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

非正则黎曼几何在台球系统动力学中的应用，高阶阿贝尔群的Anosov作用量的刚性和非交换对称空间的Banach几何。具体地说，在他对黎曼流形上的半离散台球动力学的研究中，作者计划解决以下问题：能否给出具有‘有限视界’的非退化半离散台球的第一个返回映射的拓扑熵的估计？关于没有非正曲率限制的流形上台球的拓扑熵，我们能说些什么？最基本的问题是任意非退化半离散台球在维度大于3的非正曲线流形上的万能展开空间的存在性问题，这是提出者计划用他(与D.Burago和B.Kleiner)最近发展的机器来攻击的问题。继续他(与A.Katok)对高阶Abelian群的Anosov作用量的上同调刚性的研究，作者将尝试建立不同于Weyl室流的标准Anosov作用量的上同调刚性，得到Furstberg边界上的晶格作用量的无穷小张量刚性和余紧晶格的投射作用量的可微刚性.作者研究非对易算子空间几何的主要内容是同时推广了Hilbert变换和上三角投影，即任意(UMD)-空间上紧Abelian群的强连续表示的谱投影。提案中流行的主题和工具是动力系统理论。该理论一般研究一个物理或数学系统随时间的演变。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、工程、生物学或社会学等不同领域中提取基本例子，并将其结果应用于这些领域。例如，台球系统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巨大台球系统的力学。这就是为什么物理学家在试图“证明”热力学时，最先提出了该理论的许多问题。这样的理由仍然只是一个梦想；但提出者相信，他的研究让我们更接近于它的实现。近半个世纪以来一直悬而未决的问题之一(关于气体分子之间碰撞次数的估计的存在)最近借助于某种病态空间的几何得到了解决。如上所述，这种意想不到的联系的发现为提议人想要追求的领域提供了几个机会。

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.