The Differential Geometry of Partial Differential Equations

摘要DMS - 0103884（布莱恩特，杜克）罗伯特布莱恩特计划应用微分系统理论，等效方法，以及变分法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，受天体物理学的启发，Bryant打算研究与黎曼或伪黎曼度量兼容的连接的几何，这些连接允许平行旋量场，并且与Levi-Civita连接有一个封闭的3-形式。在第二个问题中，Bryant想继续他对卡-丘流形的奇异特殊拉格朗日子变量和特殊拉格朗日叶理的性质的研究。在第三个问题中，Bryant提出了研究6-流形上的几乎复结构空间的几个自然泛函以及这些泛函的极值的几何。第四，Bryant计划继续他的研究空间的同调体积最小化循环在紧李群，目标是找到一个完整的分类的体积最小化循环在每个同调类在一个紧，简单，单连通李群。最后，Bryant计划继续他的调查芬斯勒几何，特别是常曲率空间的分类问题（在黎曼情形下，对常截面曲率的芬斯勒几何的自然推广）最优化是数学中的一个中心问题，其中人们试图在物理系统的模型的可能配置空间中选择“最佳”配置。 一个例子是在水体上航行的问题，在规划从起点到目的地的“最佳”路径时必须考虑水流，其中“最佳”意味着“最短的航行时间”。 一条在短时间内最优的路径（一个“测地线”）如果走得足够远，可能就不会保持最优。这就是所谓的不稳定性。(For例如，在一条河中，水流在中游更快，结果是下游测地线是稳定的，但上游测地线不是。）测量这种稳定性的几何量被称为“曲率”，因为它是在对地球曲率的研究中首次发现的。 布莱恩特的工作研究曲率和“超定”微分方程系统，并涉及运动规划，控制理论，机器人和高能物理中的弦理论模型的优化问题。 他研究的一些具体问题是针对数学物理的应用（例如，与平行旋量场的连接）或控制理论（例如，芬斯勒几何，这是一门研究诸如上面提到的导航问题的学科），而其他的则是针对关于极小化器的性质的更多的基础性问题（例如，李群中的体积最小化循环）或用于最小化问题的当前方法和技术中固有的限制和/或可能性（例如，特殊拉格朗日几何和几乎复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).