Geometry of diffeomorphism groups and Hamiltonian systems

洋流、河流中的流体运动、大气流动以及托卡马克和恒星中的磁场都是由流体动力学欧拉方程及其相关方程建立的数学模型。虽然欧拉型方程出现在广泛的物理现象中，但它们有许多相似之处：哈密顿结构，各种守恒定律，几何公式和与粒子重新标记（即反同态作用）相关的对称性。拟议的研究的目标是通过探索流体的几何和群论方法，有助于理解流体动力学的两个关键问题：奇点的形成和湍流的出现。它旨在了解几何无限维数学模型对各种流体动力学现象的鲁棒性，准确性和预测能力，从天气预报到冲击波的出现和洋流的行为。

Ocean currents, fluid motions in rivers, atmospheric flows, and magnetic fields in tokamaks and stars are mathematically modelled by the hydrodynamical Euler equation and its relatives. While Euler-type equations arise in a broad range of physical phenomena, they share many similarities: Hamiltonian structure, a variety of conservation laws, geometric formulation, and symmetries related to particle relabelling (i.e. diffeomorphism action). The goal of the proposed research is to contribute toward understanding of two key problems of fluid dynamics: singularity formation and appearance of turbulence by exploring the geometric and group-theoretical approach to fluids. It is aimed at understanding of the robustness, accuracy, and predicting power of geometric infinite-dimensional mathematical models for wide range of hydrodynamical phenomena, from how reliable are weather forecasts to appearance of shock waves and behaviour of ocean currents.