Geometry and Dynamics in Riemannian and Finsler Spaces

摘要奖：DMS-0412166主要研究员：Dmitri BuragoD。布拉戈提议继续他在黎曼几何和芬斯勒几何、几何起源动力系统和几何群论方面的一些长期项目的工作。这些项目包括：周期度量的几何，赋范空间和极小填充中的面积最小化曲面，以及曲面泛函的椭圆性；低维部分双曲微分同态；无界双不变拟半范和“大”群；没有共轭点的流形；奇异几何在台球系统动力学和某些几何起源的算法问题中的应用；以及PL-等距逼近。该建议所针对的问题包括：Busemann猜想，平面是赋范空间中面积最小的曲面；确定一类部分双曲系统的结构；寻找Hofer范数的弱形式(可能在某些保积同胚群上)；E.Hopf猜想在没有共轭点的环面上的各种推广。这个项目继续了作者以前的研究，包括40年代由Hedlund和Morse提出的关于无共轭点的环面上E.Hopf猜想的解，关于硬球气体模型中碰撞次数的无形式估计的存在性的“Boltzman-Sinaj”问题，上述H.Busemann问题的二维情况，H.Furstberg关于双Lipschits非等价分离网的存在性的问题，以及J.Moser的关于连续函数的存在性的问题，该连续函数不是Lipschits同胚的Jacobian行列式的存在性(都来自60年代)。提案中提出的猜想和研究方向源于提出者和他的合作者在解决这些问题时提出的想法和方法。尽管提案中的大多数主题属于相当抽象的数学领域，但它们的动机在于实际的应用问题。周期度规的大尺度不变量的物理类比是周期介质的宏观性质(如晶体物质：即辐射的传播速度等)，人们希望将这些性质与微观特征联系起来。双曲动力学确实被很好地理解了，它构成了混沌模型的第一个也是最简单的例子；然而，人们对部分双曲系统知之甚少，因为它提供了一个更现实的模型；该项目的目的是给这类具有少量自由度的系统提供新的洞察力。“Boltzman-Sinaj”问题是关于硬球气体模型中碰撞次数的统一估计的存在性问题，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的数据时，并可能导致新的计算算法。

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.