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.

摘要奖：DMS-0604195首席研究员：罗伯特·L·布莱恩特·布莱恩特将把外微分系统和微分不变量的技术应用于当前感兴趣的计量学、经济学和物理学问题。在孤子流中，将研究两个问题：刻画额外方程的完备Ricci孤子项，它们必须满足，如Ricci张量或其(对称化)协变导数的本征值具有较高的重数；在极小子流形中，Bryant将致力于构造和理解特殊完整黎曼流形中的有标子流形。在几乎复几何中，Bryant将通过研究给定6-流形上的几乎复结构空间上的泛函和几乎复结构的“拟可积”类来发展一个几乎复杂的6-流形理论，它具有可积性的许多良好的丛论性质，但更具灵活性。它有一个很好的Hermitian-Yang-Mills联络的理论，至少在局部上，人们可能希望定义、研究和应用相关的模空间。Bryant还将研究几乎复流形中伪全纯曲线的极小化性质。在Finsler几何中，Bryant将发展具有常旗曲率的Finsler流形理论，构造新的例子，了解其测地流的性质，并寻求与扭曲理论和奇异完整的关系。在较小的项目中，Bryant将继续研究度量的黑森表示及其与可积系统的密切关系，并将开始发展和推广Ekeland和Nirenberg引入的凸达布理论，以应用于计量经济学模型。(外微分系统的方法特别适合于这个问题，科比期待着在这个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来解决分析、物理、经济学和生物学中出现的问题。数学在科学中应用的许多重要进展都依赖于对这些问题的几何理解，即，一种专注于与任意选择的坐标无关的内在方面的理解。(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的曲率，而不是将其解释为一种力，其复杂的表达在观察者坐标中基本上是无关紧要的。最近，弦理论和M理论的发展也产生了物理的几何公式。几何公式经常导致数学上的退化，可以用微分系统及其不变量(科比和他的同事们正在系统地开发的非标准工具)来处理，科比正在探索这些想法在新领域的应用。