The Differential Geometry of Partial Differential Equations

（Bryant, Duke）Robert Bryant计划应用微分系统理论、等价方法和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，受数学物理的启发，Bryant打算研究与黎曼度量或伪黎曼度量兼容的连接几何，这些度量允许平行旋量场，并且与列维-奇维塔连接不同于封闭的3-形式。在第二个问题中，Bryant想继续研究Calabi-Yau流形的奇异特殊拉格朗日子变种和特殊拉格朗日叶的性质。第三个问题是关于6流形上几乎复杂结构空间的研究，Bryant建议研究这种几乎复杂结构空间上的几个自然泛函和这些泛函的极值的几何。第四，Bryant计划继续他对紧李群中同调体积极小环空间的研究，目标是找到紧、简单、单连通李群中每个同调类中体积极小环的完整分类。最后，Bryant计划继续他对芬斯勒几何的研究，特别是对常旗曲率空间的分类问题（在黎曼情况下将常截面曲率自然推广到芬斯勒几何）。优化是数学中的一个核心问题，它试图在物理系统模型的可能配置空间中选择“最佳”配置。一个例子是在水面上航行的问题，在规划从起点到目的地的“最佳”路径时，必须考虑水流，这里的“最佳”是指“最短的穿越时间”。一条短期内最优的路径（“测地线”），如果走得足够远，可能就不会一直是最优的。这就是所谓的不稳定性。（例如，在一条河流中游流速较快的地方，下游的测地线是稳定的，而上游的测地线则不是。）测量这种稳定性概念的几何量被称为“曲率”，因为它是在对地球曲率的研究中首次被确定的。Bryant的工作研究曲率和微分方程的“过度确定”系统，并且与运动规划，控制理论，机器人技术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：（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），而其他的则针对更基本的问题，即最小化的本质（例如，李群中的体积最小化环）或当前最小化问题的方法和技术固有的极限和/或可能性（例如，李群中的体积最小化环）。，特殊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).