The geometry of partial differential equations

偏微分方程的几何

• 批准号: 0848131
• 负责人: Robert Bryant
• 金额: $ 35.44万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0848131&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.

Abstractaward：DMS-0604195原理研究者：Robert L. Bryantbryant将将外部差分系统和不同不变性的技术应用于当前感兴趣的Ingeemetry，经济学和物理学的问题。 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. 在最小的亚曼福尔德（Bryant Will）中，布莱恩特（Bryant）将在理解和理解特殊自律歧管中的校准和理解的校准。 基本复合的几何形状，科比将通过在给定的6个manifold上的最大复杂结构和“ Quasi-Inflegold”类别的几乎复杂的结构上研究功能，从而发展出一种几乎复杂的6个manifolds的理论，这几乎是复杂的结构，这是对整合性酶的良好bundle bundle termenty the Integety the Integenty Properties of Flexteries of Flexteries of Flexible of Flexible flexible flexible of flexsmany的差异。 它具有至少在本地的阳性阳离子米尔斯连接的良好理论，并且可以在几乎复杂的歧管中研究，研究和应用相关的模量空间。 Infinsler几何形状，布莱恩特将以持续的旗帜曲率发展鳍式冠状动脉理论，构建新的示例，了解其地理位置流的特性，并追求与Twistor理论和异国情调的整体的关系。在较小的项目中，Bryant将继续研究与Interrics和Interge of Systems和Intermable of Systems and Inders的介绍，并将其与Intrict and Intellys Indress Indress Indrass Indress Indressign介绍，并将其与Interme的关系介绍，并开发出来，并将其与Inteltement的融合能力，并开发出来。 Ekeland和Nirenberg用于计量经济学模型。 （外部差异系统的方法特别适合这种问题问题，科比期望在此过程中与计量经济学家进行一些有趣的互动。）科比将继续开发几何方法，以解决分析，物理，经济学和生物学方面的问题。 科学夏季中数学使用的许多重要进步取决于对这些问题的几何理解，即，即专注于与内在的坐标相关的理解，这些理解是与任意选择的坐标相关的。观察者坐标的复杂表达在很大程度上是无关紧要的，弦理论和M理论的发展已在物理学的几何形式中进行。这些想法的应用到新领域。