Geometry of Riemannian and Finsler Spaces

建议：DMS-9803129首席研究员：Dmitry Burago该建议的主要主题(大尺度几何和台球系统)既属于几何系统，也属于动力学系统。D.Burago继续他(与S.Ivanov)在周期度量的大规模几何中的研究，建议研究Finsler Tori的渐近体积增长；相关的问题是分析Banach空间中仿射子空间的面积最小化性质。这可能有助于理解环面上拉格朗日系统的共轭点的缺失如何反映在其测地线流的动力学中。布拉戈还建议继续分析奥布里-马瑟理论的几何结论的高维类比。为了进一步理解BiLipschitz等价和准等距之间的关系，自然会考虑密度论证(由提出者与B.Kleiner共同开发)不起作用的情况；最引人注目的情况包括PenRose瓷砖和均匀晶格。D·布拉戈继续他(与S·费雷格，A·科诺年科)对半分散台球系统的研究，计划研究它们的拓扑熵是否可以是无穷的。提出的将奇异几何应用于台球理论的另一个问题是构造CAT(0)发展空间，其测地线表示所有的台球轨迹。作者的顾问E·Johnson致力于应用作者的方法证明曲率有限变的完备曲面的无界性，从而证明了这类嵌入平面的稳定性。该提案的主要主题有非常明显的物理类比。周期性度量的大尺度几何性质可以解释为周期性介质的全局性质，该周期性介质由以规则方式重复的相同微观图案的副本组成，例如在晶体中。特别是，周期性度量的地磁流体的动力学特性反映了光或辐射如何在这种介质中传播。由提出者和他的合作者开发的技术允许我们为最重要的情况(二次拉格朗日)解决这一领域中的许多未决问题；总的来说，这一系列问题仍然是开放的。这位提出者对台球系统理论的研究始于一个可以追溯到玻尔兹曼的问题：在一个由几个球组成的弹性碰撞系统(气体模型)中，人们能否给出在给定时间间隔内最大碰撞次数的上限？单位时间内的碰撞次数在热力学中占有重要地位。令人惊讶的是，这个问题已经通过建立与奇异几何的联系来解决；反过来，这种联系又导致了新的有趣的几何和动力学起源的问题。

Abstract Proposal: DMS-9803129 Principal Investigator: Dmitry Burago The main topics of the proposal (large-scale geometry and billiards systems) belong to both geometry and dynamical systems. Continuing his (joint with S. Ivanov) research in the large-scale geometry of periodic metrics, D. Burago proposes to study the asymptotic volume growth for Finsler tori; a related problem is to analyze area-minimizing properties of affine subspaces in Banach spaces. This may help to understand how the absence of conjugate points for a Lagrangian system on a torus is reflected in the dynamics of its geodesic flow. D. Burago also proposes to continue his analysis of higher dimensional analogs for geometric conclusions of Aubry-Mather theory. Trying to further understand the relationship between biLipschitz equivalence and quasi-isometries, it is natural to consider such cases where density arguments (developed by the proposer jointly with B. Kleiner) do not work; the most striking of such cases include Penrose tilings and uniform lattices. Continuing his (joint with S. Ferleger, A. Kononenko) study of semi-dispersing billiard systems, D. Burago plans to investigate if their topological entropy can be infinite. Another problem which arose from the proposer's method of applying singular geometry to billiard theory is constructing CAT(0) development spaces whose geodesics represent all billiard trajectories. E. Johnson, the proposer's advisee, works on applying the proposer's method to prove the unboundness for complete surfaces of finite variation of curvature to show stability of the class of embedded flat surfaces. The main subjects of the proposal have very clear physical analogs. Large-scale geometric properties of periodic metrics can be interpreted as global properties of a periodic medium consisting of copies of the same microscopic pattern repeated in a regular fashion, as in crystals. In particular, dynamical properties of geo desic flows for periodic metrics reflect how the light or radiation spreads in such media. The technique developed by the proposer and his collaborators allows us to solve many of the open problems in this area for the most important case (quadratic Lagrangians); in general, this circle of problems remains wide open. The proposer's research in the theory of billiard systems started from a problem that goes back to Boltzmann: can one give an upper bound on the maximum number of collisions in a given time interval in a system of several balls colliding elastically (gas model)? The number of collisions per time unit plays important role in thermodynamics. Surprisingly, this problem has been solved by establishing a connection with singular geometry; in its turn, this connection led to new intriguing problems of both geometric and dynamical origin.