Geometry and Dynamics in Riemannian and Finsler Spaces

主要研究方向为：边界刚度及相关逆问题、最小填充、环面渐近几何和归范空间中面积最小曲面；部分双曲微分同态的研究conjugation-invariant quasi-norms;“区域结构和空间”。它们旨在解决一些长期存在的重要问题，并提出新的问题和新的研究方向。构成提案核心的四个项目已经取得了一些重要成果。该方案所针对的问题有：Michel关于简单度量是边界刚性的猜想，Pu关于圆填充面积的猜想，Busemann关于赋范空间中的平面是面积最小化的猜想，部分双曲系统的分类，研究“面积结构”，寻找保留某些结构的微分同态的几何（共轭不变）和“非动态”（非渐近）不变量，以及E. Hopf猜想的各种推广。提案中提出的大部分猜想和研究方向都来自于PI在他之前的研究中提出的想法和方法，提案中描述的项目是对解决这些问题的研究结果的延续。所提出的研究课题中有许多与应用科学具有相当可行性的关系。边界刚性问题和相关的逆问题是由地球物理学和医学成像中的重要问题引起的。想象一下，一个人想知道地球是由什么组成的。更一般地说，人们想要找出由不同材料制成的固体内部是什么（换句话说，介质的性质随点而变化）。声音的速度取决于材料。一个人可以“轻拍”身体表面的某些点，然后“在声音到达其他点时倾听”。问题是这些信息是否足以确定里面是什么。所提出的研究已经在合理的一般情况下（以材料的性质在点与点之间变化不大为限制条件）得出了第一个此类结果，并且对一般情况有谨慎的处理希望。周期度量的大尺度不变量的物理类似物是周期介质（如晶体物质）的宏观性质，人们希望将这些性质与微观特征联系起来；同样，对部分双曲系统、测地线流和几何复杂性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数据时。

The main projects of the proposed researchinclude: boundary rigidity and related inverse problems, minimal fillings, asymptotic geometry of tori and area-minimizing surfaces in normed spaces; study of partially hyperbolic diffeomorphisms; conjugation-invariant quasi-norms; ``area structures and spaces". They are aiming at solving a number of long-standing and important problems, and formulating new problems and new directions of research. The four projects that form the core of the proposal have already yielded a number of important results.Among the problems the proposal is aimed at are Michel's Conjecture that simple metrics are boundary rigid, Pu's conjecture on the filling area of the circle, Busemann's Conjecture that flats in normed spaces are area-minimizers, classifications of partially-hyperbolic systems, studying ``area structures", finding geometric (conjugation-invariant) and ``non-dynamical" (not asymptotic) invariants of diffeomorphisms preserving certain structures, and various generalizations of the E. Hopf Conjecture.Most of the conjectures and directions of research suggested in the proposal grew from ideas and methods developed by the PI in his previous research, and projects described in the proposal continue the research resulted in solving them.Many topics of the proposed research have rather feasible relation to applied science. The Boundary Rigidity Problem and related Inverse Problems are motivated by important problems in geophysics and medical imaging.To visualize that, imagine that one wants to find out what the Earth is made of.More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material. One can "tap" at somepoints of the surface of the body and "listen when the sound gets to other points".The question is if this information is enough to determine what is inside. The proposed research already resulted in the first result of this kind for a reasonably general case (with the restriction that the properties of the material do not change to much from point to point), and there is a cautions hope to handle the general case.The physical analogs of large-scale invariants of periodic metrics are macroscopic properties of periodic media (such as a crystal substance), and one wants to relate these properties to microscopic characteristics; similarly, study of partially hyperbolic systems, geodesic flows, and geometric complexity may result in better understanding of (stability) of certain models in thermodynamics, biology, sociology, and physics, especially when dealing with imprecise data.