The geometry of partial differential equations

主要研究者：Robert L. bryant将外部微分系统和微分不变量技术应用于当前感兴趣的几何、经济学和物理学问题。在孤子流中，将研究两个问题：用它们必须满足的额外方程来描述完全Ricci孤子，例如Ricci张量的特征值或其（对称的）协变导数具有更高的多重性，以及探索具有平凡化正则束的复杂流形中cr -非退化超曲面的流动。在最小子流形中，bryant将致力于构造和理解特殊完整黎曼流形中的校准子流形。在几乎复杂几何中，Bryant将通过研究给定6-流形上的几乎复杂结构空间上的泛函和几乎复杂结构的“拟可积”类来发展一种几乎复杂的6-流形理论，这类结构具有可积情况的许多良好的束理论性质，但相当灵活。它有一个很好的hermitian - yang - mills连接理论，至少在局部，并且人们可能希望定义，研究和应用相关的模空间。Bryant还将研究几乎复杂流形中伪全纯曲线的最小化性质。在InFinsler几何中，Bryant将发展具有恒定旗曲率的finsler流形理论，构建新的例子，理解它们的测地线流的性质，并追求与扭曲理论和奇异完整的关系。在较小的项目中，Bryant将继续研究度量的hessian可表示性问题及其与可积系统的密切关系，并将开始发展和推广由Ekeland和nirenberg引入的凸达布理论，以应用于计量经济模型。（外部微分系统的方法特别适合后一个问题，布莱恩特希望在此过程中与计量经济学家进行一些有趣的互动。）布莱恩特将继续开发几何方法来解决分析、物理、经济学和生物学中的问题。科学中数学应用的许多重要进展都依赖于对这些问题的几何理解，即关注与任意选择的坐标无关的内在方面的理解。(这方面最著名的例子是爱因斯坦的广义相对论，他将引力重新解释为“空间曲率”，而不是将其作为一种力，其在观察者坐标中的复杂表达在很大程度上是无关紧要的。最近，弦理论和m理论的发展也导致了物理的几何公式。几何公式通常会导致数学中的退化，而这些退化可以通过微分系统及其不变量来处理（布莱恩特和他的同事正在系统地开发的非标准工具），布莱恩特正在探索这些思想在新领域的应用。

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.