TRR 109: Discretisation in Geometry and Dynamics

TRR 109：几何和动力学离散化

• 批准号: 195170736
• 金额: --
• 依托单位: Technische Universität München (TUM)
• 依托单位国家: 德国
• 项目类别: CRC/Transregios
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/195170736?language=en
• 关键词: TRR; 109; Discretisation; Geometry; Dynamics

The central goal of the CRC is to pursue research on the discretization of differential geometry and dynamics. In both fields of mathematics, the key objects under investigation are governed by differential equations. Generally, the term “discretization” refers to any procedure that turns a differential equation into difference equations involving only finitely many variables, whose solutions approximate those of the differential equation.In dynamics, it became apparent that obtaining locally high-accurate approximations is not enough if one is interested in the global, qualitative long-term behavior of a dynamical system. A good discretization scheme should therefore preserve important qualitative aspects of the continuous system. For example, if energy is preserved in the continuous system, then the discretized system should also exhibit some sort of energy conservation. Since the modern theory of dynamical systems is formulated in the language of geometry, the subfield that is concerned with such structure-preserving discretizations is called geometric integration.In differential geometry, structure-preserving discretizations turned out to be useful as well. For example, for many special classes of surfaces (such as minimal surfaces or surfaces with constant Gauss curvature) structure-preserving discretizations are known. These types of discrete surfaces are polyhedral surfaces with special properties defined in elementary geometric terms. However, they exhibit the same qualitative behavior as the continuous surfaces, which are governed by nonlinear partial differential equations.The common idea behind these developments in geometry and dynamics is to find and investigate discrete models that exhibit properties and structures characteristic of the corresponding smooth geometric objects and dynamical processes. Refining the discrete models by decreasing the mesh size should of course converge in the limit to the conventional description via differential equations, but in addition the important characteristic qualitative features should already be captured at the discrete level. The resulting discretization should constitute a fundamental mathematical theory, which incorporates the classical analog in the continuous limit.The CRC brings together scientists, who have joined forces in tackling the numerous problems raised by the challenge of discretizing geometry and dynamics.

CRC的中心目标是研究微分几何和动力学的离散化。在这两个数学领域中，研究的关键对象都是由微分方程控制的。一般来说，“离散化”是指将微分方程转化为仅包含100个变量的差分方程，其解近似于微分方程的解。在动力学中，如果对动力系统的全局、定性长期行为感兴趣，显然获得局部高精度近似是不够的。因此，一个好的离散化方案应该保留连续系统的重要定性方面。例如，如果能量在连续系统中保持不变，那么离散系统也应该表现出某种能量守恒。由于现代动力系统理论是用几何语言表述的，因此研究这种保持结构的离散化的子领域称为几何积分。在微分几何中，保持结构的离散化也被证明是有用的。例如，对于许多特殊类型的曲面（如极小曲面或具有常高斯曲率的曲面），结构保持离散化是已知的。这些类型的离散曲面是多面体曲面，具有用初等几何术语定义的特殊性质。然而，它们表现出与连续曲面相同的定性行为，而连续曲面是由非线性偏微分方程控制的。这些几何和动力学发展背后的共同思想是寻找和研究离散模型，这些离散模型表现出相应的光滑几何对象和动力学过程的特性和结构特征。通过减小网格尺寸来细化离散模型当然应该收敛于通过微分方程的常规描述的极限，但是另外重要的特性定性特征应该已经在离散水平上被捕获。由此产生的离散化应该构成一个基本的数学理论，它将经典的模拟在连续的极限。CRC汇集了科学家，谁已经联手解决离散化几何和动力学的挑战所提出的众多问题。