Geometry and Dynamics in Riemannian and Finsler Spaces

黎曼空间和芬斯勒空间中的几何和动力学

• 批准号: 0412166
• 负责人: Dmitri Burago
• 金额: --
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0412166&HistoricalAwards=false
• 关键词: Geometry; Dynamics; Riemannian; Finsler; Spaces

AbstractAward: DMS-0412166Principal Investigator: Dmitri BuragoD. Burago proposes to continue his work on a number of long-termprojects in Riemannian and Finsler geometry, dynamical systems ofgeometric origin, and geometric group theory. The projectsinclude: geometry of periodic metrics, area-minimizing surfacesin normed spaces and minimal fillings, and ellipticity of surfacearea functionals; low dimensional partially hyperbolicdiffeomorphisms; unbounded bi-invariant quasi-semi-norms and"large" groups; manifolds without conjugate points; applicationsof singular geometry to dynamics of billiard systems and certainalgorithmic problems of geometric origin; and approximations byPL-isometries. Among the problems the proposal is aimed at thereare: Busemann's Conjecture that flats are area-minimizingsurfaces in normed spaces; styding the structure of the class ofpartially-hyperbolic systems; finding weak versions of Hofer'snorm (possibly on some groups of volume-preservinghomeomorphisms); various generalizations of E.Hopf's conjectureon tori without conjugate points. This project continues theproposer's previous research, including a solution of the E. Hopfconjecture on tori without conjugate points posed by Hedlund andMorse in the 40s, a "Boltzman-Sinaj" problem on the existence ofuniform estimates on the number of collisions in hard ball gasmodels, the two-dimensional case of H. Busemann's problemmentioned above, H. Furstenberg's problem on the existence ofbi-Lipschits non-equivalent separated nets and J. Moser's problemon the existence of a continuous function that is not a Jacobiandeterminant of a Lipschits homeomorphism (all from the 60s). Theconjectures and directions of research suggested in the proposalgrew from ideas and methods developed by the proposer and hiscollaborators while working on these problems.Even though most topics of the proposal belong to rather abstractareas of mathematics, their motivations lie in real-wordproblems. The physical analogs of large-scale invariants ofperiodic metrics are macroscopic properties of periodic media(such as crystal substances: i.e., the rate of propagation ofradiation etc), and one wants to relate these properties tomicroscopic characteristics. Hyperbolic dynamics is really wellunderstood, and it forms the first and the simplest example ofchaotic models; however, little is know about partiallyhyperbolic systems, which offer a much more realistic model; theproject is aimed in giving new insight into such systems with asmall number of degrees of freedom. The "Boltzman-Sinaj" problemon the existence of uniform estimates on the number of collisionsin hard ball gas models originated from the most basic researchin statistical physics. Study of the geodesic flows, billiardsystems, geometric complexity, and optimal strategies may resultin better understanding of (stability) of certain models inthermodynamics, biology, sociology, and physics, especially whendealing with imprecise data, and perhaps result in newcomputational algorithms.

摘要奖：DMS-0412166主要研究者：Dmitri BuragoD。布拉戈建议继续他的工作，一些长期的项目在黎曼和芬斯勒几何，动力系统的几何起源，几何群论。这些项目包括：几何周期度量，面积最小化surfacesin赋范空间和最小的填充，和椭圆的surfacesarea泛函;低维部分双曲同态;无界双不变准半规范和“大”组;流形没有共轭点; applicationsof奇异几何的动态台球系统和某些算法问题的几何起源;和近似PL-等距。 在这些问题的建议是针对有：Busemann的猜想，单位是面积minimizingsurfaces在赋范空间; styding结构的类partially-hyperbolic系统;发现弱版本的霍费尔'snorm（可能在一些群体的体积-reducing homeomorphisms）;各种推广的E.霍普夫的constituture环面没有共枕点。本课题是作者前期工作的延续，包括E. Hedlund和Morse在40年代提出的关于环面无共轭点的Hopf猜想，硬球气体模型中碰撞数存在一致估计的“Boltzman-Sinaj”问题，二维情形的H. Busemann问题，H. Furstenberg关于双Lipschits非等价分离网的存在性问题和J. Moser关于Lipschits同胚的非Jacobian行列式的连续函数的存在性问题（都是60年代的问题）。提案中提出的猜想和研究方向是从提案人和他的合作者在研究这些问题时发展起来的想法和方法中发展起来的。尽管提案中的大多数主题属于相当抽象的数学领域，但它们的动机都是现实问题。 周期度规的大尺度不变量的物理类似物是周期介质的宏观性质（例如晶体物质：即，辐射的传播速率等），人们希望将这些性质与显微特征联系起来。双曲动力学是非常好理解的，它形成了混沌模型的第一个也是最简单的例子;然而，对部分双曲系统知之甚少，它提供了一个更现实的模型;该项目的目的是提供新的见解，以这种系统的自由度少。关于硬球气体模型中碰撞次数是否存在一致估计的“Boltzman-Sinaj”问题起源于统计物理学中最基本的研究。对测地线流、台球系统、几何复杂性和最优策略的研究可能会导致更好地理解热力学、生物学、社会学和物理学中某些模型的（稳定性），特别是在处理不精确数据时，并可能导致新的计算算法。