Dynamics of Semi-Dispersing Billiards; Geometry of Operator Symmetric Spaces; Rigidity of Higher-Rank Anosov Actions

本论文的主要研究方向是：非正则黎曼几何在台球系统动力学中的应用，高阶交换群的Anosov作用的刚性，非交换对称空间的Banach几何。具体来说，在他的研究动力学的半分散台球黎曼流形上，提议者计划解决以下问题：是否有可能给任何估计的第一个返回地图的非退化半分散台球与“有限的地平线”？在没有非正曲率限制的情况下，关于流形上台球的拓扑熵可以说些什么？最基本的问题是关于在大于3维的非正曲流形上的任意非退化半色散台球的普遍展开空间的存在性，提出者计划使用他（与D.Burago和B.Kleiner）最近开发的机器进行攻击。继续他（与A.Katok）对高阶阿贝尔群的Anosov作用的上同调刚性的研究，提议者将试图建立不同于Weyl室流的标准Anosov作用的上同调刚性;获得Furstenberg边界上的格作用的无穷小张量刚性和余紧格的投射作用的可微刚性。研究非交换算子空间几何的计划的中心是Hilbert变换和上三角投影的某些推广，上三角投影是与任意（UMD）-空间上紧阿贝尔群的强连续表示相关联的谱投影。该提案中流行的主题和工具是动力系统理论。该理论一般研究物理或数学系统随时间的演化。动力系统理论与数学的大多数主要领域紧密交织在一起，它经常从物理学，工程学，生物学或社会学等不同领域中获取基本示例并将其结果应用于这些领域。例如，台球系统的理论，一方面是几何学，另一方面是动力学和理论力学。经典热力学本质上就是一个巨大的台球系统的力学。这就是为什么这个理论的许多问题首先是由物理学家在试图“证明”热力学时提出的。这样的理由仍然只是一个梦想，但提议者认为，他的研究使我们更接近它的实现。这些问题之一，已开放了近半个世纪（关于存在的估计数的碰撞分子之间的气体）最近得到解决的帮助几何有点病态的空间。发现这种意想不到的联系，在提议者希望进行上述研究的领域中开辟了若干机会。

AbstractFerlegerThe proposer will continue his research in the following three directions: applications of non-regular Riemannian geometry to the dynamics of billiard systems, rigidity of Anosov actions of higher rank abelian groups and Banach geometry of non-commutative symmetric spaces. Specifically, in his study of dynamics of semi-dispersing billiards on Riemannian manifolds, the proposer plan to address the following questions: Is it possible to give any estimate for the topological entropy of the first return map of a non-degenerate semi-dispersing billiard with a ``finite horizon''? What can be said about the topological entropy of the billiards on manifolds without the non-positive curvature restriction? Most fundamental is the question about existence of universal unfolding space for arbitrary non-degenerate semi-dispersing billiards on non-positively curved manifolds of dimension greater than three, that the proposer plans to attack using the machinery he (jointly with D.Burago and B.Kleiner) has developed recently. Continuing his (joint with A.Katok) study of cohomological rigidity of Anosov actions of higher-rank Abelian groups, the proposer will attempt to establish cohomological rigidity of standard Anosov actions different from Weyl chamber flow; to obtain infinitesimal tensor rigidity of lattice actions on Furstenberg boundaries and differentiable rigidity of the projective actions of cocompact lattices. Central for proposer's program of studying the geometry of noncommutative operator spaces is certain generalization of Hilbert transformation and upper triangular projection at the same time, a spectral projection associated with a strongly continuous representation of a compact Abelian group on an arbitrary (UMD)-space. The prevalent theme and tool in the proposal is the theory of dynamical systems. The theory in general investigates the evolution of a physical or mathematical system over time. Theory of dynamical systems is intertwined closely with most of the major areas of mathematics and it often takes its basic examples from and applies its results to such different fields as physics, engineering, biology or sociology. Theory of billiard systems, for instance, lies at the interface of geometry on one hand and dynamics and theoretical mechanics on the other. Classical thermodynamics is essentially nothing else but mechanics of huge billiard systems. This is why a number of problems of the theory were first posed by physicists in their attempt to "justify" thermodynamics. Such justifications still is just a dream; but the proposer believes that his research brings us a little closer to its realization. One of these problems that has been open for almost half of the century (about existence of an estimate on the number of collisions between molecule of a gas) recently was solved with help of geometry of somewhat pathological spaces. Discovery of such an unexpected connection opened up several opportunities in the area that the proposer would like to pursue as outlined above.