The Differential Geometry of Partial Differential Equations

对于DMS-0103884(Bryant，Duke)，Robert Bryant计划应用微分系统理论、等价方法和变分中的方法来研究微分几何和数学物理中的一系列问题。在第一个问题中，受数学物理的启发，科比打算研究与黎曼度量或伪黎曼度量相容的连接的几何，这些连接允许平行的旋量场，并且不同于Levi-Civita连接的封闭的3-形式。在第二个问题中，Bryant想要继续研究Calabi-Yau流形的奇异特殊拉格朗日亚变分和特殊拉格朗日叶的性质。在研究六维流形上几乎复结构空间的第三个问题中，Bryant建议研究这种几乎复杂结构空间上的几个自然泛函以及这些泛函的极值的几何。第四，布莱恩特计划继续研究紧李群中同调体积最小化圈的空间，目标是在紧致、简单、单连通的李群中找到每个同调类中体积最小化圈的完全分类。最后，布莱恩特计划继续他对Finsler几何的研究，特别是对常旗曲率空间的分类问题(在黎曼情形下，恒截曲率的Finsler几何的自然推广)。最优化是数学中的一个中心问题，在这个问题中，一个人试图在一个物理系统的可能构型空间中选择一个模型的最佳构型。一个例子是在一片水域上航行的问题，在规划从起点到目的地的最佳路径时，必须考虑水流的影响，其中最佳路径的意思是“穿越的最短时间”。一条在短时间内最优的路径(测地线)，如果追求得足够远，可能不会保持最优。这就是所谓的不稳定性。(例如，在一条中游水流较快的河流中，下游测地线是稳定的，但上游测地线不稳定。)衡量稳定性这一概念的几何量被称为“曲率”，因为它最初是在研究地球曲率时发现的。布莱恩特的工作研究的是曲率和超定的微分方程组，与运动规划、控制理论、机器人和高能物理中的弦理论模型中的优化问题相关。他研究的一些具体问题旨在应用于数学物理(例如，与平行旋量场的联系)或控制理论(例如，芬斯勒几何，这是研究上述导航问题等问题的学科)，而另一些问题则针对关于极小化的本质(例如，李群中的体积极小化圈)或当前方法和技术所固有的限制和/或可能性的更基本的问题(例如，特殊的拉格朗日几何和几乎复杂的6-流形)。

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 variationsto study a collection of problems in differential geometry andmathematical physics. In the first problem, motivated by mathematicalphysics, Bryant intends to study the geometry of connections compatiblewith either a Riemannian or pseudo-Riemannian metric that admitparallel spinor fields and that differ from the Levi-Civita connectionby a closed 3-form. In the second problem, Bryant wants to continuehis investigations into the nature of singular special Lagrangian subvarietiesand special Lagrangian foliations of Calabi-Yau manifolds. In the thirdproblem, which concerns the study of the space of almost complex structures on6-manifolds, Bryant proposes to investigate several natural functionals on the spaceof such almost complex structures and the geometry of the extremaof these functionals. Fourth, Bryant plans to continue his studyof the space of homologically volume minimizing cycles in compact Lie groups,with the goal of finding a complete classification of the volume minimizingcycles 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 constantflag curvature (the natural generalization to Finsler geometryof constant sectional curvature in the Riemannian case).Optimization is a central problem in mathematics, in which one triesto select the 'best' configuration in a space of possible configurationsof a model for a physical system. An example is the problem of navigatingon a body of water in which one must take water currents into accountin planning the 'best' path from origin to destination, where 'best' istaken to mean 'shortest time of traverse'. A path that is optimal fora 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 isfaster in midstream it turns out that downstream geodesics are stable, butthat upstream geodesics are not.) The geometric quantity that measures thisnotion of stability is known as 'curvature', since it was first identifiedin studies of the curvature of the Earth. Bryant's work studies curvatureand 'over-determined' systems of differential equations, and is relevantto optimization problems in motion planning, control theory, robotics,and string theory models in high energy physics. Some of the specificproblems 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 asthe navigation problem mentioned above), while others are aimed at morefoundational questions about the nature of minimizers (e.g., volumeminimizing cycles in Lie groups) or the limits and/or possibilities inherentin the current methods and techniques for minimization problems(e.g., special Lagrangian geometry and almost complex 6-manifolds).