Topics in Dynamics, Differential Topology and Differential Geometry

这一建议涉及各种具体的研究问题，从动力系统理论，微分拓扑和微分几何，主要是出于物理或工程应用的动机。它们包括估计弹性碰撞系统（台球，理想气体模型）中周期运动的数量以下，可积台球的刚性，磁场中电荷的运动（特别是希尔伯特第四问题的磁性模拟），轮胎轨迹几何和浮选问题，流体动力学中丝方程的比哈密顿方法及其离散版本，拓扑机器人运动规划问题的复杂性，流形的非简并浸入和嵌入（后者与微分拓扑的h原理和光滑流形构型空间的拓扑密切相关）。有两个主题，统一了提案中考虑的各种问题。第一个是连续和离散之间的相互作用（例如，描述理想流体中漩涡的曲线的演变与多边形的类似演变）。第二个统一的主题是显著的动态或几何现象（例如，一个圆圆柱体是唯一均匀的圆柱形固体-一个圆木-在所有位置平衡浮动？）的刚性和灵活性之间的相互作用？答案取决于对数的密度。)

This proposal concerns a variety of concrete research problems from the theory of dynamical systems, differential topology and differential geometry, mostly motivated by physical or engineering applications. They include estimating below the number of periodic motions in systems with elastic collisions (billiards, ideal gas models), rigidity of integrable billiards, motion of an electric charge in magnetic field (in particular, the magnetic analog of Hilbert's fourth problem), tire track geometry and flotation problems, bihamiltonian approach to the filament equation in hydrodynamics and its discrete versions, complexity of the motion planning problem in topological robotics, non-degenerate immersions and embeddings of manifolds (the latter are closely related with H-principle of differential topology and with the topology of configuration spaces of smooth manifolds).There are two themes, unifying the diverse problems considered in the proposal. The first is the interplay between continuous and discrete (for example, evolution of curves describing vortices in ideal fluid versus similar evolution of polygons). The second unifying theme is the interplay between rigidity and flexibility of remarkable dynamical or geometrical phenomena (for example, is a round cylinder the only uniform cylindrical solid - a log - that floats in equilibrium in all positions? The answer depends on the density of the log.)