The geometry of partial differential equations

偏微分方程的几何

• 批准号: 0604195
• 负责人: Robert Bryant
• 金额: $ 55.9万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2008-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604195&HistoricalAwards=false
• 关键词: geometry; partial; differential; equations

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.

摘要奖：DMS-0604195主要研究者：Robert L. BryantBryant将应用外部微分系统和微分不变量的技术来解决当前感兴趣的几何学、经济学和物理学问题。 在孤子流中，我们将研究两个问题：用Ricci张量或其（对称化的）协变导数的特征值的重数更大的额外方程来刻画完备Ricci孤子，以及探索具有平凡化正则丛的复流形中CR-非退化超曲面的流。 在极小子流形中，Bryanson致力于构造和理解特殊完整黎曼流形中的校准子流形。 在几乎复杂的几何，布莱恩特将制定一个理论almostcomplex 6-流形通过研究空间上的泛函几乎复杂的结构在一个给定的6-流形和“准可积”类几乎复杂的结构，其中有许多良好的斗争理论性质的integrablecase，但相当灵活。 它有一个很好的理论厄米-杨-米尔斯连接，至少在局部，人们可能希望定义，研究和应用相关的模空间。布莱恩特还将研究极小化性质的伪全纯曲线在几乎复杂的流形。 在芬斯勒几何，布莱恩特将发展芬斯勒流形的理论与恒定的旗帜曲率，构建新的例子，了解其测地线流的性质，并追求与扭量理论和异国holonomy的关系。在较小的项目，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，还将开始发展和推广由Ekeland和Nirenberg，计量经济学模型的应用。 (The外微分系统的方法特别适合于后一个问题，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。）布莱恩特将继续发展几何方法来解决分析、物理、经济和生物学中出现的问题。 许多重要的进展，在使用数学的科学依赖于发展的几何理解这些问题，即，一种专注于内在方面的理解，这种理解不受任意选择的坐标的约束。(The这方面最著名的例子是爱因斯坦的广义相对论，他通过将引力重新解释为“空间曲率”而不是一种在观察者坐标中复杂表达基本上无关紧要的力来发现这一理论。 最近，弦理论和M理论的发展也导致了物理学的几何表述。） 几何公式通常会导致数学中的退化，这些退化可以通过微分系统及其不变量（Bryant和他的同事系统开发的非标准工具）来处理，Bryant正在探索这些想法在新领域的应用。