.g., special Lagrangian geometry and almost complex 6-manifolds).